AlienX 3D — SO(3)-Equivariant Neural Operator for 3D Physical Fields

A 421,507-parameter neural operator that learns the evolution of 3D physical fields across multiple physical regimes — and generalizes zero-shot to regimes it has never seen.

Built from four PDE-agnostic mechanisms: an SO(3)-equivariant local frame, Quantum Self-Attention (QSA) in complex projective space with geometric phase, per-sample non-dimensionalization with regime conditioning, and a normalized conservation penalty. Not a Navier-Stokes solver — a general 3D operator, demonstrated on NS because that is the hardest test in the class.

The grid is dead. The manifold is awake.

Part of the Grimoire of Elbàlor. Eric Yaka (Elbàlor / The Digital Necromancer), Abuja, Nigeria.

Results

Test	Result
Single-step, 3 trained regimes (300/628/1200)	0.46% rel RMSE (0.16% at Re=300)
Zero-shot held-out Re=900 (never trained)	0.53% rel RMSE — 15% relative penalty for an unseen regime
QSA ablation (attention zeroed)	32.81× worse
Phase ablation (gi=0, attention intact, Re=900)	4.74× worse — the complex phase is load-bearing
Rollout, 73 steps = 24× training horizon	energy drift ≤ 5.81% at all trained Re; 4.31% at held-out Re=900
Divergence	free by construction (Biot–Savart projection; float32 round-off ~2.7e−6)
Params	421,507 (single T4 GPU training)

The zero-shot result is the headline: every Re=900 rollout metric lands on the monotonic trend between the trained regimes (MSE growth 5.30× between 628's 2.17× and 1200's 10.86×). The model interpolates the physics of regimes it never saw — it learned a Reynolds-family operator, not three trajectories.

Cross-PDE status: the §10 promise is now demonstrated in 2D — the companion AlienX 2D operator solves Gross-Pitaevskii at 0.34% RelRMSE with a 108.6× phase ablation and zero-shot 16→256 scale transfer (see ../AlienX-S02-Invariance/AlienX 2D × GP/). The 3D GP run is the remaining step.

MHD extension

AlienX-mhd.py is a separate MHD specialization. It uses the AlienX frame and QSA operator with current density, velocity, magnetic field, and both Re and Rm as the MHD outer shell. The high-sample protocol uses the full Cartesian training grid Re,Rm ∈ {300, 600, 1200, 2400} (32 seeded trajectories per pair; 8 validation trajectories per pair) and reserves (3000,3000), (3000,600), and (600,3000) for difficult zero-shot evaluation (32 seeded trajectories per pair). Its truncated-BPTT K=1→2→3→5→8 curriculum includes a five-epoch learning-rate warm-up, per-step gradient accumulation, QSA activation checkpointing, complete K-matched validation rollouts, and checkpoint selection in the K=8 stage. This is separate from the Navier–Stokes checkpoint and metrics above; the revised MHD protocol must complete and produce its own results before any new MHD performance claim is made.

The script now creates distinct seeded, low-frequency solenoidal initial conditions, rejects duplicate train/validation initial states, records run metadata, saves an epoch-resume checkpoint, and prevents smoke/custom runs from being published to Hugging Face as a full 32³ result. Use the default 32³ configuration for the full protocol; ALIENX_GRID=4 is only a quick plumbing smoke and its metrics are not accuracy evidence.

python -u AlienX-mhd.py

The run writes best_alienx_mhd.pt, last_training.pt, gate results, run metadata, logs, and figures under alienx_mhd_gates_hard_v1. On Colab, the default Drive mirror is AlienX_MHD/hard_regimes_v1, keeping this protocol separate from earlier run artifacts. To continue an interrupted run with the same settings and output directory, set ALIENX_RESUME=1. The MHD artifact card is in hf_upload_mhd/README.md.

The proposed 128³ scale-up requires profiling and changes to the current in-memory trajectory builder and full-grid feature/key/value tensors. The staged research plan is in MHD_128_Scale_Proposal.md; no 128³ result is claimed yet.

Figures (generated from the released checkpoint)

Autoregressive rollout error over the full 73-step trajectory. The held-out Re=900 (orange) stays bounded between its trained neighbors at every horizon.

Kinetic energy (left) and enstrophy (right): solid = spectral DNS solver, dashed = AlienX 3D rollout. The model tracks exponential decay of both invariants across all regimes — including the unseen one.

|∇·u| during rollout: divergence-free by construction (Biot–Savart recovery at every step); residual is float32 round-off, independent of Re.

