Daily Papers

Generative models inspired by dynamical transport of measure -- such as flows and diffusions -- construct a continuous-time map between two probability densities. Conventionally, one of these is the target density, only accessible through samples, while the other is taken as a simple base density that is data-agnostic. In this work, using the framework of stochastic interpolants, we formalize how to couple the base and the target densities. This enables us to incorporate information about class labels or continuous embeddings to construct dynamical transport maps that serve as conditional generative models. We show that these transport maps can be learned by solving a simple square loss regression problem analogous to the standard independent setting. We demonstrate the usefulness of constructing dependent couplings in practice through experiments in super-resolution and in-painting.

Typical generative diffusion models rely on a Gaussian diffusion process for training the backward transformations, which can then be used to generate samples from Gaussian noise. However, real world data often takes place in discrete-state spaces, including many scientific applications. Here, we develop a theoretical formulation for arbitrary discrete-state Markov processes in the forward diffusion process using exact (as opposed to variational) analysis. We relate the theory to the existing continuous-state Gaussian diffusion as well as other approaches to discrete diffusion, and identify the corresponding reverse-time stochastic process and score function in the continuous-time setting, and the reverse-time mapping in the discrete-time setting. As an example of this framework, we introduce ``Blackout Diffusion'', which learns to produce samples from an empty image instead of from noise. Numerical experiments on the CIFAR-10, Binarized MNIST, and CelebA datasets confirm the feasibility of our approach. Generalizing from specific (Gaussian) forward processes to discrete-state processes without a variational approximation sheds light on how to interpret diffusion models, which we discuss.

Diffusion- and flow-based models have emerged as state-of-the-art generative modeling approaches, but they require many sampling steps. Consistency models can distill these models into efficient one-step generators; however, unlike flow- and diffusion-based methods, their performance inevitably degrades when increasing the number of steps, which we show both analytically and empirically. Flow maps generalize these approaches by connecting any two noise levels in a single step and remain effective across all step counts. In this paper, we introduce two new continuous-time objectives for training flow maps, along with additional novel training techniques, generalizing existing consistency and flow matching objectives. We further demonstrate that autoguidance can improve performance, using a low-quality model for guidance during distillation, and an additional boost can be achieved by adversarial finetuning, with minimal loss in sample diversity. We extensively validate our flow map models, called Align Your Flow, on challenging image generation benchmarks and achieve state-of-the-art few-step generation performance on both ImageNet 64x64 and 512x512, using small and efficient neural networks. Finally, we show text-to-image flow map models that outperform all existing non-adversarially trained few-step samplers in text-conditioned synthesis.

Multi-Agent Pathfinding (MAPF) is the problem of finding paths for multiple agents such that every agent reaches its goal and the agents do not collide. Most prior work on MAPF was on grids, assumed agents' actions have uniform duration, and that time is discretized into timesteps. We propose a MAPF algorithm that does not rely on these assumptions, is complete, and provides provably optimal solutions. This algorithm is based on a novel adaptation of Safe interval path planning (SIPP), a continuous time single-agent planning algorithm, and a modified version of Conflict-based search (CBS), a state of the art multi-agent pathfinding algorithm. We analyze this algorithm, discuss its pros and cons, and evaluate it experimentally on several standard benchmarks.

Channel prediction is critical to address the channel aging issue in mobile scenarios. Existing channel prediction techniques are mainly designed for discrete channel prediction, which can only predict the future channel in a fixed time slot per frame, while the other intra-frame channels are usually recovered by interpolation. However, these approaches suffer from a serious interpolation loss, especially for mobile millimeter wave communications. To solve this challenging problem, we propose a tensor neural ordinary differential equation (TN-ODE) based continuous-time channel prediction scheme to realize the direct prediction of intra-frame channels. Specifically, inspired by the recently developed continuous mapping model named neural ODE in the field of machine learning, we first utilize the neural ODE model to predict future continuous-time channels. To improve the channel prediction accuracy and reduce computational complexity, we then propose the TN-ODE scheme to learn the structural characteristics of the high-dimensional channel by low dimensional learnable transform. Simulation results show that the proposed scheme is able to achieve higher intra-frame channel prediction accuracy than existing schemes.

Recurrent neural networks (RNNs), temporal convolutions, and neural differential equations (NDEs) are popular families of deep learning models for time-series data, each with unique strengths and tradeoffs in modeling power and computational efficiency. We introduce a simple sequence model inspired by control systems that generalizes these approaches while addressing their shortcomings. The Linear State-Space Layer (LSSL) maps a sequence u mapsto y by simply simulating a linear continuous-time state-space representation x = Ax + Bu, y = Cx + Du. Theoretically, we show that LSSL models are closely related to the three aforementioned families of models and inherit their strengths. For example, they generalize convolutions to continuous-time, explain common RNN heuristics, and share features of NDEs such as time-scale adaptation. We then incorporate and generalize recent theory on continuous-time memorization to introduce a trainable subset of structured matrices A that endow LSSLs with long-range memory. Empirically, stacking LSSL layers into a simple deep neural network obtains state-of-the-art results across time series benchmarks for long dependencies in sequential image classification, real-world healthcare regression tasks, and speech. On a difficult speech classification task with length-16000 sequences, LSSL outperforms prior approaches by 24 accuracy points, and even outperforms baselines that use hand-crafted features on 100x shorter sequences.

Flow-based generative models achieve state-of-the-art sample quality, but require the expensive solution of a differential equation at inference time. Flow map models, commonly known as consistency models, encompass many recent efforts to improve inference-time efficiency by learning the solution operator of this differential equation. Yet despite their promise, these models lack a unified description that clearly explains how to learn them efficiently in practice. Here, building on the methodology proposed in Boffi et. al. (2024), we present a systematic algorithmic framework for directly learning the flow map associated with a flow or diffusion model. By exploiting a relationship between the velocity field underlying a continuous-time flow and the instantaneous rate of change of the flow map, we show how to convert any distillation scheme into a direct training algorithm via self-distillation, eliminating the need for pre-trained teachers. We introduce three algorithmic families based on different mathematical characterizations of the flow map: Eulerian, Lagrangian, and Progressive methods, which we show encompass and extend all known distillation and direct training schemes for consistency models. We find that the novel class of Lagrangian methods, which avoid both spatial derivatives and bootstrapping from small steps by design, achieve significantly more stable training and higher performance than more standard Eulerian and Progressive schemes. Our methodology unifies existing training schemes under a single common framework and reveals new design principles for accelerated generative modeling. Associated code is available at https://github.com/nmboffi/flow-maps.

This paper presents the first photo-realistic LiDAR-Inertial-Camera Gaussian Splatting SLAM system that simultaneously addresses visual quality, geometric accuracy, and real-time performance. The proposed method performs robust and accurate pose estimation within a continuous-time trajectory optimization framework, while incrementally reconstructing a 3D Gaussian map using camera and LiDAR data, all in real time. The resulting map enables high-quality, real-time novel view rendering of both RGB images and depth maps. To effectively address under-reconstruction in regions not covered by the LiDAR, we employ a lightweight zero-shot depth model that synergistically combines RGB appearance cues with sparse LiDAR measurements to generate dense depth maps. The depth completion enables reliable Gaussian initialization in LiDAR-blind areas, significantly improving system applicability for sparse LiDAR sensors. To enhance geometric accuracy, we use sparse but precise LiDAR depths to supervise Gaussian map optimization and accelerate it with carefully designed CUDA-accelerated strategies. Furthermore, we explore how the incrementally reconstructed Gaussian map can improve the robustness of odometry. By tightly incorporating photometric constraints from the Gaussian map into the continuous-time factor graph optimization, we demonstrate improved pose estimation under LiDAR degradation scenarios. We also showcase downstream applications via extending our elaborate system, including video frame interpolation and fast 3D mesh extraction. To support rigorous evaluation, we construct a dedicated LiDAR-Inertial-Camera dataset featuring ground-truth poses, depth maps, and extrapolated trajectories for assessing out-of-sequence novel view synthesis. Both the dataset and code will be made publicly available on project page https://xingxingzuo.github.io/gaussian_lic2.

Histopathology whole-slide images (WSIs) are routinely acquired in clinical practice and contain rich tissue morphology but lack direct molecular architecture and functional programs defining pathological states, whereas RNA sequencing (RNA-seq) provides genome-wide transcriptional profiles at substantial cost, thereby motivating WSI-based genome-wide transcriptomic prediction. Existing approaches for predicting gene expression from WSIs predominantly rely on deterministic regression with one-to-one mapping, limiting their ability to capture biological heterogeneity and predictive uncertainty. We propose RNA-FM, a flow-matching generative framework for genome-wide bulk RNA-seq prediction from WSIs. RNA-FM formulates transcriptomic prediction as a continuous-time conditional transport problem, learning a velocity field that maps a simple prior to the target gene expression distribution conditioned on morphologies. By integrating pathway-level structure, RNA-FM enables scalable and biologically interpretable genome-wide gene expression imputation. Extensive experiments demonstrate that RNA-FM consistently outperforms state-of-the-art approaches while maintaining biological meaningfulness. Code is available at https://github.com/YXSong000/RNA-FM.

Diffusion models have found widespread adoption in various areas. However, their sampling process is slow because it requires hundreds to thousands of network evaluations to emulate a continuous process defined by differential equations. In this work, we use neural operators, an efficient method to solve the probability flow differential equations, to accelerate the sampling process of diffusion models. Compared to other fast sampling methods that have a sequential nature, we are the first to propose parallel decoding method that generates images with only one model forward pass. We propose diffusion model sampling with neural operator (DSNO) that maps the initial condition, i.e., Gaussian distribution, to the continuous-time solution trajectory of the reverse diffusion process. To model the temporal correlations along the trajectory, we introduce temporal convolution layers that are parameterized in the Fourier space into the given diffusion model backbone. We show our method achieves state-of-the-art FID of 4.12 for CIFAR-10 and 8.35 for ImageNet-64 in the one-model-evaluation setting.

Per-ticker forecasting models dominate financial time-series work yet remain blind to cross-company propagation: a foundry disruption in Taiwan does not register in a single-asset model until Apple's own price has already moved. To address this limitation, we introduce a heterogeneous Rust-Python streaming architecture that maps cross-company attention as a continuous-time graph driven directly from text. We show that on the ingestion side, a zero-copy Rust edge parses news records in sim100 ns and scans the target equity universe in sim1.2 μs. On the inference end, a multivariate Neural Hawkes Process featuring per-node continuous-time LSTM states and a bilinear latent projection propagates directed excitation, while an adaptive pruning rule bounds the computational cost of dynamic neighborhood updates. Combining these stages, we demonstrate an end-to-end processing latency of sim13 ms per incoming news record on a single commodity CPU. Evaluated on a one-month temporal holdout of the FNSPID corpus (638 articles across 47 tickers), the system delivers a 1.70times precision lift over random at the 90th-percentile next-day return threshold, and 3.36times over a same-sector baseline. Crucially, removing the graph topology collapses precision to zero, confirming that the dynamic attention network is the sole driver of cross-company signal in this architecture.

In this paper we introduce token time continuous diffusion (TTCD), a new diffusion language model which (a) operates in continuous space, deterministically mapping Gaussian noise to a final token canvas with no further sampling, and crucially (b) incorporates a new notion of per-token times, with some tokens proceeding from noise to token at a faster rate than others. Continuous space modeling helps TTCD avoid the parallel sampling of multiple tokens, which is a key source of inaccuracy at high speedups for models that iterate purely in discrete space. The notion of per-token times helps TTCD to better model conditional generation, allows for more sure tokens to proceed at a faster rate, and allows for differentiated inter-token influences during refinement. TTCD outperforms discrete models at high speedups. We train a 160M parameter TTCD model on OpenWebText, and then self-distill it; we find that at high speedups we are comparable in unconditional generation quality, and outperform in conditional generation, several existing models of similar size trained, on the same data, and self-distilled. We achieve similar gains in Sudoku solving as well.

A demonstration of a real-time and continuous turn-taking prediction system is presented. The system is based on a voice activity projection (VAP) model, which directly maps dialogue stereo audio to future voice activities. The VAP model includes contrastive predictive coding (CPC) and self-attention transformers, followed by a cross-attention transformer. We examine the effect of the input context audio length and demonstrate that the proposed system can operate in real-time with CPU settings, with minimal performance degradation.

Structured memory representations such as knowledge graphs are central to autonomous agents and other long-lived systems. However, most existing approaches model time as discrete metadata, either sorting by recency (burying old-yet-permanent knowledge), simply overwriting outdated facts, or requiring an expensive LLM call at every ingestion step, leaving them unable to distinguish persistent facts from evolving ones. To address this, we introduce RoMem, a drop-in temporal knowledge graph module for structured memory systems, applicable to agentic memory and beyond. A pretrained Semantic Speed Gate maps each relation's text embedding to a volatility score, learning from data that evolving relations (e.g., "president of") should rotate fast while persistent ones (e.g., "born in") should remain stable. Combined with continuous phase rotation, this enables geometric shadowing: obsolete facts are rotated out of phase in complex vector space, so temporally correct facts naturally outrank contradictions without deletion. On temporal knowledge graph completion, RoMem achieves state-of-the-art results on ICEWS05-15 (72.6 MRR). Applied to agentic memory, it delivers 2-3x MRR and answer accuracy on temporal reasoning (MultiTQ), dominates hybrid benchmark (LoCoMo), preserves static memory with zero degradation (DMR-MSC), and generalises zero-shot to unseen financial domains (FinTMMBench).

Relative Geologic Time (RGT) estimation from seismic data is a cornerstone of subsurface structural modeling, depositional evolution analysis, and reservoir characterization, supporting horizon correlation and depositional system reconstruction. Yet accurate RGT estimation remains challenging: RGT is intrinsically a topologically constrained continuous field, in which local errors readily propagate globally and distort the overall result. Conventional methods rely heavily on priors, attribute extraction, and manual interaction, leading to cumbersome workflows. Existing deep-learning approaches mostly use a regression formulation with pixel-wise MSE/MAE losses, which struggle to capture thin horizons and fail to model the stratigraphic semantics of the RGT field, yielding limited generalization and unstable ordering across diverse structural and depositional settings. We propose RGT-Est, a deep-learning framework that transfers the optimization target from the topologically constrained continuous field into a differentiable sinusoidal space, which explicitly encodes the periodic stratigraphic semantics of RGT and alleviates over-smoothing of fine horizons. Pointwise, perceptual, and adversarial losses are jointly imposed in this space to enforce local fidelity, inter-layer consistency, and global structural plausibility, providing both fine-horizon discrimination and global stratigraphic awareness. An optional horizon-guidance module further accepts sparse 2D or 3D horizons as priors. Trained on synthetic data and evaluated on field surveys with densely faulted zones, large unconformities, steeply dipping strata, folded deformations, and clinoforms, RGT-Est achieves state-of-the-art performance among AI-based methods without horizon constraints, and attains substantially higher horizon-correlation accuracy and global topological consistency once sparse priors are incorporated.

