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SPDEBench: An Extensive Benchmark for Learning Stochastic PDEs

NeurIPS 2026 · Probabilistic and scientific modeling › Differential equations and PDEs · approach: Physics-informed operator learning · 0 citations (OpenAlex)

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Introduces SPDEBench, a unified benchmark with controlled, renormalized data generation for regular and singular SPDEs (Phi^4_d, KPZ, KdV, wave, NSE), plus baseline neural operators, 7 metrics, and systematic robustness/OOD evaluation.

Problem

ML surrogate models for stochastic PDEs lack a unified benchmark; singular SPDE datasets require renormalization and data-generation choices (noise truncation, basis, renormalization) strongly affect results and evaluation.

Approach

A benchmark that treats SPDE data-generation parameters (noise truncation degree, basis, renormalization constants) as controlled variables, includes renormalized solvers for singular SPDEs, and evaluates models with Sobolev and distributional (signature-based) metrics beyond relative L2.

Key results

• On Phi^4_1 (Dirichlet, J=32) DLR-Net reaches relative L2 about 0.002 vs 0.022 for FNO and 0.125 for DeepONet.
• NSPDE-S achieves lower relative L2 than NSPDE under distribution shift on Phi^4_2 (e.g., 0.205 vs 0.245 at J=128 with in-distribution training).
• ML models give about 22x speedup on Phi^4_1 and over 270x on Phi^4_2 relative to the numerical solver.

Surprising findings

• Increasing training set size from 1,000 to 10,000 yields only marginal gains, suggesting model design is the bottleneck rather than data availability.
• FNO consistently outperforms WNO even when data are generated in the wavelet basis.

Limitations

• Benchmark currently uses synthetic datasets, limiting direct applicability to real-world systems.
• High-dimensional SPDEs (d>3) and large transformer-based models are not fully explored due to compute constraints.

Experimental setup

Datasets: Dynamic Phi^4_1 Dynamic Phi^4_2 Dynamic Phi^4_3 KdV Wave NSE KPZ

Benchmarks: SPDEBench Phi^4_1 SPDEBench Phi^4_2 KdV NSE

Metrics: relative L2 absolute Lp Sobolev W^{1,2} norm RMSE ACF cross-correlation Sig-W1

Baselines: NCDE NRDE NCDE-FNO DeepONet FNO WNO GT NORS DLR-Net NSPDE

Builds on: Neural SPDE DLR-Net Fourier Neural Operator Galerkin Transformer

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