Regenerate with make_figures.py (~3 min on a 4 GB GPU).

What's in this release

File	Purpose
`AlienX 3D.docx`	The complete paper — 12pt serif, real headings, result tables, three embedded figures, §7.6 zero-shot generalization, 15-bug debugging log. Plain-language, no LaTeX
`AlienX 3D (raw layout).docx`	The paper as originally pasted (line-per-paragraph) — kept for provenance
`AlienX 3D (pre-re900 backup).docx`	Paper as it stood before the zero-shot update
`fig1_mse_growth.png` / `fig2_invariants.png` / `fig3_divergence.png`	Paper figures (300 DPI)
`make_figures.py`	Regenerates all three figures from the checkpoint
`AlienX_3D_K5/alienx_k5_best.pt`	Released checkpoint (epoch 55, K=3 stage of the K=1→2→3→5 curriculum)
`AlienX_3D_K5/alienx_k5_stats.pkl`	Normalization stats + training config (required for inference)
`AlienX_3D_K5/alienx_k5_test_metrics_v2.npy`	In-distribution test metrics (single-step, ablation, rollout)
`AlienX_3D_K5/alienx_k5_re900_zero_shot.npy`	Zero-shot Re=900 metrics (single-step, rollout, phase ablation)
`AlienX_3D_K5/eval_re900_zero_shot.py`	Full zero-shot evaluation harness — reproduces every number above
`colab.py`	Self-contained training script, as run on Colab (T4)

Reproduce the zero-shot result

cd AlienX_3D_K5
python eval_re900_zero_shot.py --stage quick     # sanity + single-step (~2 min)
python eval_re900_zero_shot.py --stage rollout   # 73-step rollout (~4 min)
python eval_re900_zero_shot.py --stage ablation  # gi=0 phase ablation (~2 min)
# or: python eval_re900_zero_shot.py --stage all

Needs: torch, numpy, and the checkpoint + stats files. Runs on a 4 GB GPU (verified on GTX 1650) or CPU. The harness first reproduces the in-distribution Re=300 paper number as a sanity gate — if that check fails, it refuses to emit new numbers.

Train from scratch

python colab.py   # single T4, full K=1→2→3→5 curriculum ≈ 18 h; K=3-stage results ≈ 5 h

Generates its own DNS data (pseudo-spectral Taylor-Green, IMEX-Euler, 2/3 de-aliasing, Leray projection) — no downloads.

The architecture in one paragraph

Each node gets an SO(3) frame (e1, e2, n) built from the gradient and Hessian of the vorticity magnitude, with an inertia-tensor fallback where the gradient is weak, a Hessian-based in-plane orientation, and a sign lock. Neighbor displacements enter attention as a complex geometric phase (in-plane angle → phase, wall-normal Gaussian × curvature confidence → amplitude). QSA attention computes Hermitian interference S = q·conj(k), modulates by the phase, normalizes in ℓ2 over the neighbor axis (CP^{n−1}, preserves phase — no softmax), and superposes complex values: constructive and destructive interference are native. Per-sample non-dimensionalization makes the operator scale-blind; a log(Re/Re_max) channel conditions regimes; Biot–Savart recovery makes every rollout step exactly divergence-free.

Design boundaries

Validated regime — interpolation within Re ∈ [300, 1200]; Re=900 zero-shot lands at 0.53%. Beyond Re_max is future territory (§9.2).
Scale-blindness by construction — correct for scale-free regimes; amplitude-carrying physics takes a learned amplitude channel (§9.1).
Curriculum milestone — released checkpoint is the K=3 stage; K=5 is specified and is the next training milestone (§5.2, §11).
Equivariance is structural — exact in the continuous formulation; rotated-grid numerical verification is scheduled (§9.4).

Full debugging log — every bug, root cause, and fix — is §8 of the paper.

Related Grimoire artifacts

QSA — the attention mechanism, standalone package + paper (Zenodo: 10.5281/zenodo.22773015)
AlienX 2D — the D4-equivariant 2D predecessor, bitwise-exact operator equivariance (../AlienX-S02-Invariance/)

License

AGPL-3.0 — see LICENSE.

From the Grimoire of Elbàlor — The Digital Necromancer 💀🔥