Understanding and forecasting audience reactions to video content are crucial for improving content creation, recommendation systems, and media analysis. To enable audience reaction prediction and other content engagement applications, we introduce Video2Reaction, a multimodal dataset that maps short movie segments to a distribution of induced emotions of viewers in the wild, as expressed through social media. Video2Reaction spans more than 10,000 videos and serves as a reliable benchmark as well as a training resource for audience reaction prediction. To enable cost-effective continuous annotations as reactions may change over time, we develop a two-stage multi-agent pipeline using only open-source LLMs, achieving 86% correctness under blind human verification despite the inherently noisy and subjective nature of the task. We establish the first benchmark for video-to-reaction-distribution prediction in the wild and show that pretrained foundation video models fail in zero-shot settings, while finetuning transforms them into state-of-the-art predictors capable of modeling both full reaction distributions and dominant responses from video alone. However, the task remains challenging: even the strongest methods achieve only 77% Top-3 F1 in dominant reaction prediction (LLaVA-Next), highlighting a substantial gap in modeling collective audience reaction. \modification{Dataset and code are available at our project page: https://information-fusion-lab-umass.github.io/video2reaction-bench.github.io

While 3D Scene Graphs (3DSGs) provide crucial structured representations for embodied agents, conventional Ahead-of-Time, build-everything-then-filter pipelines conflict with the real-time, low-latency demands of edge platforms, inducing a perceptual saturation effect via severe observation redundancy. To resolve this, we present JITOMA (Just-In-Time On-demand Memory Activation), a closed-loop framework that unifies task reasoning, perception, and memory into a just-in-time growth process. Instead of exhaustively mapping the entire environment, JITOMA leverages a top-down task heatmap at the frontend to filter continuous observations, routing minimal streams to maintain a global foundation of low-cost, dormant anchors. Upon a cognitive query, the backend Large Language Model (LLM) parses the robotic intent to dynamically awaken task-relevant anchors, triggering resource-intensive operations -- such as dense node captioning and functional inference -- exclusively within the activated local subgraph. To evaluate these dynamic capabilities and study perceptual saturation trade-offs, we introduce JITOMA-Bench, a comprehensive suite for long-horizon multi-tasking and complex multi-step reasoning. Extensive experiments demonstrate that JITOMA substantially reduces active graph size and captioning latency, while maintaining stable processing time under long-horizon task switching.

Irregular multivariate time series impose a trade-off for long-horizon forecasting: discrete methods can distort temporal structure via re-gridding, while continuous-time models often require sequential solvers prone to drift. To bridge this gap, we present Latent Laplace Diffusion (LLapDiff), a generative framework that models the target as a low-dimensional latent trajectory, enabling horizon-wide generation without step-by-step integration over physical time. We guide the reverse process utilizing a stable modal parameterization motivated by stochastic port-Hamiltonian dynamics, and parameterize its mean evolution in the Laplace domain via learnable complex-conjugate poles, enabling direct evaluation over irregular timestamps. We also link continuous dynamics to irregular observations through renewal-averaging analysis, which maps sampling gaps to effective event-domain poles and motivates a gap-aware history summarizer. Extensive experiments show that LLapDiff improves over baselines in long-horizon forecasting, and its continuous-time generative nature supports missing-value imputation by querying the same model at historical timestamps. Code is available at https://github.com/pixelhero98/LLapDiffusion.

This paper addresses the task of space-time video super-resolution (ST-VSR). Existing methods generally suffer from inaccurate motion estimation and motion compensation (MEMC) problems for large motions. Inspired by recent progress in physics-informed neural networks, we model the challenges of MEMC in ST-VSR as a mapping between two continuous function spaces. Specifically, our approach transforms independent low-resolution representations in the coarse-grained continuous function space into refined representations with enriched spatiotemporal details in the fine-grained continuous function space. To achieve efficient and accurate MEMC, we design a Galerkin-type attention function to perform frame alignment and temporal interpolation. Due to the linear complexity of the Galerkin-type attention mechanism, our model avoids patch partitioning and offers global receptive fields, enabling precise estimation of large motions. The experimental results show that the proposed method surpasses state-of-the-art techniques in both fixed-size and continuous space-time video super-resolution tasks.

X-ray observing facilities, such as the Chandra X-ray Observatory and the eROSITA, have detected millions of astronomical sources associated with high-energy phenomena. The arrival of photons as a function of time follows a Poisson process and can vary by orders-of-magnitude, presenting obstacles for common tasks such as source classification, physical property derivation, and anomaly detection. Previous work has either failed to directly capture the Poisson nature of the data or only focuses on Poisson rate function reconstruction. In this work, we present Poisson Process AutoDecoder (PPAD). PPAD is a neural field decoder that maps fixed-length latent features to continuous Poisson rate functions across energy band and time via unsupervised learning. PPAD reconstructs the rate function and yields a representation at the same time. We demonstrate the efficacy of PPAD via reconstruction, regression, classification and anomaly detection experiments using the Chandra Source Catalog.

Diffusion and flow-based models have become the de facto approaches for generating continuous data, e.g., in domains such as images and videos. Their success has attracted growing interest in applying them to language modeling. Unlike their image-domain counterparts, today's leading diffusion language models (DLMs) primarily operate over discrete tokens. In this paper, we show that continuous DLMs can be made effective with minimal adaptation to the discrete domain. We propose Embedded Language Flows (ELF), a class of diffusion models in continuous embedding space based on continuous-time Flow Matching. Unlike existing DLMs, ELF predominantly stays within the continuous embedding space until the final time step, where it maps to discrete tokens using a shared-weight network. This formulation makes it straightforward to adapt established techniques from image-domain diffusion models, e.g., classifier-free guidance (CFG). Experiments show that ELF substantially outperforms leading discrete and continuous DLMs, achieving better generation quality with fewer sampling steps. These results suggest that ELF offers a promising path toward effective continuous DLMs.

Effective spatio-temporal representation is fundamental to modeling, understanding, and predicting dynamics in videos. The atomic unit of a video, the pixel, traces a continuous 3D trajectory over time, serving as the primitive element of dynamics. Based on this principle, we propose representing any video as a Trajectory Field: a dense mapping that assigns a continuous 3D trajectory function of time to each pixel in every frame. With this representation, we introduce Trace Anything, a neural network that predicts the entire trajectory field in a single feed-forward pass. Specifically, for each pixel in each frame, our model predicts a set of control points that parameterizes a trajectory (i.e., a B-spline), yielding its 3D position at arbitrary query time instants. We trained the Trace Anything model on large-scale 4D data, including data from our new platform, and our experiments demonstrate that: (i) Trace Anything achieves state-of-the-art performance on our new benchmark for trajectory field estimation and performs competitively on established point-tracking benchmarks; (ii) it offers significant efficiency gains thanks to its one-pass paradigm, without requiring iterative optimization or auxiliary estimators; and (iii) it exhibits emergent abilities, including goal-conditioned manipulation, motion forecasting, and spatio-temporal fusion. Project page: https://trace-anything.github.io/.

Retrieval-Augmented Generation (RAG) enhances Large Language Models (LLMs) by integrating external knowledge, leading to improved accuracy and relevance. However, scaling RAG pipelines remains computationally expensive as retrieval sizes grow. To address this, we introduce OSCAR, a novel query-dependent online soft compression method that reduces computational overhead while preserving performance. Unlike traditional hard compression methods, which shorten retrieved texts, or soft compression approaches, which map documents to continuous embeddings offline, OSCAR dynamically compresses retrieved information at inference time, eliminating storage overhead and enabling higher compression rates. Additionally, we extend OSCAR to simultaneously perform reranking, further optimizing the efficiency of the RAG pipeline. Our experiments demonstrate state-of-the-art performance with a 2-5x speed-up in inference and minimal to no loss in accuracy for LLMs ranging from 1B to 24B parameters. The models are available at: https://huggingface.co/collections/naver/oscar-67d446a8e3a2551f57464295.

Continuous-time neural processes are performant sequential decision-makers that are built by differential equations (DE). However, their expressive power when they are deployed on computers is bottlenecked by numerical DE solvers. This limitation has significantly slowed down the scaling and understanding of numerous natural physical phenomena such as the dynamics of nervous systems. Ideally, we would circumvent this bottleneck by solving the given dynamical system in closed form. This is known to be intractable in general. Here, we show it is possible to closely approximate the interaction between neurons and synapses -- the building blocks of natural and artificial neural networks -- constructed by liquid time-constant networks (LTCs) efficiently in closed-form. To this end, we compute a tightly-bounded approximation of the solution of an integral appearing in LTCs' dynamics, that has had no known closed-form solution so far. This closed-form solution substantially impacts the design of continuous-time and continuous-depth neural models; for instance, since time appears explicitly in closed-form, the formulation relaxes the need for complex numerical solvers. Consequently, we obtain models that are between one and five orders of magnitude faster in training and inference compared to differential equation-based counterparts. More importantly, in contrast to ODE-based continuous networks, closed-form networks can scale remarkably well compared to other deep learning instances. Lastly, as these models are derived from liquid networks, they show remarkable performance in time series modeling, compared to advanced recurrent models.

Extending discrete-time causal Prior-data Fitted Networks for time series to continuous time invites writing the mechanism as a stochastic differential equation (SDE) -- but if the SDE is integrated once per observation gap, the trajectory law depends on when it is observed, and the prior remains a discrete-time Markov model in SDE clothing. We propose a precise continuity criterion -- trajectory-law invariance to the observation schedule -- together with a three-tier taxonomy (discrete; naive observation-grid integration; fine-grid integration with decoupled observation) and a construction realising the top tier on a random DAG with OU or small-MLP nonlinear drifts, irregular observation schedules, and hard / soft / time-varying interventions. A 2 times 2 encoder times integrator ablation, run independently on a linear and a nonlinear prior, finds fine-grid integration beats naive on 8/8 cells (sign-consistency p < 1/256) with the gap growing as the eval grid refines; the encoder axis is null with fine integration but time-aware-leading with naive. We release the prior and a preliminary zero-shot protocol on pharmacokinetic and physical-system data.

Sequential VAEs have been successfully considered for many high-dimensional time series modelling problems, with many variant models relying on discrete-time mechanisms such as recurrent neural networks (RNNs). On the other hand, continuous-time methods have recently gained attraction, especially in the context of irregularly-sampled time series, where they can better handle the data than discrete-time methods. One such class are Gaussian process variational autoencoders (GPVAEs), where the VAE prior is set as a Gaussian process (GP). However, a major limitation of GPVAEs is that it inherits the cubic computational cost as GPs, making it unattractive to practioners. In this work, we leverage the equivalent discrete state space representation of Markovian GPs to enable linear time GPVAE training via Kalman filtering and smoothing. We show on a variety of high-dimensional temporal and spatiotemporal tasks that our method performs favourably compared to existing approaches whilst being computationally highly scalable.

Learning accurate predictive models of real-world dynamic phenomena (e.g., climate, biological) remains a challenging task. One key issue is that the data generated by both natural and artificial processes often comprise time series that are irregularly sampled and/or contain missing observations. In this work, we propose the Neural Continuous-Discrete State Space Model (NCDSSM) for continuous-time modeling of time series through discrete-time observations. NCDSSM employs auxiliary variables to disentangle recognition from dynamics, thus requiring amortized inference only for the auxiliary variables. Leveraging techniques from continuous-discrete filtering theory, we demonstrate how to perform accurate Bayesian inference for the dynamic states. We propose three flexible parameterizations of the latent dynamics and an efficient training objective that marginalizes the dynamic states during inference. Empirical results on multiple benchmark datasets across various domains show improved imputation and forecasting performance of NCDSSM over existing models.

We propose a new class of parameterizations for spatio-temporal point processes which leverage Neural ODEs as a computational method and enable flexible, high-fidelity models of discrete events that are localized in continuous time and space. Central to our approach is a combination of continuous-time neural networks with two novel neural architectures, i.e., Jump and Attentive Continuous-time Normalizing Flows. This approach allows us to learn complex distributions for both the spatial and temporal domain and to condition non-trivially on the observed event history. We validate our models on data sets from a wide variety of contexts such as seismology, epidemiology, urban mobility, and neuroscience.

This paper presents a continuous-time optimal control framework for the generation of reference trajectories in driving scenarios with uncertainty. A previous work presented a discrete-time stochastic generator for autonomous vehicles; those results are extended to continuous time to ensure the robustness of the generator in a real-time setting. We show that the stochastic model in continuous time can capture the uncertainty of information by producing better results, limiting the risk of violating the problem's constraints compared to a discrete approach. Dynamic solvers provide faster computation and the continuous-time model is more robust to a wider variety of driving scenarios than the discrete-time model, as it can handle further time horizons, which allows trajectory planning outside the framework of urban driving scenarios.

Generating continuous-time, continuous-space stochastic processes (e.g., videos, weather forecasts) conditioned on partial observations (e.g., first and last frames) is a fundamental challenge. Existing approaches, (e.g., diffusion models), suffer from key limitations: (1) noise-to-data evolution fails to capture structural similarity between states close in physical time and has unstable integration in low-step regimes; (2) random noise injected is insensitive to the physical process's time elapsed, resulting in incorrect dynamics; (3) they overlook conditioning on arbitrary subsets of states (e.g., irregularly sampled timesteps, future observations). We propose ABC: Any-Subset Autoregressive Models via Non-Markovian Diffusion Bridges in Continuous Time and Space. Crucially, we model the process with one continual SDE whose time variable and intermediate states track the real time and process states. This has provable advantages: (1) the starting point for generating future states is the already-close previous state, rather than uninformative noise; (2) random noise injection scales with physical time elapsed, encouraging physically plausible dynamics with similar time-adjacent states. We derive SDE dynamics via changes-of-measure on path space, yielding another advantage: (3) path-dependent conditioning on arbitrary subsets of the state history and/or future. To learn these dynamics, we derive a path- and time-dependent extension of denoising score matching. Our experiments show ABC's superiority to competing methods on multiple domains, including video generation and weather forecasting.

