[Paper Review] PDEBENCH: An Extensive Benchmark for Scientific Machine Learning

Makoto Takamoto, Timothy Praditia+5 more|arXiv (Cornell University)|Oct 13, 2022

Model Reduction and Neural Networks56 citations

TL;DR

PDEBench provides a large, extensible benchmark suite for Scientific ML with 11 PDEs, large datasets, baseline models (FNO, U-Net, PINN), and new evaluation metrics to assess physics-aware performance and generalization.

ABSTRACT

Machine learning-based modeling of physical systems has experienced increased interest in recent years. Despite some impressive progress, there is still a lack of benchmarks for Scientific ML that are easy to use but still challenging and representative of a wide range of problems. We introduce PDEBench, a benchmark suite of time-dependent simulation tasks based on Partial Differential Equations (PDEs). PDEBench comprises both code and data to benchmark the performance of novel machine learning models against both classical numerical simulations and machine learning baselines. Our proposed set of benchmark problems contribute the following unique features: (1) A much wider range of PDEs compared to existing benchmarks, ranging from relatively common examples to more realistic and difficult problems; (2) much larger ready-to-use datasets compared to prior work, comprising multiple simulation runs across a larger number of initial and boundary conditions and PDE parameters; (3) more extensible source codes with user-friendly APIs for data generation and baseline results with popular machine learning models (FNO, U-Net, PINN, Gradient-Based Inverse Method). PDEBench allows researchers to extend the benchmark freely for their own purposes using a standardized API and to compare the performance of new models to existing baseline methods. We also propose new evaluation metrics with the aim to provide a more holistic understanding of learning methods in the context of Scientific ML. With those metrics we identify tasks which are challenging for recent ML methods and propose these tasks as future challenges for the community. The code is available at https://github.com/pdebench/PDEBench.

Motivation & Objective

• Provide a comprehensive, easy-to-use benchmark for Scientific ML across a wide range of time-dependent PDEs.
• Offer large, diverse datasets (forward and inverse problems) with ready-to-use data and code to generate more.
• Establish baseline models (FNO, U-Net, PINN) and a standardized API for fair comparisons.
• Introduce physics-aware evaluation metrics beyond RMSE to capture conservation laws and boundary behavior.
• Encourage community extended benchmarks and reproducible experiments with FAIR data principles.

Proposed method

• Define a forward propagator for discretised PDEs and learn a surrogate that approximates the discretised forward operator.
• Provide datasets across 11 PDEs (1–3D) with varying parameters, initial/boundary conditions, and resolutions.
• Implement and evaluate baseline models: Fourier Neural Operator (FNO), U-Net, and Physics-Informed Neural Networks (PINN).
• Propose a suite of metrics including RMSE, normalised RMSE, max error, and physics-centric metrics (conserved quantities, boundary, and Fourier-regime errors).
• Offer data access via DaRUS with DOIs, a uniform data interface, and code to generate new data via a configurable API (Hydra).
• Explore inverse problems by using learned forward surrogates to infer unobserved initial conditions or PDE parameters.

Experimental results

Research questions

• RQ1 What is the breadth of PDEs and problem settings needed to meaningfully benchmark Scientific ML methods?
• RQ2 How do popular ML baselines (FNO, U-Net, PINN) perform across forward and inverse PDE tasks under realistic 1–3D settings?
• RQ3 Do standard ML metrics (e.g., RMSE) adequately reflect physics-consistent performance, especially in regimes with shocks, sparsity, or high-frequency content?
• RQ4 What limitations do current ML surrogates exhibit (e.g., temporal extrapolation, high-frequency errors) and how do these depend on PDE type and parameters?
• RQ5 Can PDEBench facilitate fair, extensible comparisons and rapid data access for researchers to extend benchmarks with new PDEs or models?

Key findings

• FNO generally provides the best overall predictive accuracy across many PDEs and respects conserved quantities and boundary behavior well.
• FNO shows stable error across frequency content for many problems, but high-frequency errors (fRMSE high) can rise in regimes with strong discontinuities or small diffusion.
• Autoregressive training for U-Net can be unstable; the pushforward trick improves stability and long-horizon accuracy.
• For temporal extrapolation beyond training steps, ML models tend to fail to reliably extend dynamics, indicating challenges in capturing long-time behavior.
• Compressible Navier–Stokes and high-resolution, higher-dimensional problems remain challenging, with performance sensitive to PDE parameters (e.g., diffusion coefficients, Reynolds-like numbers).
• PINNs may handle some high-frequency features better than expected in certain tasks, but memory constraints limit some 2D/3D experiments.