Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

SPDEBench: An Extensive Benchmark for Learning Regular and Singular Stochastic PDEs

Zheyan Li, Yuantu Zhu, Hao Ni, Siran Li, Bingguang Chen, Qi Meng

TL;DR

SPDEs driven by space-time white noise model rough spatio-temporal dynamics and pose challenges for ML surrogates. SPDEBench provides an extensible, open-source benchmark with both nonsingular and singular SPDE datasets (notably the dynamical $\Phi^4_2$ model) and renormalization-aware data generation, enabling rigorous evaluation of neural operators and surrogate models such as FNO, NSPDE, and DLR-Net. A key contribution is showing how renormalization and noise-generation choices affect learning: models like NSPDE-S that ingest renormalization information can outperform standard pipelines, while high singularity requires robust architectures. The benchmark promotes reproducible, fair comparisons and will guide the development of ML methods capable of capturing rough SPDE dynamics at scale.

Abstract

Stochastic Partial Differential Equations (SPDEs) driven by random noise play a central role in modelling physical processes whose spatio-temporal dynamics can be rough, such as turbulence flows, superconductors, and quantum dynamics. To efficiently model these processes and make predictions, machine learning (ML)-based surrogate models are proposed, with their network architectures incorporating the spatio-temporal roughness in their design. However, it lacks an extensive and unified datasets for SPDE learning; especially, existing datasets do not account for the computational error introduced by noise sampling and the necessary renormalization required for handling singular SPDEs. We thus introduce SPDEBench, which is designed to solve typical SPDEs of physical significance (e.g., the $Φ^4_d$, wave, incompressible Navier--Stokes, and KdV equations) on 1D or 2D tori driven by white noise via ML methods. New datasets for singular SPDEs based on the renormalization process have been constructed, and novel ML models achieving the best results to date have been proposed. In particular, we investigate the impact of computational error introduced by noise sampling and renormalization on the performance comparison of ML models and highlight the importance of selecting high-quality test data for accurate evaluation. Results are benchmarked with traditional numerical solvers and ML-based models, including FNO, NSPDE and DLR-Net, etc. It is shown that, for singular SPDEs, naively applying ML models on data without specifying the numerical schemes can lead to significant errors and misleading conclusions. Our SPDEBench provides an open-source codebase that ensures full reproducibility of benchmarking across a variety of SPDE datasets while offering the flexibility to incorporate new datasets and machine learning baselines, making it a valuable resource for the community.

SPDEBench: An Extensive Benchmark for Learning Regular and Singular Stochastic PDEs

TL;DR

SPDEs driven by space-time white noise model rough spatio-temporal dynamics and pose challenges for ML surrogates. SPDEBench provides an extensible, open-source benchmark with both nonsingular and singular SPDE datasets (notably the dynamical model) and renormalization-aware data generation, enabling rigorous evaluation of neural operators and surrogate models such as FNO, NSPDE, and DLR-Net. A key contribution is showing how renormalization and noise-generation choices affect learning: models like NSPDE-S that ingest renormalization information can outperform standard pipelines, while high singularity requires robust architectures. The benchmark promotes reproducible, fair comparisons and will guide the development of ML methods capable of capturing rough SPDE dynamics at scale.

Abstract

Stochastic Partial Differential Equations (SPDEs) driven by random noise play a central role in modelling physical processes whose spatio-temporal dynamics can be rough, such as turbulence flows, superconductors, and quantum dynamics. To efficiently model these processes and make predictions, machine learning (ML)-based surrogate models are proposed, with their network architectures incorporating the spatio-temporal roughness in their design. However, it lacks an extensive and unified datasets for SPDE learning; especially, existing datasets do not account for the computational error introduced by noise sampling and the necessary renormalization required for handling singular SPDEs. We thus introduce SPDEBench, which is designed to solve typical SPDEs of physical significance (e.g., the , wave, incompressible Navier--Stokes, and KdV equations) on 1D or 2D tori driven by white noise via ML methods. New datasets for singular SPDEs based on the renormalization process have been constructed, and novel ML models achieving the best results to date have been proposed. In particular, we investigate the impact of computational error introduced by noise sampling and renormalization on the performance comparison of ML models and highlight the importance of selecting high-quality test data for accurate evaluation. Results are benchmarked with traditional numerical solvers and ML-based models, including FNO, NSPDE and DLR-Net, etc. It is shown that, for singular SPDEs, naively applying ML models on data without specifying the numerical schemes can lead to significant errors and misleading conclusions. Our SPDEBench provides an open-source codebase that ensures full reproducibility of benchmarking across a variety of SPDE datasets while offering the flexibility to incorporate new datasets and machine learning baselines, making it a valuable resource for the community.

Paper Structure

This paper contains 25 sections, 17 equations, 3 figures, 14 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mild solution of SPDE
  4. Cylindrical Wiener process
  5. Formulation of the learning problem
  6. SPDEBench: A Benchmark for Learning SPDEs
  7. Overview of Datasets
  8. Nonsingular SPDE datasets
  9. A novel singular SPDE dataset
  10. Baseline ML Surrogate Models
  11. Data Format, Benchmark Access, Maintenance, and Extensibility
  12. Experiments
  13. The effect of renormalization on accuracy
  14. The effect of noise simulation on accuracy
  15. The effect of noise truncation degree
  16. ...and 10 more sections

Figures (3)

  • Figure 1: Comparison of two data generation methods of $\Phi^4_2$ when truncation degree $J=128$ at time step $t=249$. The left panel is mean of 1200 samples generated through the renormalization procedure, while the right panel is mean of 1200 samples generated through an explicit scheme without implementing the renormalization.
  • Figure 2: Comparison of FNO and NSPDE trained on KdV datasets with noise generated by cylindrical Wiener process ($\sigma=0.5$) and Q-Wiener process ($\sigma=1$).
  • Figure 3: Illustration of the time evolution of the NSE($J=$256) data and NSPDE's predictions. The model is trained on a $16\times 16$ mesh and evaluated on a $64\times 64$ mesh for visualization. Top panel: Ground truth generated by the numerical solver. Bottom panel: Predictions from the model with input $(u_0,\xi)$.

Theorems & Definitions (2)

  • Definition 2.1
  • Definition A.1: $Q$-Wiener process