Learning the continuous dynamics of a system from snapshots of its temporal marginals is a problem which appears throughout natural sciences and machine learning, including in quantum systems, single-cell biological data, and generative modeling. In these settings, we assume access to cross-sectional samples that are uncorrelated over time, rather than full trajectories of samples. In order to better understand the systems under observation, we would like to learn a model of the underlying process that allows us to propagate samples in time and thereby simulate entire individual trajectories. In this work, we propose Action Matching, a method for learning a rich family of dynamics using only independent samples from its time evolution. We derive a tractable training objective, which does not rely on explicit assumptions about the underlying dynamics and does not require back-propagation through differential equations or optimal transport solvers. Inspired by connections with optimal transport, we derive extensions of Action Matching to learn stochastic differential equations and dynamics involving creation and destruction of probability mass. Finally, we showcase applications of Action Matching by achieving competitive performance in a diverse set of experiments from biology, physics, and generative modeling.

Recent work on the sequence universality of State Space Models (SSMs) has introduced efficient, maximally expressive continuous-time approaches for time-series modelling. While these works focus on discriminative settings, we extend this perspective to generative time-series modelling by proving that maximally expressive Structured Linear Controlled Differential Equations (SLiCEs) are universal time-series generators, in the sense that they can approximate the induced path laws of continuous causal pushforwards on compact latent sets in W_infty. Building on these theoretical results, we propose Generative SLiCEs (G-SLiCEs), a maximally expressive continuous-time model for flow matching on path-space. Empirically, we show that expressivity improves performance in probabilistic forecasting and downstream tasks, while retaining the advantages of continuous-time models such as generalising to arbitrary observation grids. This is particularly beneficial for irregular grids, where fixed-grid models often struggle.

We introduce a new class of time-continuous recurrent neural network models. Instead of declaring a learning system's dynamics by implicit nonlinearities, we construct networks of linear first-order dynamical systems modulated via nonlinear interlinked gates. The resulting models represent dynamical systems with varying (i.e., liquid) time-constants coupled to their hidden state, with outputs being computed by numerical differential equation solvers. These neural networks exhibit stable and bounded behavior, yield superior expressivity within the family of neural ordinary differential equations, and give rise to improved performance on time-series prediction tasks. To demonstrate these properties, we first take a theoretical approach to find bounds over their dynamics and compute their expressive power by the trajectory length measure in latent trajectory space. We then conduct a series of time-series prediction experiments to manifest the approximation capability of Liquid Time-Constant Networks (LTCs) compared to classical and modern RNNs. Code and data are available at https://github.com/raminmh/liquid_time_constant_networks

This paper explores the expressive power of deep neural networks through the framework of function compositions. We demonstrate that the repeated compositions of a single fixed-size ReLU network exhibit surprising expressive power, despite the limited expressive capabilities of the individual network itself. Specifically, we prove by construction that L_2circ g^{circ r}circ mathcal{L}_1 can approximate 1-Lipschitz continuous functions on [0,1]^d with an error O(r^{-1/d}), where g is realized by a fixed-size ReLU network, mathcal{L}_1 and L_2 are two affine linear maps matching the dimensions, and g^{circ r} denotes the r-times composition of g. Furthermore, we extend such a result to generic continuous functions on [0,1]^d with the approximation error characterized by the modulus of continuity. Our results reveal that a continuous-depth network generated via a dynamical system has immense approximation power even if its dynamics function is time-independent and realized by a fixed-size ReLU network.

This paper reformulates the Greenwood and Guner (2009) marriage and divorce model in continuous time using the HACT methods of Achdou et al. (2022). Replacing the AR(1) match quality process with an Ornstein-Uhlenbeck process yields a tridiagonal generator, reducing the computational complexity of both the value function and stationary distribution calculations from quadratic to linear in the number of grid points. The continuous-time model closely replicates the discrete-time equilibrium across all key outcomes, including the share of married households, the marriage rate, and the divorce rate, while achieving substantial gains in computation time and memory usage.

Discovering the underlying relationships among variables from temporal observations has been a longstanding challenge in numerous scientific disciplines, including biology, finance, and climate science. The dynamics of such systems are often best described using continuous-time stochastic processes. Unfortunately, most existing structure learning approaches assume that the underlying process evolves in discrete-time and/or observations occur at regular time intervals. These mismatched assumptions can often lead to incorrect learned structures and models. In this work, we introduce a novel structure learning method, SCOTCH, which combines neural stochastic differential equations (SDE) with variational inference to infer a posterior distribution over possible structures. This continuous-time approach can naturally handle both learning from and predicting observations at arbitrary time points. Theoretically, we establish sufficient conditions for an SDE and SCOTCH to be structurally identifiable, and prove its consistency under infinite data limits. Empirically, we demonstrate that our approach leads to improved structure learning performance on both synthetic and real-world datasets compared to relevant baselines under regular and irregular sampling intervals.

Probabilistic completeness is an important property in motion planning. Although it has been established with clear assumptions for geometric planners, the panorama of completeness results for kinodynamic planners is still incomplete, as most existing proofs rely on strong assumptions that are difficult, if not impossible, to verify on practical systems. In this paper, we focus on an important class of kinodynamic planners, namely those that interpolate trajectories in the state space. We provide a proof of probabilistic completeness for these planners under assumptions that can be readily verified from the system's equations of motion and the user-defined interpolation function. Our proof relies crucially on a property of interpolated trajectories, termed second-order continuity (SOC), which we show is tightly related to the ability of a planner to benefit from denser sampling. We analyze the impact of this property in simulations on a low-torque pendulum. Our results show that a simple RRT using a second-order continuous interpolation swiftly finds solution, while it is impossible for the same planner using standard Bezier curves (which are not SOC) to find any solution.

We analyse linear ensemble sampling (ES) with standard Gaussian perturbations in stochastic linear bandits. We show that for ensemble size m=Θ(dlog n), ES attains tilde O(d^{3/2}sqrt n) high-probability regret, closing the gap to the Thompson sampling benchmark while keeping computation comparable. The proof brings a new perspective on randomized exploration in linear bandits by reducing the analysis to a time-uniform exceedance problem for m independent Brownian motions. Intriguingly, this continuous-time lens is not forced; it appears natural--and perhaps necessary: the discrete-time problem seems to be asking for a continuous-time solution, and we know of no other way to obtain a sharp ES bound.

Continuous-time generative models, such as diffusion models, flow matching, and rectified flow, learn time-dependent vector fields but are typically trained with objectives that treat timesteps independently, leading to high estimator variance and inefficient sampling. Prior approaches mitigate this via explicit smoothness penalties, trajectory regularization, or modified probability paths and solvers. We introduce Temporal Pair Consistency (TPC), a lightweight variance-reduction principle that couples velocity predictions at paired timesteps along the same probability path, operating entirely at the estimator level without modifying the model architecture, probability path, or solver. We provide a theoretical analysis showing that TPC induces a quadratic, trajectory-coupled regularization that provably reduces gradient variance while preserving the underlying flow-matching objective. Instantiated within flow matching, TPC improves sample quality and efficiency across CIFAR-10 and ImageNet at multiple resolutions, achieving lower FID at identical or lower computational cost than prior methods, and extends seamlessly to modern SOTA-style pipelines with noise-augmented training, score-based denoising, and rectified flow.

We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box differential equation solver. These continuous-depth models have constant memory cost, adapt their evaluation strategy to each input, and can explicitly trade numerical precision for speed. We demonstrate these properties in continuous-depth residual networks and continuous-time latent variable models. We also construct continuous normalizing flows, a generative model that can train by maximum likelihood, without partitioning or ordering the data dimensions. For training, we show how to scalably backpropagate through any ODE solver, without access to its internal operations. This allows end-to-end training of ODEs within larger models.

Koopman representations aim to learn features of nonlinear dynamical systems (NLDS) which lead to linear dynamics in the latent space. Theoretically, such features can be used to simplify many problems in modeling and control of NLDS. In this work we study autoencoder formulations of this problem, and different ways they can be used to model dynamics, specifically for future state prediction over long horizons. We discover several limitations of predicting future states in the latent space and propose an inference-time mechanism, which we refer to as Periodic Reencoding, for faithfully capturing long term dynamics. We justify this method both analytically and empirically via experiments in low and high dimensional NLDS.

The analysis of non-stationary time-series data requires insight into its local and global patterns with physical interpretability. However, traditional smoothing algorithms, such as B-splines, Savitzky-Golay filtering, and Empirical Mode Decomposition (EMD), lack the ability to perform parametric optimization with guaranteed continuity. In this paper, we propose Functional Continuous Decomposition (FCD), a JAX-accelerated framework that performs parametric, continuous optimization on a wide range of mathematical functions. By using Levenberg-Marquardt optimization to achieve up to C^1 continuous fitting, FCD transforms raw time-series data into M modes that capture different temporal patterns from short-term to long-term trends. Applications of FCD include physics, medicine, financial analysis, and machine learning, where it is commonly used for the analysis of signal temporal patterns, optimized parameters, derivatives, and integrals of decomposition. Furthermore, FCD can be applied for physical analysis and feature extraction with an average SRMSE of 0.735 per segment and a speed of 0.47s on full decomposition of 1,000 points. Finally, we demonstrate that a Convolutional Neural Network (CNN) enhanced with FCD features, such as optimized function values, parameters, and derivatives, achieved 16.8% faster convergence and 2.5% higher accuracy over a standard CNN.

Standard continuous-depth models, such as Neural Ordinary Differential Equations (NODEs), offer significant advantages in modeling physical systems by learning continuous vector fields rather than discrete temporal steps. However, when applied to complex dynamical systems, standard NODEs frequently struggle with highly nonlinear dynamics. This paper investigates the Frequency-domain Neural ODE (FNODE), an architecture that projects continuous temporal dynamics into the frequency domain using the Fast Fourier Transform (FFT). By operating in the frequency domain, the model provides better generalization to the dynamical system. The architecture is empirically evaluated against discrete models, specifically Gated Recurrent Units (GRUs) and Long Short-Term Memory (LSTMs), and other continuous-depth variants, including Augmented Neural ODE (ANODE), across four distinct dynamical systems: the Lotka-Volterra model, the forced Duffing oscillator, the Van der Pol oscillator, and the Lorenz system. To rigorously assess generalization and robustness, curriculum and ensemble learning are used to evaluate the model's convergence by estimating confidence intervals across different ensemble models. The empirical results demonstrate that the FNODE architecture achieves better generalization while exhibiting remarkable convergence stability.

We introduce Lagrangian Flow Networks (LFlows) for modeling fluid densities and velocities continuously in space and time. By construction, the proposed LFlows satisfy the continuity equation, a PDE describing mass conservation in its differentiable form. Our model is based on the insight that solutions to the continuity equation can be expressed as time-dependent density transformations via differentiable and invertible maps. This follows from classical theory of the existence and uniqueness of Lagrangian flows for smooth vector fields. Hence, we model fluid densities by transforming a base density with parameterized diffeomorphisms conditioned on time. The key benefit compared to methods relying on numerical ODE solvers or PINNs is that the analytic expression of the velocity is always consistent with changes in density. Furthermore, we require neither expensive numerical solvers, nor additional penalties to enforce the PDE. LFlows show higher predictive accuracy in density modeling tasks compared to competing models in 2D and 3D, while being computationally efficient. As a real-world application, we model bird migration based on sparse weather radar measurements.

Step distillation has become a leading technique for accelerating diffusion models, among which Distribution Matching Distillation (DMD) and Consistency Distillation are two representative paradigms. While consistency methods enforce self-consistency along the full PF-ODE trajectory to steer it toward the clean data manifold, vanilla DMD relies on sparse supervision at a few predefined discrete timesteps. This restricted discrete-time formulation and mode-seeking nature of the reverse KL divergence tends to exhibit visual artifacts and over-smoothed outputs, often necessitating complex auxiliary modules -- such as GANs or reward models -- to restore visual fidelity. In this work, we introduce Continuous-Time Distribution Matching (CDM), migrating the DMD framework from discrete anchoring to continuous optimization for the first time. CDM achieves this through two continuous-time designs. First, we replace the fixed discrete schedule with a dynamic continuous schedule of random length, so that distribution matching is enforced at arbitrary points along sampling trajectories rather than only at a few fixed anchors. Second, we propose a continuous-time alignment objective that performs active off-trajectory matching on latents extrapolated via the student's velocity field, improving generalization and preserving fine visual details. Extensive experiments on different architectures, including SD3-Medium and Longcat-Image, demonstrate that CDM provides highly competitive visual fidelity for few-step image generation without relying on complex auxiliary objectives. Code is available at https://github.com/byliutao/cdm.

Irregularly-sampled time series occur in many domains including healthcare. They can be challenging to model because they do not naturally yield a fixed-dimensional representation as required by many standard machine learning models. In this paper, we consider irregular sampling from the perspective of missing data. We model observed irregularly-sampled time series data as a sequence of index-value pairs sampled from a continuous but unobserved function. We introduce an encoder-decoder framework for learning from such generic indexed sequences. We propose learning methods for this framework based on variational autoencoders and generative adversarial networks. For continuous irregularly-sampled time series, we introduce continuous convolutional layers that can efficiently interface with existing neural network architectures. Experiments show that our models are able to achieve competitive or better classification results on irregularly-sampled multivariate time series compared to recent RNN models while offering significantly faster training times.

We consider the problem of estimating the optimal transport map between two probability distributions, P and Q in mathbb R^d, on the basis of i.i.d. samples. All existing statistical analyses of this problem require the assumption that the transport map is Lipschitz, a strong requirement that, in particular, excludes any examples where the transport map is discontinuous. As a first step towards developing estimation procedures for discontinuous maps, we consider the important special case where the data distribution Q is a discrete measure supported on a finite number of points in mathbb R^d. We study a computationally efficient estimator initially proposed by Pooladian and Niles-Weed (2021), based on entropic optimal transport, and show in the semi-discrete setting that it converges at the minimax-optimal rate n^{-1/2}, independent of dimension. Other standard map estimation techniques both lack finite-sample guarantees in this setting and provably suffer from the curse of dimensionality. We confirm these results in numerical experiments, and provide experiments for other settings, not covered by our theory, which indicate that the entropic estimator is a promising methodology for other discontinuous transport map estimation problems.

