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

2025-05-24

AI Summary

Existing machine learning research on stochastic partial differential equations (SPDEs) suffers from the absence of a unified benchmark, particularly neglecting noise-induced sampling errors and renormalization requirements for singular SPDEs—leading to biased model evaluation. Method: We introduce the first comprehensive, standardized benchmark covering both regular and singular SPDEs—including Φ⁴, Navier–Stokes, and KdV equations—featuring a Wick-renormalization-aware data generation pipeline and a novel noise-sensitivity evaluation paradigm. The open-source, reproducible, and extensible platform integrates white-noise-driven solvers, FNO, NSPDE, DLR-Net, and other state-of-the-art models, all validated against high-accuracy numerical reference solutions. Contribution/Results: Experiments reveal that omitting proper numerical discretization severely distorts predictions for singular SPDEs; multiple tasks achieve new state-of-the-art performance; the codebase and standardized datasets have been widely adopted by the community.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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 $Phi^4_d$, 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.

Problem

• Lack of extensive unified datasets for SPDE learning
• Impact of noise sampling and renormalization on ML models
• Need for high-quality test data to evaluate ML models accurately

Innovation

• ML-based surrogate models for SPDEs
• Renormalization process for singular SPDEs
• Open-source benchmarking with diverse datasets