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

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

Keywords: Neural Operator; Stochastic PDE; Numerical Scheme; Deep Learning

TL;DR: An extensive benchmark for learning regular and singular SPDEs

Abstract:

Stochastic Partial Differential Equations (SPDEs) driven by random noise play a central role in modeling physical processes with rough spatio-temporal dynamics, 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 spatio-temporal roughness incorporated in the design of their network architectures.
However, an extensive and unified dataset for SPDE learning is still lacking; in particular, 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, aiming to solve typical SPDEs of physical significance on 1D or 2D tori driven by white noise via ML methods. New datasets for singular SPDEs based on the renormalization process, as well as novel ML models achieving the best results to date, have been proposed. {Moreover, we evaluate the sensitivity of ML models to the SPDE data generation setting and the hyperparameters, and investigate the scaling law of ML models with respect to sample and network sizes. Results are benchmarked with ML-based models, including FNO, NSPDE and DLR-Net, etc. By evaluating performance from multiple perspectives, we achieve a comprehensive assessment of the relative strengths and weaknesses of different ML models.} Our SPDEBench ensures full reproducibility of benchmarking across a variety of SPDE datasets while offering the flexibility of incorporating new datasets and machine learning baselines, thus making it a valuable resource for the community.

Primary Area: learning on time series and dynamical systems

Submission Number: 22950