Most contemporary neural learning systems rely on epoch-based optimization and repeated access to historical data, implicitly assuming reversible computation. In contrast, real-world environments often present information as irreversible streams, where inputs cannot be replayed or revisited. Under such conditions, conventional architectures degrade into reactive filters lacking long-horizon coherence. This paper introduces Stream Neural Networks (StNN), an execution paradigm designed for irreversible input streams. StNN operates through a stream-native execution algorithm, the Stream Network Algorithm (SNA), whose fundamental unit is the stream neuron. Each stream neuron maintains a persistent temporal state that evolves continuously across inputs. We formally establish three structural guarantees: (1) stateless mappings collapse under irreversibility and cannot encode temporal dependencies; (2) persistent state dynamics remain bounded under mild activation constraints; and (3) the state transition operator is contractive for λ < 1, ensuring stable long-horizon execution. Empirical phase-space analysis and continuous tracking experiments validate these theoretical results. The execution principles introduced in this work define a minimal substrate for neural computation under irreversible streaming constraints.

Tucker decomposition is a powerful tensor model to handle multi-aspect data. It demonstrates the low-rank property by decomposing the grid-structured data as interactions between a core tensor and a set of object representations (factors). A fundamental assumption of such decomposition is that there are finite objects in each aspect or mode, corresponding to discrete indexes of data entries. However, real-world data is often not naturally posed in this setting. For example, geographic data is represented as continuous indexes of latitude and longitude coordinates, and cannot fit tensor models directly. To generalize Tucker decomposition to such scenarios, we propose Functional Bayesian Tucker Decomposition (FunBaT). We treat the continuous-indexed data as the interaction between the Tucker core and a group of latent functions. We use Gaussian processes (GP) as functional priors to model the latent functions. Then, we convert each GP into a state-space prior by constructing an equivalent stochastic differential equation (SDE) to reduce computational cost. An efficient inference algorithm is developed for scalable posterior approximation based on advanced message-passing techniques. The advantage of our method is shown in both synthetic data and several real-world applications. We release the code of FunBaT at https://github.com/xuangu-fang/Functional-Bayesian-Tucker-Decomposition.

We consider the general class of time-homogeneous stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to learning the transfer operator or the generator of the system, which in turn can be used for numerous tasks, such as forecasting and interpreting the system dynamics. We show that the search for a good representation can be cast as an optimization problem over neural networks. Our approach is supported by recent results in statistical learning theory, highlighting the role of approximation error and metric distortion in the learning problem. The objective function we propose is associated with projection operators from the representation space to the data space, overcomes metric distortion, and can be empirically estimated from data. In the discrete-time setting, we further derive a relaxed objective function that is differentiable and numerically well-conditioned. We compare our method against state-of-the-art approaches on different datasets, showing better performance across the board.

This work explores the in-context learning capabilities of State Space Models (SSMs) and presents, to the best of our knowledge, the first theoretical explanation of a possible underlying mechanism. We introduce a novel weight construction for SSMs, enabling them to predict the next state of any dynamical system after observing previous states without parameter fine-tuning. This is accomplished by extending the HiPPO framework to demonstrate that continuous SSMs can approximate the derivative of any input signal. Specifically, we find an explicit weight construction for continuous SSMs and provide an asymptotic error bound on the derivative approximation. The discretization of this continuous SSM subsequently yields a discrete SSM that predicts the next state. Finally, we demonstrate the effectiveness of our parameterization empirically. This work should be an initial step toward understanding how sequence models based on SSMs learn in context.

Generating realistic time series data is important for many engineering and scientific applications. Existing work tackles this problem using generative adversarial networks (GANs). However, GANs are often unstable during training, and they can suffer from mode collapse. While variational autoencoders (VAEs) are known to be more robust to these issues, they are (surprisingly) less often considered for time series generation. In this work, we introduce Koopman VAE (KVAE), a new generative framework that is based on a novel design for the model prior, and that can be optimized for either regular and irregular training data. Inspired by Koopman theory, we represent the latent conditional prior dynamics using a linear map. Our approach enhances generative modeling with two desired features: (i) incorporating domain knowledge can be achieved by leverageing spectral tools that prescribe constraints on the eigenvalues of the linear map; and (ii) studying the qualitative behavior and stablity of the system can be performed using tools from dynamical systems theory. Our results show that KVAE outperforms state-of-the-art GAN and VAE methods across several challenging synthetic and real-world time series generation benchmarks. Whether trained on regular or irregular data, KVAE generates time series that improve both discriminative and predictive metrics. We also present visual evidence suggesting that KVAE learns probability density functions that better approximate empirical ground truth distributions.

Spiking Neural Networks (SNNs) can achieve competitive performance by converting already existing well-trained Artificial Neural Networks (ANNs), avoiding further costly training. This property is particularly attractive in Reinforcement Learning (RL), where training through environment interaction is expensive and potentially unsafe. However, existing conversion methods perform poorly in continuous control, where suitable baselines are largely absent. We identify error amplification as the key cause: small action approximation errors become temporally correlated across decision steps, inducing cumulative state distribution shift and severe performance degradation. To address this issue, we propose Cross-Step Residual Potential Initialization (CRPI), a lightweight gradient-free mechanism that carries over residual membrane potentials across decision steps to suppress temporally correlated errors. Experiments on continuous control benchmarks with both vector and visual observations demonstrate that CRPI can be integrated into existing conversion pipelines and substantially recovers lost performance. Our results highlight continuous control as a critical and challenging benchmark for ANN-to-SNN conversion, where small errors can be strongly amplified and impact performance. Code is available at https://github.com/xuzijie32/ANN2SNN-CRPI.

Commuting families of contractions or contractive C_{0}-semigroups on Hilbert spaces often fail to admit power dilations resp, simultaneous unitary dilations which are themselves commutative (see [45, 13, 15]). In the non-commutative setting, Sz.-Nagy [60] and Bożejko [5] provided means to dilate arbitrary families of contractions. The present work extends these discrete-time results to families {T_{i}}_{i in I} of contractive C_{0}-semigroups. We refer to these dilations as continuous-time free unitary dilations and present three distinct approaches to obtain them: 1) An explicit derivation applicable to semigroups that arise as interpolations; 2) A full proof with an explicit construction, via the theory of co-generators à la Słociński [54, 55]; and 3) A second full proof based on the abstract structure of semigroups, which admits a natural reformulation to semigroups defined over topological free products of R_{geq 0} and leads to various residuality results. In 2) a IInd free dilation theorem for topologised index sets is developed via a reformulation of the Trotter--Kato theorem for co-generators. As an application of this we demonstrate how evolution families can be reduced to continuously monitored processes subject to temporal change, à la the quantum Zeno effect [22, 23, 24, 30, 37].

Diffusion models for continuous data gained widespread adoption owing to their high quality generation and control mechanisms. However, controllable diffusion on discrete data faces challenges given that continuous guidance methods do not directly apply to discrete diffusion. Here, we provide a straightforward derivation of classifier-free and classifier-based guidance for discrete diffusion, as well as a new class of diffusion models that leverage uniform noise and that are more guidable because they can continuously edit their outputs. We improve the quality of these models with a novel continuous-time variational lower bound that yields state-of-the-art performance, especially in settings involving guidance or fast generation. Empirically, we demonstrate that our guidance mechanisms combined with uniform noise diffusion improve controllable generation relative to autoregressive and diffusion baselines on several discrete data domains, including genomic sequences, small molecule design, and discretized image generation.

In this paper we show some remarkable consequences of the method which proves that every bisymmetric, symmetric, reflexive, strictly monotonic binary map on a proper interval is continuous, in particular it is a quasi-arithmetic mean. Now we demonstrate that this result can be refined in the way that the symmetry condition can be weakened by assuming symmetry only for a pair of distinct points of an interval.

In this work, we aim to reconstruct a time-varying 3D model, capable of rendering photo-realistic renderings with independent control of viewpoint, illumination, and time, from Internet photos of large-scale landmarks. The core challenges are twofold. First, different types of temporal changes, such as illumination and changes to the underlying scene itself (such as replacing one graffiti artwork with another) are entangled together in the imagery. Second, scene-level temporal changes are often discrete and sporadic over time, rather than continuous. To tackle these problems, we propose a new scene representation equipped with a novel temporal step function encoding method that can model discrete scene-level content changes as piece-wise constant functions over time. Specifically, we represent the scene as a space-time radiance field with a per-image illumination embedding, where temporally-varying scene changes are encoded using a set of learned step functions. To facilitate our task of chronology reconstruction from Internet imagery, we also collect a new dataset of four scenes that exhibit various changes over time. We demonstrate that our method exhibits state-of-the-art view synthesis results on this dataset, while achieving independent control of viewpoint, time, and illumination.

In neural networks, it is often desirable to work with various representations of the same space. For example, 3D rotations can be represented with quaternions or Euler angles. In this paper, we advance a definition of a continuous representation, which can be helpful for training deep neural networks. We relate this to topological concepts such as homeomorphism and embedding. We then investigate what are continuous and discontinuous representations for 2D, 3D, and n-dimensional rotations. We demonstrate that for 3D rotations, all representations are discontinuous in the real Euclidean spaces of four or fewer dimensions. Thus, widely used representations such as quaternions and Euler angles are discontinuous and difficult for neural networks to learn. We show that the 3D rotations have continuous representations in 5D and 6D, which are more suitable for learning. We also present continuous representations for the general case of the n-dimensional rotation group SO(n). While our main focus is on rotations, we also show that our constructions apply to other groups such as the orthogonal group and similarity transforms. We finally present empirical results, which show that our continuous rotation representations outperform discontinuous ones for several practical problems in graphics and vision, including a simple autoencoder sanity test, a rotation estimator for 3D point clouds, and an inverse kinematics solver for 3D human poses.

Networks have become indispensable and ubiquitous structures in many fields to model the interactions among different entities, such as friendship in social networks or protein interactions in biological graphs. A major challenge is to understand the structure and dynamics of these systems. Although networks evolve through time, most existing graph representation learning methods target only static networks. Whereas approaches have been developed for the modeling of dynamic networks, there is a lack of efficient continuous time dynamic graph representation learning methods that can provide accurate network characterization and visualization in low dimensions while explicitly accounting for prominent network characteristics such as homophily and transitivity. In this paper, we propose the Piecewise-Velocity Model (PiVeM) for the representation of continuous-time dynamic networks. It learns dynamic embeddings in which the temporal evolution of nodes is approximated by piecewise linear interpolations based on a latent distance model with piecewise constant node-specific velocities. The model allows for analytically tractable expressions of the associated Poisson process likelihood with scalable inference invariant to the number of events. We further impose a scalable Kronecker structured Gaussian Process prior to the dynamics accounting for community structure, temporal smoothness, and disentangled (uncorrelated) latent embedding dimensions optimally learned to characterize the network dynamics. We show that PiVeM can successfully represent network structure and dynamics in ultra-low two-dimensional spaces. It outperforms relevant state-of-art methods in downstream tasks such as link prediction. In summary, PiVeM enables easily interpretable dynamic network visualizations and characterizations that can further improve our understanding of the intrinsic dynamics of time-evolving networks.

Many events occur in the world. Some event types are stochastically excited or inhibited---in the sense of having their probabilities elevated or decreased---by patterns in the sequence of previous events. Discovering such patterns can help us predict which type of event will happen next and when. We model streams of discrete events in continuous time, by constructing a neurally self-modulating multivariate point process in which the intensities of multiple event types evolve according to a novel continuous-time LSTM. This generative model allows past events to influence the future in complex and realistic ways, by conditioning future event intensities on the hidden state of a recurrent neural network that has consumed the stream of past events. Our model has desirable qualitative properties. It achieves competitive likelihood and predictive accuracy on real and synthetic datasets, including under missing-data conditions.

A fundamental dilemma in generative modeling persists: iterative diffusion models achieve outstanding fidelity, but at a significant computational cost, while efficient few-step alternatives are constrained by a hard quality ceiling. This conflict between generation steps and output quality arises from restrictive training objectives that focus exclusively on either infinitesimal dynamics (PF-ODEs) or direct endpoint prediction. We address this challenge by introducing an exact, continuous-time dynamics equation that analytically defines state transitions across any finite time interval. This leads to a novel generative paradigm, Transition Models (TiM), which adapt to arbitrary-step transitions, seamlessly traversing the generative trajectory from single leaps to fine-grained refinement with more steps. Despite having only 865M parameters, TiM achieves state-of-the-art performance, surpassing leading models such as SD3.5 (8B parameters) and FLUX.1 (12B parameters) across all evaluated step counts. Importantly, unlike previous few-step generators, TiM demonstrates monotonic quality improvement as the sampling budget increases. Additionally, when employing our native-resolution strategy, TiM delivers exceptional fidelity at resolutions up to 4096x4096.

Biological brains demonstrate complex neural activity, where the timing and interplay between neurons is critical to how brains process information. Most deep learning architectures simplify neural activity by abstracting away temporal dynamics. In this paper we challenge that paradigm. By incorporating neuron-level processing and synchronization, we can effectively reintroduce neural timing as a foundational element. We present the Continuous Thought Machine (CTM), a model designed to leverage neural dynamics as its core representation. The CTM has two core innovations: (1) neuron-level temporal processing, where each neuron uses unique weight parameters to process a history of incoming signals; and (2) neural synchronization employed as a latent representation. The CTM aims to strike a balance between oversimplified neuron abstractions that improve computational efficiency, and biological realism. It operates at a level of abstraction that effectively captures essential temporal dynamics while remaining computationally tractable for deep learning. We demonstrate the CTM's strong performance and versatility across a range of challenging tasks, including ImageNet-1K classification, solving 2D mazes, sorting, parity computation, question-answering, and RL tasks. Beyond displaying rich internal representations and offering a natural avenue for interpretation owing to its internal process, the CTM is able to perform tasks that require complex sequential reasoning. The CTM can also leverage adaptive compute, where it can stop earlier for simpler tasks, or keep computing when faced with more challenging instances. The goal of this work is to share the CTM and its associated innovations, rather than pushing for new state-of-the-art results. To that end, we believe the CTM represents a significant step toward developing more biologically plausible and powerful artificial intelligence systems.

