README.md

metadata

license: mit
task_categories:
  - tabular-regression
  - time-series-forecasting
tags:
  - physics-informed-neural-networks
  - pinn
  - partial-differential-equations
  - quantitative-finance
  - scientific-machine-learning
  - sobolev-space
pretty_name: High-Dimensional PDE Collocation & Sobolev Benchmarks
size_categories:
  - 1K<n<10K

📊 High-Dimensional PDE Collocation & Sobolev Benchmarks

An institutional scientific machine learning (SciML) benchmark dataset curated by Ramprasad Atluri for training and benchmarking Physics-Informed Neural Networks (PINNs) on parabolic and elliptic partial differential equations.

📌 Mathematical Operator Formulation

Physics-Informed Neural Networks approximate the solution $u_\theta(S, t)$ by directly penalizing the residual of the differential operator $\mathcal{L}[u]$ in Sobolev space $\mathcal{H}^2(\Omega)$:

$$\mathcal{L}[u]=-\frac{\mathrm{\partial }u}{\mathrm{\partial }\tau }+\frac{1}{2}{\sigma }^2{S}^2\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{S}^2}+rS\frac{\mathrm{\partial }u}{\mathrm{\partial }S}-ru=0\backslash mathcal\{L\}[u]\; =\; -\backslash frac\{\backslash partial\; u\}\{\backslash partial\; \backslash tau\}\; +\; \backslash frac\{1\}\{2\}\; \backslash sigma^2\; S^2\; \backslash frac\{\backslash partial^2\; u\}\{\backslash partial\; S^2\}\; +\; r\; S\; \backslash frac\{\backslash partial\; u\}\{\backslash partial\; S\}\; -\; r\; u\; =\; 0$$

Subject to the terminal payoff condition on the boundary $\partial \Omega$:

$$u(S,0)=\max (S-K,0)u(S,\; 0)\; =\; \backslash max(S\; -\; K,\; 0)$$

And Dirichlet / Neumann asymptotic boundary conditions:

$$\underset{S\to 0}{\lim }u(S,\tau )=0,\underset{S\to \mathrm{\infty }}{\lim }u(S,\tau )=S-K{e}^{-r\tau }\backslash lim\_\{S\; \backslash to\; 0\}\; u(S,\; \backslash tau)\; =\; 0,\; \backslash quad\; \backslash lim\_\{S\; \backslash to\; \backslash infty\}\; u(S,\; \backslash tau)\; =\; S\; -\; K\; e^\{-r\; \backslash tau\}$$

🔬 Collocation Mesh Specifications

This dataset contains 5,000 Latin-hypercube spatio-temporal collocation nodes $(S_i, \tau_i) \in [20.0, 250.0] \times [0.01, 1.5]$ equipped with analytical ground truths:

1. Exact Functional Solution: $$u(S, \tau) = S \Phi(d_1) - K e^{-r \tau} \Phi(d_2)$$
2. First Spatial Derivative ($\Delta = \nabla_S u$): $$\Delta = \frac{\partial u}{\partial S} = \Phi(d_1)$$
3. Second Spatial Derivative ($\Gamma = \nabla_S^2 u$): $$\Gamma = \frac{\partial^2 u}{\partial S^2} = \frac{\phi(d_1)}{S \sigma \sqrt{\tau}}$$
4. Temporal Derivative ($\Theta = \partial u / \partial \tau$): $$\Theta = \frac{\partial u}{\partial \tau} = \frac{S \phi(d_1) \sigma}{2 \sqrt{\tau}} + r K e^{-r \tau} \Phi(d_2)$$
5. Differential Residual ($\mathcal{R}_{\text{PDE}}$): $$\mathcal{R}(S_i, \tau_i) = -\Theta_i + \frac{1}{2} \sigma^2 S_i^2 \Gamma_i + r S_i \Delta_i - r u_i \equiv 0$$

🚀 Loading in Python

import pandas as pd

url = "https://huggingface.co/datasets/At1uri/high-dimensional-pde-collocation-benchmarks/raw/main/pde_collocation_mesh_benchmarks.csv"
df = pd.read_csv(url)

print(df.head())
print("Maximum Analytical Residual Error:", df["pde_operator_residual"].abs().max())

🛠️ Schema Definition

Feature	Type	Description
point_id	Integer	Unique collocation node index
spot_s	Float	Spatial coordinate $S \in \mathbb{R}^+$
time_to_maturity_tau	Float	Temporal coordinate $\tau = T - t$
exact_solution_u	Float	Analytical valuation $u(S, \tau)$
first_spatial_derivative_delta	Float	First derivative $\Delta = \partial u / \partial S$
second_spatial_derivative_gamma	Float	Second derivative $\Gamma = \partial^2 u / \partial S^2$
temporal_derivative_theta	Float	Time decay $\Theta = \partial u / \partial \tau$
volatility_sensitivity_vega	Float	Volatility sensitivity $\nu = \partial u / \partial \sigma$
pde_operator_residual	Float	Verification residual $\mathcal{L}[u](S_i, \tau_i)$

⚖️ License

MIT License. Free for scientific machine learning, quantitative finance, and deep neural PDE research.