Recent 4D Gaussian Splatting (4DGS) methods achieve impressive dynamic scene reconstruction but often rely on piecewise linear velocity approximations and short temporal windows. This disjointed modeling leads to severe temporal fragmentation, forcing primitives to be repeatedly eliminated and regenerated to track complex nonlinear dynamics. This makeshift approximation eliminates the long-term temporal identity of objects and causes an inevitable proliferation of Gaussians, hindering scalability to extended video sequences. To address this, we propose TRiGS, a novel 4D representation that utilizes unified, continuous geometric transformations. By integrating SE(3) transformations, hierarchical Bezier residuals, and learnable local anchors, TRiGS models geometrically consistent rigid motions for individual primitives. This continuous formulation preserves temporal identity and effectively mitigates unbounded memory growth. Extensive experiments demonstrate that TRiGS achieves high fidelity rendering on standard benchmarks while uniquely scaling to extended video sequences (e.g., 600 to 1200 frames) without severe memory bottlenecks, significantly outperforming prior works in temporal stability.

Large language model (LLM) agents increasingly operate over long and recurring external contexts, like document corpora and code repositories. Across invocations, existing approaches preserve either the agent's trajectory, passive access to raw material, or task-level strategies. None of them preserves what we argue is most needed for repeated same-context workloads: reusable orientation knowledge (e.g., what the context contains, how it is organized, and which entities, constants, and schemas have historically been useful) about the recurring context itself. We introduce PEEK, a system that caches and maintains this orientation knowledge as a context map: a small, constant-sized artifact in the agent's prompt that gives it a persistent peek into the external context. The map is maintained by a programmable cache policy with three modules: a Distiller that extracts transferable knowledge from inference-time signals, a Cartographer that translates it into structured edits, and a priority-based Evictor that enforces a fixed token budget. On long-context reasoning and information aggregation, PEEK improves over strong baselines by 6.3-34.0% while using 93-145 fewer iterations and incurring 1.7-5.8x lower cost than the state-of-the-art prompt-learning framework, ACE. On context learning, PEEK improves solving rate and rubric accuracy by 6.0-14.0% and 7.8-12.1%, respectively, at 1.4x lower cost than ACE. These gains generalize across LMs and agent architectures, including OpenAI Codex, a production-grade coding agent. Together, these results show that a context map helps long-context LLM agents interact with recurring external contexts more accurately and efficiently.

The accurate modeling of dynamics in interactive environments is critical for successful long-range prediction. Such a capability could advance Reinforcement Learning (RL) and Planning algorithms, but achieving it is challenging. Inaccuracies in model estimates can compound, resulting in increased errors over long horizons. We approach this problem from the lens of Koopman theory, where the nonlinear dynamics of the environment can be linearized in a high-dimensional latent space. This allows us to efficiently parallelize the sequential problem of long-range prediction using convolution while accounting for the agent's action at every time step. Our approach also enables stability analysis and better control over gradients through time. Taken together, these advantages result in significant improvement over the existing approaches, both in the efficiency and the accuracy of modeling dynamics over extended horizons. We also show that this model can be easily incorporated into dynamics modeling for model-based planning and model-free RL and report promising experimental results.

We study a class of Z^{d}-substitutive subshifts, including a large family of constant-length substitutions, and homomorphisms between them, i.e., factors modulo isomorphisms of Z^{d}. We prove that any measurable factor map and even any homomorphism associated to a matrix commuting with the expansion matrix, induces a continuous one. We also get strong restrictions on the normalizer group, proving that any endomorphism is invertible, the normalizer group is virtually generated by the shift action and the quotient of the normalizer group by the automorphisms is restricted by the digit tile of the substitution.

Temporal causal representation learning methods assume that causal mechanisms switch instantaneously between discrete domains, yet real-world systems often exhibit continuous mechanism transitions. For example, a vehicle's dynamics evolve gradually through a turning maneuver, and human gait shifts smoothly from walking to running. We formalize this setting by modeling transitional mechanisms as convex combinations of finitely many atomic mechanisms, governed by time-varying mixing coefficients. Our theoretical contributions establish that both the latent causal variables and the continuous mixing trajectory are jointly identifiable. We further propose TRACE, a Mixture-of-Experts framework where each expert learns one atomic mechanism during training, enabling recovery of mechanism trajectories at test time. This formulation generalizes to intermediate mechanism states never observed during training. Experiments on synthetic and real-world data demonstrate that TRACE recovers mixing trajectories with up to 0.99 correlation, substantially outperforming discrete-switching baselines.

We present metrics for measuring state similarity in Markov decision processes (MDPs) with infinitely many states, including MDPs with continuous state spaces. Such metrics provide a stable quantitative analogue of the notion of bisimulation for MDPs, and are suitable for use in MDP approximation. We show that the optimal value function associated with a discounted infinite horizon planning task varies continuously with respect to our metric distances.

Methods based on ordinary differential equations (ODEs) are widely used to build generative models of time-series. In addition to high computational overhead due to explicitly computing hidden states recurrence, existing ODE-based models fall short in learning sequence data with sharp transitions - common in many real-world systems - due to numerical challenges during optimization. In this work, we propose LS4, a generative model for sequences with latent variables evolving according to a state space ODE to increase modeling capacity. Inspired by recent deep state space models (S4), we achieve speedups by leveraging a convolutional representation of LS4 which bypasses the explicit evaluation of hidden states. We show that LS4 significantly outperforms previous continuous-time generative models in terms of marginal distribution, classification, and prediction scores on real-world datasets in the Monash Forecasting Repository, and is capable of modeling highly stochastic data with sharp temporal transitions. LS4 sets state-of-the-art for continuous-time latent generative models, with significant improvement of mean squared error and tighter variational lower bounds on irregularly-sampled datasets, while also being x100 faster than other baselines on long sequences.

Remote sensing offers an unparalleled vantage point for observing the Earth's long-term surface evolution, yet it demands that a model not only perceive land cover at isolated moments, but also track changes, memorize evolution histories, and reason across time and space. However, existing studies lack a systematic evaluation that dissects these distinct competencies. To fill this gap, we introduce ChronoBench, a multidimensional benchmark that decomposes this task into four progressive cognitive levels (i.e., Land Cover Perception, Temporal Recognition, Long-Term Memory, and Spatio-Temporal Reasoning). The ChronoBench comprises 12 sub-tasks and 17,689 rigorously validated QA (Question-Answer) pairs. Extensive evaluations reveal that mainstream MLLMs fall drastically behind human experts, with Long-Term Memory emerging as the most critical bottleneck. Motivated by this finding, we further propose GeoChrono, an MLLM with enhanced capabilities for tracing, memorizing, and reasoning about long-term geographic evolution. Leveraging the physical prior that geographic parcels remain spatially fixed while their semantics evolve, we design a Temporal Trajectory Encoder~(TempEnc) that constructs per-location temporal trajectories for dedicated land cover evolution modeling, and we introduce a Coarse-to-Fine Token Compressor~(C2FComp) that adaptively preserves dynamic regions while compressing the static background. To support training, we also construct ChronoInstruct, a 104K-sample instruction-tuning dataset spanning all competency levels for training. GeoChrono achieves state-of-the-art performance on ChronoBench, surpassing the leading commercial MLLMs by over 20%, while C2FComp reduces visual tokens by over 56% while retaining GeoChrono's 94.6% performance. The code and data will be available at https://github.com/IntelliSensing/GeoChrono

Convergent Cross Mapping (CCM) is a powerful method for detecting causality in coupled nonlinear dynamical systems, providing a model-free approach to capture dynamic causal interactions. Partial Cross Mapping (PCM) was introduced as an extension of CCM to address indirect causality in three-variable systems by comparing cross-mapping quality between direct cause-effect mapping and indirect mapping through an intermediate conditioning variable. However, PCM remains limited to univariate delay embeddings in its cross-mapping processes. In this work, we extend PCM to the multivariate setting, introducing multiPCM, which leverages multivariate embeddings to more effectively distinguish indirect causal relationships. We further propose a multivariate cross-mapping framework (MXMap) for causal discovery in dynamical systems. This two-phase framework combines (1) pairwise CCM tests to establish an initial causal graph and (2) multiPCM to refine the graph by pruning indirect causal connections. Through experiments on simulated data and the ERA5 Reanalysis weather dataset, we demonstrate the effectiveness of MXMap. Additionally, MXMap is compared against several baseline methods, showing advantages in accuracy and causal graph refinement.

Few-step video generation has been significantly advanced by consistency distillation. However, the performance of consistency-distilled models often degrades as more sampling steps are allocated at test time, limiting their effectiveness for any-step video diffusion. This limitation arises because consistency distillation replaces the original probability-flow ODE trajectory with a consistency-sampling trajectory, weakening the desirable test-time scaling behavior of ODE sampling. To address this limitation, we introduce AnyFlow, the first any-step video diffusion distillation framework based on flow maps. Instead of distilling a model for only a few fixed sampling steps, AnyFlow optimizes the full ODE sampling trajectory. To this end, we shift the distillation target from endpoint consistency mapping (z_{t}rightarrow z_{0}) to flow-map transition learning (z_{t}rightarrow z_{r}) over arbitrary time intervals. We further propose Flow Map Backward Simulation, which decomposes a full Euler rollout into shortcut flow-map transitions, enabling efficient on-policy distillation that reduces test-time errors (i.e., discretization error in few-step sampling and exposure bias in causal generation). Extensive experiments across both bidirectional and causal architectures, at scales ranging from 1.3B to 14B parameters, demonstrate that AnyFlow achieves performance matches or surpasses consistency-based counterparts in the few-step regime, while scaling with sampling step budgets.

Markov processes are widely used mathematical models for describing dynamic systems in various fields. However, accurately simulating large-scale systems at long time scales is computationally expensive due to the short time steps required for accurate integration. In this paper, we introduce an inference process that maps complex systems into a simplified representational space and models large jumps in time. To achieve this, we propose Time-lagged Information Bottleneck (T-IB), a principled objective rooted in information theory, which aims to capture relevant temporal features while discarding high-frequency information to simplify the simulation task and minimize the inference error. Our experiments demonstrate that T-IB learns information-optimal representations for accurately modeling the statistical properties and dynamics of the original process at a selected time lag, outperforming existing time-lagged dimensionality reduction methods.

Current interactive LLM agents rely on goal-conditioned stepwise planning, where environmental understanding is acquired reactively during execution rather than established beforehand. This temporal inversion leads to Delayed Environmental Perception: agents must infer environmental constraints through trial-and-error, resulting in an Epistemic Bottleneck that traps them in inefficient failure cycles. Inspired by human affordance perception and cognitive map theory, we propose the Map-then-Act Paradigm (MAP), a plug-and-play framework that shifts environment understanding before execution. MAP consists of three stages: (1) Global Exploration, acquiring environment-general priors; (2) Task-Specific Mapping, constructing a structured cognitive map; and (3) Knowledge-Augmented Execution, solving tasks grounded on the map. Experiments show consistent gains across benchmarks and LLMs. On ARC-AGI-3, MAP enables frontier models to surpass near-zero baseline performance in 22 of 25 game environments. We further introduce MAP-2K, a dataset of map-then-act trajectories, and show that training on it outperforms expert execution traces, suggesting that understanding environments is more fundamental than imitation.

Effectively modeling long spatiotemporal sequences is challenging due to the need to model complex spatial correlations and long-range temporal dependencies simultaneously. ConvLSTMs attempt to address this by updating tensor-valued states with recurrent neural networks, but their sequential computation makes them slow to train. In contrast, Transformers can process an entire spatiotemporal sequence, compressed into tokens, in parallel. However, the cost of attention scales quadratically in length, limiting their scalability to longer sequences. Here, we address the challenges of prior methods and introduce convolutional state space models (ConvSSM) that combine the tensor modeling ideas of ConvLSTM with the long sequence modeling approaches of state space methods such as S4 and S5. First, we demonstrate how parallel scans can be applied to convolutional recurrences to achieve subquadratic parallelization and fast autoregressive generation. We then establish an equivalence between the dynamics of ConvSSMs and SSMs, which motivates parameterization and initialization strategies for modeling long-range dependencies. The result is ConvS5, an efficient ConvSSM variant for long-range spatiotemporal modeling. ConvS5 significantly outperforms Transformers and ConvLSTM on a long horizon Moving-MNIST experiment while training 3X faster than ConvLSTM and generating samples 400X faster than Transformers. In addition, ConvS5 matches or exceeds the performance of state-of-the-art methods on challenging DMLab, Minecraft and Habitat prediction benchmarks and enables new directions for modeling long spatiotemporal sequences.

A complex system with cluttered observations may be a coupled mixture of multiple simple sub-systems corresponding to latent entities. Such sub-systems may hold distinct dynamics in the continuous-time domain; therein, complicated interactions between sub-systems also evolve over time. This setting is fairly common in the real world but has been less considered. In this paper, we propose a sequential learning approach under this setting by decoupling a complex system for handling irregularly sampled and cluttered sequential observations. Such decoupling brings about not only subsystems describing the dynamics of each latent entity but also a meta-system capturing the interaction between entities over time. Specifically, we argue that the meta-system evolving within a simplex is governed by projected differential equations (ProjDEs). We further analyze and provide neural-friendly projection operators in the context of Bregman divergence. Experimental results on synthetic and real-world datasets show the advantages of our approach when facing complex and cluttered sequential data compared to the state-of-the-art.

We develop a discipline-agnostic emergence calculus that treats theories as fixed points of idempotent operators acting on descriptions. We show that, once processes are composable but access to the underlying system is mediated by a bounded observational interface, a canonical toolkit of six closure-changing primitives (P1--P6) is unavoidable. The framework unifies order-theoretic closure operators with dynamics-induced endomaps E_{τ,f} built from a Markov kernel, a coarse-graining lens, and a time scale τ. We introduce a computable total-variation idempotence defect for E_{τ,f}; small retention error implies approximate idempotence and yields stable "objects" packaged at the chosen τ within a fixed lens. For directionality, we define an arrow-of-time functional as the path-space KL divergence between forward and time-reversed trajectories and prove it is monotone under coarse-graining (data processing); we also formalize a protocol-trap audit showing that protocol holonomy alone cannot sustain asymmetry without a genuine affinity in the lifted dynamics. Finally, we prove a finite forcing-style counting lemma: relative to a partition-based theory, definable predicate extensions are exponentially rare, giving a clean anti-saturation mechanism for strict ladder climbing.

Temporal Point Processes (TPP) are probabilistic generative frameworks. They model discrete event sequences localized in continuous time. Generally, real-life events reveal descriptive information, known as marks. Marked TPPs model time and marks of the event together for practical relevance. Conditioned on past events, marked TPPs aim to learn the joint distribution of the time and the mark of the next event. For simplicity, conditionally independent TPP models assume time and marks are independent given event history. They factorize the conditional joint distribution of time and mark into the product of individual conditional distributions. This structural limitation in the design of TPP models hurt the predictive performance on entangled time and mark interactions. In this work, we model the conditional inter-dependence of time and mark to overcome the limitations of conditionally independent models. We construct a multivariate TPP conditioning the time distribution on the current event mark in addition to past events. Besides the conventional intensity-based models for conditional joint distribution, we also draw on flexible intensity-free TPP models from the literature. The proposed TPP models outperform conditionally independent and dependent models in standard prediction tasks. Our experimentation on various datasets with multiple evaluation metrics highlights the merit of the proposed approach.

Continuous-time event sequences play a vital role in real-world domains such as healthcare, finance, online shopping, social networks, and so on. To model such data, temporal point processes (TPPs) have emerged as the most natural and competitive models, making a significant impact in both academic and application communities. Despite the emergence of many powerful models in recent years, there hasn't been a central benchmark for these models and future research endeavors. This lack of standardization impedes researchers and practitioners from comparing methods and reproducing results, potentially slowing down progress in this field. In this paper, we present EasyTPP, the first central repository of research assets (e.g., data, models, evaluation programs, documentations) in the area of event sequence modeling. Our EasyTPP makes several unique contributions to this area: a unified interface of using existing datasets and adding new datasets; a wide range of evaluation programs that are easy to use and extend as well as facilitate reproducible research; implementations of popular neural TPPs, together with a rich library of modules by composing which one could quickly build complex models. All the data and implementation can be found at https://github.com/ant-research/EasyTemporalPointProcess. We will actively maintain this benchmark and welcome contributions from other researchers and practitioners. Our benchmark will help promote reproducible research in this field, thus accelerating research progress as well as making more significant real-world impacts.

We address the problem of learning the dynamics of an unknown non-parametric system linking a target and a feature time series. The feature time series is measured on a sparse and irregular grid, while we have access to only a few points of the target time series. Once learned, we can use these dynamics to predict values of the target from the previous values of the feature time series. We frame this task as learning the solution map of a controlled differential equation (CDE). By leveraging the rich theory of signatures, we are able to cast this non-linear problem as a high-dimensional linear regression. We provide an oracle bound on the prediction error which exhibits explicit dependencies on the individual-specific sampling schemes. Our theoretical results are illustrated by simulations which show that our method outperforms existing algorithms for recovering the full time series while being computationally cheap. We conclude by demonstrating its potential on real-world epidemiological data.

Selective state space models (SSM), such as Mamba, have gained prominence for their effectiveness in modeling sequential data. Despite their outstanding empirical performance, a comprehensive theoretical understanding of deep selective SSM remains elusive, hindering their further development and adoption for applications that need high fidelity. In this paper, we investigate the dynamical properties of tokens in a pre-trained Mamba model. In particular, we derive the dynamical system governing the continuous-time limit of the Mamba model and characterize the asymptotic behavior of its solutions. In the one-dimensional case, we prove that only one of the following two scenarios happens: either all tokens converge to zero, or all tokens diverge to infinity. We provide criteria based on model parameters to determine when each scenario occurs. For the convergent scenario, we empirically verify that this scenario negatively impacts the model's performance. For the divergent scenario, we prove that different tokens will diverge to infinity at different rates, thereby contributing unequally to the updates during model training. Based on these investigations, we propose two refinements for the model: excluding the convergent scenario and reordering tokens based on their importance scores, both aimed at improving practical performance. Our experimental results validate these refinements, offering insights into enhancing Mamba's effectiveness in real-world applications.

This work represents the first effort to scale up continuous-time consistency distillation to general application-level image and video diffusion models. Although continuous-time consistency model (sCM) is theoretically principled and empirically powerful for accelerating academic-scale diffusion, its applicability to large-scale text-to-image and video tasks remains unclear due to infrastructure challenges in Jacobian-vector product (JVP) computation and the limitations of standard evaluation benchmarks. We first develop a parallelism-compatible FlashAttention-2 JVP kernel, enabling sCM training on models with over 10 billion parameters and high-dimensional video tasks. Our investigation reveals fundamental quality limitations of sCM in fine-detail generation, which we attribute to error accumulation and the "mode-covering" nature of its forward-divergence objective. To remedy this, we propose the score-regularized continuous-time consistency model (rCM), which incorporates score distillation as a long-skip regularizer. This integration complements sCM with the "mode-seeking" reverse divergence, effectively improving visual quality while maintaining high generation diversity. Validated on large-scale models (Cosmos-Predict2, Wan2.1) up to 14B parameters and 5-second videos, rCM matches or surpasses the state-of-the-art distillation method DMD2 on quality metrics while offering notable advantages in diversity, all without GAN tuning or extensive hyperparameter searches. The distilled models generate high-fidelity samples in only 1sim4 steps, accelerating diffusion sampling by 15timessim50times. These results position rCM as a practical and theoretically grounded framework for advancing large-scale diffusion distillation.

We propose a unified framework to study policy evaluation (PE) and the associated temporal difference (TD) methods for reinforcement learning in continuous time and space. We show that PE is equivalent to maintaining the martingale condition of a process. From this perspective, we find that the mean--square TD error approximates the quadratic variation of the martingale and thus is not a suitable objective for PE. We present two methods to use the martingale characterization for designing PE algorithms. The first one minimizes a "martingale loss function", whose solution is proved to be the best approximation of the true value function in the mean--square sense. This method interprets the classical gradient Monte-Carlo algorithm. The second method is based on a system of equations called the "martingale orthogonality conditions" with test functions. Solving these equations in different ways recovers various classical TD algorithms, such as TD(lambda), LSTD, and GTD. Different choices of test functions determine in what sense the resulting solutions approximate the true value function. Moreover, we prove that any convergent time-discretized algorithm converges to its continuous-time counterpart as the mesh size goes to zero, and we provide the convergence rate. We demonstrate the theoretical results and corresponding algorithms with numerical experiments and applications.

The Brownian sphere is a random metric space, homeomorphic to the two-dimensional sphere, which arises as the universal scaling limit of many types of random planar maps. The direct construction of the Brownian sphere is via a continuous analogue of the Cori--Vauquelin--Schaeffer (CVS) bijection. The CVS bijection maps labeled trees to planar maps, and the continuous version maps Aldous' continuum random tree with Brownian labels (the Brownian snake) to the Brownian sphere. In this work, we describe the inverse of the continuous CVS bijection, by constructing the Brownian snake as a measurable function of the Brownian sphere. Special care is needed to work with the orientation of the Brownian sphere.

Maps are fundamental medium to visualize and represent the real word in a simple and 16 philosophical way. The emergence of the 3rd wave information has made a proportion of maps are available to be generated ubiquitously, which would significantly enrich the dimensions and perspectives to understand the characteristics of the real world. However, a majority of map dataset have never been discovered, acquired and effectively used, and the map data used in many applications might not be completely fitted for the authentic demands of these applications. This challenge is emerged due to the lack of numerous well-labelled benchmark datasets for implementing the deep learning approaches into identifying complicated map content. Thus, we develop a large-scale benchmark dataset that includes well-labelled dataset for map text annotation recognition, map scene classification, map super-resolution reconstruction, and map style transferring. Furthermore, these well-labelled datasets would facilitate the state-of-the-art machine intelligence technologies to conduct map feature detection, map pattern recognition and map content retrieval. We hope our efforts would be useful for AI-enhanced cartographical applications.

Pairwise dependence measures such as correlation and causality are fundamental to temporal data mining, yet there is still no principled and robust way to quantify dependence between heterogeneous data types, especially between continuous time series and discrete temporal event sequences. Existing approaches rely on ad hoc transformations or mutual-information estimators that are highly sensitive to quantization, repeated values, and event redundancy, leading to biased or unstable results in practice. We propose a nonparametric mutual information estimator that directly measures the dependence between time series and event sequences without data transformation, learning, or ad hoc discretization. Our method models the continuous-discrete duality of real-world time series to handle quantization and repeated-value artifacts and introduces a latent event clustering strategy to mitigate bias from event co-occurrence and redundancy. Together, these yield a robust and unified framework that bridges discrete and continuous mutual information. We evaluate the proposed estimator on four representative tasks: discrete-continuous time-delayed mutual information for causality analysis, global and local temporal repetition discovery, discrete covariate selection for time series forecasting, and continuous feature selection for classification. Experiments on synthetic and real-world datasets show consistent improvements over existing methods in accuracy, robustness, and interpretability, positioning our approach as a general-purpose dependence operator for heterogeneous temporal data, similar to Pearson correlation for homogeneous time series. Code available at: https://github.com/HaojiHu/Multimodal-Temporal-Data-Quantification

Event cameras are dynamic vision sensors inspired by the biological retina, characterized by their high dynamic range, high temporal resolution, and low power consumption. These features make them capable of perceiving 3D environments even in extreme conditions. Event data is continuous across the time dimension, which allows a detailed description of each pixel's movements. To fully utilize the temporally dense and continuous nature of event cameras, we propose a novel temporal event stereo, a framework that continuously uses information from previous time steps. This is accomplished through the simultaneous training of an event stereo matching network alongside stereoscopic flow, a new concept that captures all pixel movements from stereo cameras. Since obtaining ground truth for optical flow during training is challenging, we propose a method that uses only disparity maps to train the stereoscopic flow. The performance of event-based stereo matching is enhanced by temporally aggregating information using the flows. We have achieved state-of-the-art performance on the MVSEC and the DSEC datasets. The method is computationally efficient, as it stacks previous information in a cascading manner. The code is available at https://github.com/mickeykang16/TemporalEventStereo.

We propose a unified, few-step generative modeling framework based on cumulative flow maps for long-range transport in probability space, inspired by flow-map techniques for physical transport and dynamics. At its core is a cumulative-flow abstraction that connects local, instantaneous updates with finite-time transport, enabling generative models to reason about global state transitions. This perspective yields a unified few-step framework built on cumulative transport and cumulative parameterization that applies broadly to existing diffusion- and flow-based models without being tied to a specific prediction instantiation. Our formulation supports few-step and even one-step generation while preserving synthesis quality, requiring only minimal changes to time embeddings and training objectives, and no increase in model capacity. We demonstrate its effectiveness across diverse tasks, including image generation, geometric distribution modeling, joint prediction, and SDF generation, with reduced inference cost.

Koopman theory turns nonlinear dynamics into a linear spectral problem. In computation, however, everything depends on a hard finite-dimensional choice: the observables must be expressive, nearly invariant under the dynamics, and, ideally, compatible with composition. Deep Koopman methods learn flexible coordinates, whereas structure-preserving methods enforce operator identities on fixed dictionaries. We combine these ideas by introducing Deep Embedded Multiplicative Dynamic Mode Decomposition (DeepMDMD), a method that learns a latent space and a partition of it, while enforcing the Koopman product rule as an exact algebraic constraint. Training alternates between an exact multiplicative operator update and a differentiable latent-clustering step that promotes Koopman closure. The result is a finite transition map on learned latent cells. Its nonzero spectrum lies on the unit circle, its dictionary is shaped by the dynamics rather than by ambient geometry, and forecasts are made in latent coordinates before being decoded to physical space. Across Hamiltonian, chaotic, and fluid examples, DeepMDMD learns dictionaries that are far more compact and dynamically coherent than those produced by geometric MDMD partitions. It reduces spectral pollution, reveals richer continuous-spectrum structure, and gives stable forecasts under severe noise. In high-dimensional flows, including a 158,624-dimensional cylinder wake and a noisy Re=20,000 lid-driven cavity, it preserves coherent structures and long-time spectral statistics where state-space MDMD fails. These results suggest a practical rule for Koopman learning: learn the coordinates, constrain the algebra.

Goal representation affects the performance of Hierarchical Reinforcement Learning (HRL) algorithms by decomposing the complex learning problem into easier subtasks. Recent studies show that representations that preserve temporally abstract environment dynamics are successful in solving difficult problems and provide theoretical guarantees for optimality. These methods however cannot scale to tasks where environment dynamics increase in complexity i.e. the temporally abstract transition relations depend on larger number of variables. On the other hand, other efforts have tried to use spatial abstraction to mitigate the previous issues. Their limitations include scalability to high dimensional environments and dependency on prior knowledge. In this paper, we propose a novel three-layer HRL algorithm that introduces, at different levels of the hierarchy, both a spatial and a temporal goal abstraction. We provide a theoretical study of the regret bounds of the learned policies. We evaluate the approach on complex continuous control tasks, demonstrating the effectiveness of spatial and temporal abstractions learned by this approach.

Climate and weather prediction traditionally relies on complex numerical simulations of atmospheric physics. Deep learning approaches, such as transformers, have recently challenged the simulation paradigm with complex network forecasts. However, they often act as data-driven black-box models that neglect the underlying physics and lack uncertainty quantification. We address these limitations with ClimODE, a spatiotemporal continuous-time process that implements a key principle of advection from statistical mechanics, namely, weather changes due to a spatial movement of quantities over time. ClimODE models precise weather evolution with value-conserving dynamics, learning global weather transport as a neural flow, which also enables estimating the uncertainty in predictions. Our approach outperforms existing data-driven methods in global and regional forecasting with an order of magnitude smaller parameterization, establishing a new state of the art.

Conventional neural architectures for sequential data present important limitations. Recurrent networks suffer from exploding and vanishing gradients, small effective memory horizons, and must be trained sequentially. Convolutional networks are unable to handle sequences of unknown size and their memory horizon must be defined a priori. In this work, we show that all these problems can be solved by formulating convolutional kernels in CNNs as continuous functions. The resulting Continuous Kernel Convolution (CKConv) allows us to model arbitrarily long sequences in a parallel manner, within a single operation, and without relying on any form of recurrence. We show that Continuous Kernel Convolutional Networks (CKCNNs) obtain state-of-the-art results in multiple datasets, e.g., permuted MNIST, and, thanks to their continuous nature, are able to handle non-uniformly sampled datasets and irregularly-sampled data natively. CKCNNs match or perform better than neural ODEs designed for these purposes in a faster and simpler manner.

Modern techniques for physical simulations rely on numerical schemes and mesh-refinement methods to address trade-offs between precision and complexity, but these handcrafted solutions are tedious and require high computational power. Data-driven methods based on large-scale machine learning promise high adaptivity by integrating long-range dependencies more directly and efficiently. In this work, we focus on fluid dynamics and address the shortcomings of a large part of the literature, which are based on fixed support for computations and predictions in the form of regular or irregular grids. We propose a novel setup to perform predictions in a continuous spatial and temporal domain while being trained on sparse observations. We formulate the task as a double observation problem and propose a solution with two interlinked dynamical systems defined on, respectively, the sparse positions and the continuous domain, which allows to forecast and interpolate a solution from the initial condition. Our practical implementation involves recurrent GNNs and a spatio-temporal attention observer capable of interpolating the solution at arbitrary locations. Our model not only generalizes to new initial conditions (as standard auto-regressive models do) but also performs evaluation at arbitrary space and time locations. We evaluate on three standard datasets in fluid dynamics and compare to strong baselines, which are outperformed both in classical settings and in the extended new task requiring continuous predictions.

We present a unified framework capable of solving a broad range of 3D tasks. Our approach features a stateful recurrent model that continuously updates its state representation with each new observation. Given a stream of images, this evolving state can be used to generate metric-scale pointmaps (per-pixel 3D points) for each new input in an online fashion. These pointmaps reside within a common coordinate system, and can be accumulated into a coherent, dense scene reconstruction that updates as new images arrive. Our model, called CUT3R (Continuous Updating Transformer for 3D Reconstruction), captures rich priors of real-world scenes: not only can it predict accurate pointmaps from image observations, but it can also infer unseen regions of the scene by probing at virtual, unobserved views. Our method is simple yet highly flexible, naturally accepting varying lengths of images that may be either video streams or unordered photo collections, containing both static and dynamic content. We evaluate our method on various 3D/4D tasks and demonstrate competitive or state-of-the-art performance in each. Project Page: https://cut3r.github.io/

We introduce a modern Hopfield network with continuous states and a corresponding update rule. The new Hopfield network can store exponentially (with the dimension of the associative space) many patterns, retrieves the pattern with one update, and has exponentially small retrieval errors. It has three types of energy minima (fixed points of the update): (1) global fixed point averaging over all patterns, (2) metastable states averaging over a subset of patterns, and (3) fixed points which store a single pattern. The new update rule is equivalent to the attention mechanism used in transformers. This equivalence enables a characterization of the heads of transformer models. These heads perform in the first layers preferably global averaging and in higher layers partial averaging via metastable states. The new modern Hopfield network can be integrated into deep learning architectures as layers to allow the storage of and access to raw input data, intermediate results, or learned prototypes. These Hopfield layers enable new ways of deep learning, beyond fully-connected, convolutional, or recurrent networks, and provide pooling, memory, association, and attention mechanisms. We demonstrate the broad applicability of the Hopfield layers across various domains. Hopfield layers improved state-of-the-art on three out of four considered multiple instance learning problems as well as on immune repertoire classification with several hundreds of thousands of instances. On the UCI benchmark collections of small classification tasks, where deep learning methods typically struggle, Hopfield layers yielded a new state-of-the-art when compared to different machine learning methods. Finally, Hopfield layers achieved state-of-the-art on two drug design datasets. The implementation is available at: https://github.com/ml-jku/hopfield-layers

Continuous convolution has recently gained prominence due to its ability to handle irregularly sampled data and model long-term dependency. Also, the promising experimental results of using large convolutional kernels have catalyzed the development of continuous convolution since they can construct large kernels very efficiently. Leveraging neural networks, more specifically multilayer perceptrons (MLPs), is by far the most prevalent approach to implementing continuous convolution. However, there are a few drawbacks, such as high computational costs, complex hyperparameter tuning, and limited descriptive power of filters. This paper suggests an alternative approach to building a continuous convolution without neural networks, resulting in more computationally efficient and improved performance. We present self-moving point representations where weight parameters freely move, and interpolation schemes are used to implement continuous functions. When applied to construct convolutional kernels, the experimental results have shown improved performance with drop-in replacement in the existing frameworks. Due to its lightweight structure, we are first to demonstrate the effectiveness of continuous convolution in a large-scale setting, e.g., ImageNet, presenting the improvements over the prior arts. Our code is available on https://github.com/sangnekim/SMPConv

We study the SAM (Sharpness-Aware Minimization) optimizer which has recently attracted a lot of interest due to its increased performance over more classical variants of stochastic gradient descent. Our main contribution is the derivation of continuous-time models (in the form of SDEs) for SAM and two of its variants, both for the full-batch and mini-batch settings. We demonstrate that these SDEs are rigorous approximations of the real discrete-time algorithms (in a weak sense, scaling linearly with the learning rate). Using these models, we then offer an explanation of why SAM prefers flat minima over sharp ones~--~by showing that it minimizes an implicitly regularized loss with a Hessian-dependent noise structure. Finally, we prove that SAM is attracted to saddle points under some realistic conditions. Our theoretical results are supported by detailed experiments.

Continuous perception, the ability to integrate visual observations over time in a continuous stream fashion, is essential for robust real-world understanding, yet remains largely untested in current multimodal models. We introduce CP-Bench, a minimal and fully controlled benchmark designed to isolate this capability using an extremely simple task: counting identical cubes in a synthetic scene while the camera moves and only reveals subsets of objects at any moment. Despite the simplicity of the setting, we find that state-of-the-art open-source and commercial models, including Qwen-3-VL, InternVL3, GPT-5, and Gemini-3-Pro, fail dramatically. A static-camera control variant confirms that the failure arises not from object recognition but from an inability to accumulate evidence across time. Further experiments show that neither higher sampling FPS, perception- or spatial-enhanced models, nor finetuning with additional videos leads to meaningful cross-temporal generalization. Our results reveal a fundamental limitation in modern multimodal architectures and training paradigms. CP-Bench provides a simple yet powerful diagnostic tool and establishes a clean testbed for developing models capable of genuine time-consistent visual reasoning.

Real-world time series are characterized by intrinsic non-stationarity that poses a principal challenge for deep forecasting models. While previous models suffer from complicated series variations induced by changing temporal distribution, we tackle non-stationary time series with modern Koopman theory that fundamentally considers the underlying time-variant dynamics. Inspired by Koopman theory of portraying complex dynamical systems, we disentangle time-variant and time-invariant components from intricate non-stationary series by Fourier Filter and design Koopman Predictor to advance respective dynamics forward. Technically, we propose Koopa as a novel Koopman forecaster composed of stackable blocks that learn hierarchical dynamics. Koopa seeks measurement functions for Koopman embedding and utilizes Koopman operators as linear portraits of implicit transition. To cope with time-variant dynamics that exhibits strong locality, Koopa calculates context-aware operators in the temporal neighborhood and is able to utilize incoming ground truth to scale up forecast horizon. Besides, by integrating Koopman Predictors into deep residual structure, we ravel out the binding reconstruction loss in previous Koopman forecasters and achieve end-to-end forecasting objective optimization. Compared with the state-of-the-art model, Koopa achieves competitive performance while saving 77.3% training time and 76.0% memory.

Despite the unprecedented volume of multimodal data provided by modern Earth observation systems, our ability to model atmospheric dynamics remains constrained. Traditional modeling frameworks force heterogeneous measurements into predefined spatial grids, inherently limiting the full exploitation of raw sensor data and creating severe computational bottlenecks. Here we present Earth-o1, an observation-native atmospheric world model that overcomes these structural limitations. Rather than relying on conventional atmospheric dynamical modeling systems or traditional data assimilation, Earth-o1 directly learns the continuous, three-dimensional physical evolution of the Earth system from ungridded observational data. By integrating diverse sensor inputs into a unified, grid-free dynamical field, the model autonomously advances the atmospheric state in space and time. We show that this fundamentally distinct paradigm enables direct, real-time forecasting and cross-sensor inference without the overhead of explicit numerical solvers. In hindcast evaluations, Earth-o1 achieves surface forecast skill comparable to the operational Integrated Forecasting System (IFS). These results establish that continuous, observation-driven world models -- a new class of fully observation-native geophysical simulators -- can match the fidelity of established physical frameworks, providing a scalable data-driven foundation for a digital twin of the Earth.

Mapping is crucial for spatial reasoning, planning and robot navigation. Existing approaches range from metric, which require precise geometry-based optimization, to purely topological, where image-as-node based graphs lack explicit object-level reasoning and interconnectivity. In this paper, we propose a novel topological representation of an environment based on "image segments", which are semantically meaningful and open-vocabulary queryable, conferring several advantages over previous works based on pixel-level features. Unlike 3D scene graphs, we create a purely topological graph with segments as nodes, where edges are formed by a) associating segment-level descriptors between pairs of consecutive images and b) connecting neighboring segments within an image using their pixel centroids. This unveils a "continuous sense of a place", defined by inter-image persistence of segments along with their intra-image neighbours. It further enables us to represent and update segment-level descriptors through neighborhood aggregation using graph convolution layers, which improves robot localization based on segment-level retrieval. Using real-world data, we show how our proposed map representation can be used to i) generate navigation plans in the form of "hops over segments" and ii) search for target objects using natural language queries describing spatial relations of objects. Furthermore, we quantitatively analyze data association at the segment level, which underpins inter-image connectivity during mapping and segment-level localization when revisiting the same place. Finally, we show preliminary trials on segment-level `hopping' based zero-shot real-world navigation. Project page with supplementary details: oravus.github.io/RoboHop/

Visual navigation ability is strongly tied to its underlying representation of the world. Unlike classical 3D maps that require globally-consistent geometry, image- or object-relative topological graphs almost entirely do away with geometric understanding. But, this comes at the cost of navigation capability, often limiting it to merely teach-and-repeat. In this work, we propose a novel map representation in the form of pixel-relative connectivity, which is geometrically accurate but does not require global geometric consistency. Inspired by recent progress in 3D grounded image matching, we construct a map from an image sequence through inter-image connectivity based on pixel correspondences in the relative 3D coordinate systems of individual image pairs. We then use this pixel-level graph to perform global path planning by approximating and sparsifying intra-image pixel connectivity. Through this, we derive a ''WayPixel Costmap'' representation and train a controller conditioned on it to predict a trajectory rollout. We show that this dense pixel-level costmap based on relative geometry is a more accurate conditioning variable for control prediction than its image- and object-level counterparts. This enables a highly capable navigation system, as validated on four types of navigation tasks in the simulator and through real world demonstrations.

We consider the problem of learning in a non-stationary reinforcement learning (RL) environment, where the setting can be fully described by a piecewise stationary discrete-time Markov decision process (MDP). We introduce a variant of the Restarted Bayesian Online Change-Point Detection algorithm (R-BOCPD) that operates on input streams originating from the more general multinomial distribution and provides near-optimal theoretical guarantees in terms of false-alarm rate and detection delay. Based on this, we propose an improved version of the UCRL2 algorithm for MDPs with state transition kernel sampled from a multinomial distribution, which we call R-BOCPD-UCRL2. We perform a finite-time performance analysis and show that R-BOCPD-UCRL2 enjoys a favorable regret bound of Oleft(D O A T K_T logleft (frac{T{delta} right) + K_T log frac{K_T{delta}}{minlimits_ell : KLleft( {theta^{(ell+1)}}midmathbf{theta^{(ell)}}right)}}right), where D is the largest MDP diameter from the set of MDPs defining the piecewise stationary MDP setting, O is the finite number of states (constant over all changes), A is the finite number of actions (constant over all changes), K_T is the number of change points up to horizon T, and theta^{(ell)} is the transition kernel during the interval [c_ell, c_{ell+1}), which we assume to be multinomially distributed over the set of states O. Interestingly, the performance bound does not directly scale with the variation in MDP state transition distributions and rewards, ie. can also model abrupt changes. In practice, R-BOCPD-UCRL2 outperforms the state-of-the-art in a variety of scenarios in synthetic environments. We provide a detailed experimental setup along with a code repository (upon publication) that can be used to easily reproduce our experiments.

Continuous diffusion has been the foundation of high-fidelity, controllable, and few-step generation of many data modalities such as images. However, in language modeling, prior continuous diffusion language models (DLMs) lag behind discrete counterparts due to the sparse data space and the underexplored design space. In this work, we close this gap with LangFlow, the first continuous DLM to rival discrete diffusion, by connecting embedding-space DLMs to Flow Matching via Bregman divergence, alongside three key innovations: (1) we derive a novel ODE-based NLL bound for principled evaluation of continuous flow-based language models; (2) we propose an information-uniform principle for setting the noise schedule, which motivates a learnable noise scheduler based on a Gumbel distribution; and (3) we revise prior training protocols by incorporating self-conditioning, as we find it improves both likelihood and sample quality of embedding-space DLMs with effects substantially different from discrete diffusion. Putting everything together, LangFlow rivals top discrete DLMs on both the perplexity (PPL) and the generative perplexity (Gen. PPL), reaching a PPL of 30.0 on LM1B and 24.6 on OpenWebText. It even exceeds autoregressive baselines in zero-shot transfer on 4 out of 7 benchmarks. LangFlow provides the first clear evidence that continuous diffusion is a promising paradigm for language modeling. Homepage: https://github.com/nealchen2003/LangFlow

Robust reinforcement learning is essential for deploying reinforcement learning algorithms in real-world scenarios where environmental uncertainty predominates. Traditional robust reinforcement learning often depends on rectangularity assumptions, where adverse probability measures of outcome states are assumed to be independent across different states and actions. This assumption, rarely fulfilled in practice, leads to overly conservative policies. To address this problem, we introduce a new time-constrained robust MDP (TC-RMDP) formulation that considers multifactorial, correlated, and time-dependent disturbances, thus more accurately reflecting real-world dynamics. This formulation goes beyond the conventional rectangularity paradigm, offering new perspectives and expanding the analytical framework for robust RL. We propose three distinct algorithms, each using varying levels of environmental information, and evaluate them extensively on continuous control benchmarks. Our results demonstrate that these algorithms yield an efficient tradeoff between performance and robustness, outperforming traditional deep robust RL methods in time-constrained environments while preserving robustness in classical benchmarks. This study revisits the prevailing assumptions in robust RL and opens new avenues for developing more practical and realistic RL applications.

Vectorized high-definition map online construction has garnered considerable attention in the field of autonomous driving research. Most existing approaches model changeable map elements using a fixed number of points, or predict local maps in a two-stage autoregressive manner, which may miss essential details and lead to error accumulation. Towards precise map element learning, we propose a simple yet effective architecture named PivotNet, which adopts unified pivot-based map representations and is formulated as a direct set prediction paradigm. Concretely, we first propose a novel point-to-line mask module to encode both the subordinate and geometrical point-line priors in the network. Then, a well-designed pivot dynamic matching module is proposed to model the topology in dynamic point sequences by introducing the concept of sequence matching. Furthermore, to supervise the position and topology of the vectorized point predictions, we propose a dynamic vectorized sequence loss. Extensive experiments and ablations show that PivotNet is remarkably superior to other SOTAs by 5.9 mAP at least. The code will be available soon.

We propose the convergent graph solver (CGS), a deep learning method that learns iterative mappings to predict the properties of a graph system at its stationary state (fixed point) with guaranteed convergence. CGS systematically computes the fixed points of a target graph system and decodes them to estimate the stationary properties of the system without the prior knowledge of existing solvers or intermediate solutions. The forward propagation of CGS proceeds in three steps: (1) constructing the input dependent linear contracting iterative maps, (2) computing the fixed-points of the linear maps, and (3) decoding the fixed-points to estimate the properties. The contractivity of the constructed linear maps guarantees the existence and uniqueness of the fixed points following the Banach fixed point theorem. To train CGS efficiently, we also derive a tractable analytical expression for its gradient by leveraging the implicit function theorem. We evaluate the performance of CGS by applying it to various network-analytic and graph benchmark problems. The results indicate that CGS has competitive capabilities for predicting the stationary properties of graph systems, irrespective of whether the target systems are linear or non-linear. CGS also shows high performance for graph classification problems where the existence or the meaning of a fixed point is hard to be clearly defined, which highlights the potential of CGS as a general graph neural network architecture.

Geo-temporal understanding, the ability to infer location, time, and contextual properties from visual input alone, underpins applications such as disaster management, traffic planning, embodied navigation, world modeling, and geography education. Although recent vision-language models (VLMs) have advanced image geo-localization using cues like landmarks and road signs, their ability to reason about temporal signals and physically grounded spatial cues remains limited. To address this gap, we introduce TimeSpot, a benchmark for evaluating real-world geo-temporal reasoning in VLMs. TimeSpot comprises 1,455 ground-level images from 80 countries and requires structured prediction of temporal attributes (season, month, time of day, daylight phase) and geographic attributes (continent, country, climate zone, environment type, latitude-longitude) directly from visual evidence. It also includes spatial-temporal reasoning tasks that test physical plausibility under real-world uncertainty. Evaluations of state-of-the-art open- and closed-source VLMs show low performance, particularly for temporal inference. While supervised fine-tuning yields improvements, results remain insufficient, highlighting the need for new methods to achieve robust, physically grounded geo-temporal understanding. TimeSpot is available at: https://TimeSpot-GT.github.io.

In lifelong learning, an agent learns throughout its entire life without resets, in a constantly changing environment, as we humans do. Consequently, lifelong learning comes with a plethora of research problems such as continual domain shifts, which result in non-stationary rewards and environment dynamics. These non-stationarities are difficult to detect and cope with due to their continuous nature. Therefore, exploration strategies and learning methods are required that are capable of tracking the steady domain shifts, and adapting to them. We propose Reactive Exploration to track and react to continual domain shifts in lifelong reinforcement learning, and to update the policy correspondingly. To this end, we conduct experiments in order to investigate different exploration strategies. We empirically show that representatives of the policy-gradient family are better suited for lifelong learning, as they adapt more quickly to distribution shifts than Q-learning. Thereby, policy-gradient methods profit the most from Reactive Exploration and show good results in lifelong learning with continual domain shifts. Our code is available at: https://github.com/ml-jku/reactive-exploration.

Reliably reconstructing physical fields from sparse sensor data is a challenge that frequently arises in many scientific domains. In practice, the process generating the data often is not understood to sufficient accuracy. Therefore, there is a growing interest in using the deep neural network route to address the problem. This work presents a novel approach that learns a continuous representation of the physical field using implicit neural representations (INRs). Specifically, after factorizing spatiotemporal variability into spatial and temporal components using the separation of variables technique, the method learns relevant basis functions from sparsely sampled irregular data points to develop a continuous representation of the data. In experimental evaluations, the proposed model outperforms recent INR methods, offering superior reconstruction quality on simulation data from a state-of-the-art climate model and a second dataset that comprises ultra-high resolution satellite-based sea surface temperature fields.

State-space models (SSMs) offer a powerful framework for dynamical system analysis, wherein the temporal dynamics of the system are assumed to be captured through the evolution of the latent states, which govern the values of the observations. This paper provides a selective review of recent advancements in deep neural network-based approaches for SSMs, and presents a unified perspective for discrete time deep state space models and continuous time ones such as latent neural Ordinary Differential and Stochastic Differential Equations. It starts with an overview of the classical maximum likelihood based approach for learning SSMs, reviews variational autoencoder as a general learning pipeline for neural network-based approaches in the presence of latent variables, and discusses in detail representative deep learning models that fall under the SSM framework. Very recent developments, where SSMs are used as standalone architectural modules for improving efficiency in sequence modeling, are also examined. Finally, examples involving mixed frequency and irregularly-spaced time series data are presented to demonstrate the advantage of SSMs in these settings.

Continuous normalizing flows (CNFs) construct invertible mappings between an arbitrary complex distribution and an isotropic Gaussian distribution using Neural Ordinary Differential Equations (neural ODEs). It has not been tractable on large datasets due to the incremental complexity of the neural ODE training. Optimal Transport theory has been applied to regularize the dynamics of the ODE to speed up training in recent works. In this paper, a temporal optimization is proposed by optimizing the evolutionary time for forward propagation of the neural ODE training. In this appoach, we optimize the network weights of the CNF alternately with evolutionary time by coordinate descent. Further with temporal regularization, stability of the evolution is ensured. This approach can be used in conjunction with the original regularization approach. We have experimentally demonstrated that the proposed approach can significantly accelerate training without sacrifying performance over baseline models.

Continuous-time Consistency Models (CMs) promise efficient few-step generation but face significant challenges with training instability. We argue this instability stems from a fundamental conflict: by training a network to learn only a shortcut across a probability flow, the model loses its grasp on the instantaneous velocity field that defines the flow. Our solution is to explicitly anchor the model in the underlying flow during training. We introduce the Flow-Anchored Consistency Model (FACM), a simple but effective training strategy that uses a Flow Matching (FM) task as an anchor for the primary CM shortcut objective. This Flow-Anchoring approach requires no architectural modifications and is broadly compatible with standard model architectures. By distilling a pre-trained LightningDiT model, our method achieves a state-of-the-art FID of 1.32 with two steps (NFE=2) and 1.76 with just one step (NFE=1) on ImageNet 256x256, significantly outperforming previous methods. This provides a general and effective recipe for building high-performance, few-step generative models. Our code and pretrained models: https://github.com/ali-vilab/FACM.

Transformer architectures have dramatically advanced representation learning and inference in deep models through self-attention mechanisms. In parallel,associative memory (AM) frameworks map representations onto energy landscapes, offering interpretable retrieval mechanisms. However, their continuous-time inference dynamics lack the biological plausibility of classical Continuous Attractor Neural Networks (CANNs). To bridge this gap, we propose Controlled Dynamics Attractor Transformer (CDAT), which couples a mixture von Mises-Fisher (Mo-vMF) attention energy with a Hopfield refinement energy, while augmenting energy descent with a CANN-inspired excitation-inhibition modulation. CDAT instantiates a topology-constrained dynamical system whose couplings encode relational structure among tokens, thereby linking attractor-style dynamics to modern energy-based attention. We further provide a constructive dissipation analysis to formally establish their controlled inference dynamics. Benefiting from these robust and structured dynamics, CDAT achieves state-of-the-art performance across multiple benchmarks in graph anomaly detection and graph classification.

Time series prototype learning is fundamentally challenged by observational ambiguity. Discrete architectures fail to resolve this, as they lack the capacity to decouple stochastic noise from continuous dynamics. Furthermore, rigid closed-set assumptions fail to capture unseen diversity. To address these limitations, we propose a hierarchical ordinary differential equation clustering network, which utilizes neural ordinary differential equation to model latent state evolution as a continuous integral curve. This formulation enforces temporal continuity to effectively disentangle smooth feature trends from stochastic noise, while our adaptive hierarchical mechanism autonomously determines the appropriate number of prototypes without rigid prior constraints. Validated on the early link failure detection task with irregularly sampled time series, the proposed method effectively extracts underlying physical prototypes, thereby enabling robust failure detection. Our code is available at https://github.com/NJ-LNN/Hierarchical-ODE.

Multivariate time-series data often exhibit complex temporal dependencies, irregular sampling, and heterogeneous dynamics across multiple time scales, making accurate sequence modeling particularly challenging. Traditional recurrent neural networks (RNNs), such as Long Short-Term Memory (LSTM) networks, operate in discrete time and may struggle to effectively capture continuous and irregular temporal behaviors. Liquid Neural Networks (LNNs) address some of these limitations through continuous-time dynamics, but standard LNN architectures typically rely on a single dynamical system, limiting their ability to model heterogeneous temporal patterns. To address these challenges, we propose a Multi-Rate Mixture-of-Experts (MR-MoE) framework built on top of Liquid Neural Networks. In the proposed architecture, multiple LNN-based experts operate at distinct time scales, enabling the model to explicitly separate fast-changing dynamics from slow-evolving temporal trends. A gating network further enables adaptive expert specialization based on input conditions. In addition, we incorporate both feature-level and temporal attention mechanisms to improve robustness, interpretability, and long-range dependency modeling. Feature-level attention suppresses noisy or irrelevant variables, while temporal attention selectively focuses on informative historical states. We evaluate the proposed framework on a complex multivariate time-series prediction task and compare it against strong baselines, including LSTM, monolithic LNN, and standard MoE models. Experimental results demonstrate that the proposed MR-MoE framework consistently achieves improved AUROC and AUPRC performance while maintaining favorable computational efficiency. These results highlight the effectiveness of combining continuous-time dynamics, multi-scale expert decomposition, and adaptive attention mechanisms for time-series modeling.

Long-term forecasting of chaotic systems from short-term observations remains a fundamental and underexplored challenge due to the intrinsic sensitivity to initial conditions and the complex geometry of strange attractors. Existing approaches often rely on long-term training data or focus on short-term sequence correlations, struggling to maintain predictive stability and dynamical coherence over extended horizons. We propose PhyxMamba, a novel framework that integrates a Mamba-based state-space model with physics-informed principles to capture the underlying dynamics of chaotic systems. By reconstructing the attractor manifold from brief observations using time-delay embeddings, PhyxMamba extracts global dynamical features essential for accurate forecasting. Our generative training scheme enables Mamba to replicate the physical process, augmented by multi-token prediction and attractor geometry regularization for physical constraints, enhancing prediction accuracy and preserving key statistical invariants. Extensive evaluations on diverse simulated and real-world chaotic systems demonstrate that PhyxMamba delivers superior long-term forecasting and faithfully captures essential dynamical invariants from short-term data. This framework opens new avenues for reliably predicting chaotic systems under observation-scarce conditions, with broad implications across climate science, neuroscience, epidemiology, and beyond. Our code is open-source at https://github.com/tsinghua-fib-lab/PhyxMamba.

Neural Temporal Point Processes (TPPs) are the prevalent paradigm for modeling continuous-time event sequences, such as user activities on the web and financial transactions. In real-world applications, event data is typically received in a streaming manner, where the distribution of patterns may shift over time. Additionally, privacy and memory constraints are commonly observed in practical scenarios, further compounding the challenges. Therefore, the continuous monitoring of a TPP to learn the streaming event sequence is an important yet under-explored problem. Our work paper addresses this challenge by adopting Continual Learning (CL), which makes the model capable of continuously learning a sequence of tasks without catastrophic forgetting under realistic constraints. Correspondingly, we propose a simple yet effective framework, PromptTPPOur code is available at {\small \url{ https://github.com/yanyanSann/PromptTPP}}, by integrating the base TPP with a continuous-time retrieval prompt pool. The prompts, small learnable parameters, are stored in a memory space and jointly optimized with the base TPP, ensuring that the model learns event streams sequentially without buffering past examples or task-specific attributes. We present a novel and realistic experimental setup for modeling event streams, where PromptTPP consistently achieves state-of-the-art performance across three real user behavior datasets.

Novel view synthesis aims to render unseen views given a set of calibrated images. In practical applications, the coverage, appearance or geometry of the scene may change over time, with new images continuously being captured. Efficiently incorporating such continuous change is an open challenge. Standard NeRF benchmarks only involve scene coverage expansion. To study other practical scene changes, we propose a new dataset, World Across Time (WAT), consisting of scenes that change in appearance and geometry over time. We also propose a simple yet effective method, CLNeRF, which introduces continual learning (CL) to Neural Radiance Fields (NeRFs). CLNeRF combines generative replay and the Instant Neural Graphics Primitives (NGP) architecture to effectively prevent catastrophic forgetting and efficiently update the model when new data arrives. We also add trainable appearance and geometry embeddings to NGP, allowing a single compact model to handle complex scene changes. Without the need to store historical images, CLNeRF trained sequentially over multiple scans of a changing scene performs on-par with the upper bound model trained on all scans at once. Compared to other CL baselines CLNeRF performs much better across standard benchmarks and WAT. The source code, and the WAT dataset are available at https://github.com/IntelLabs/CLNeRF. Video presentation is available at: https://youtu.be/nLRt6OoDGq0?si=8yD6k-8MMBJInQPs

With the emergence of powerful representations of continuous data in the form of neural fields, there is a need for discretization invariant learning: an approach for learning maps between functions on continuous domains without being sensitive to how the function is sampled. We present a new framework for understanding and designing discretization invariant neural networks (DI-Nets), which generalizes many discrete networks such as convolutional neural networks as well as continuous networks such as neural operators. Our analysis establishes upper bounds on the deviation in model outputs under different finite discretizations, and highlights the central role of point set discrepancy in characterizing such bounds. This insight leads to the design of a family of neural networks driven by numerical integration via quasi-Monte Carlo sampling with discretizations of low discrepancy. We prove by construction that DI-Nets universally approximate a large class of maps between integrable function spaces, and show that discretization invariance also describes backpropagation through such models. Applied to neural fields, convolutional DI-Nets can learn to classify and segment visual data under various discretizations, and sometimes generalize to new types of discretizations at test time. Code: https://github.com/clintonjwang/DI-net.