PDEInvBench: A Comprehensive Dataset and Design Space Exploration of Neural Networks for PDE Inverse Problems
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Abstract (arXiv)
Inverse problems in partial differential equations (PDEs) involve estimating the physical parameters of a system from observed spatiotemporal solution fields. Neural networks are well-suited for PDE parameter estimation due to their capability to model function-to-function space transformations. While existing benchmarks of machine learning methods for PDEs primarily focus on the forward problem, there are no similar comprehensive studies and benchmark datasets on PDE inverse problems, i.e., mapping solution fields to underlying physical parameters. We fill this gap by introducing PDEInvBench, a comprehensive benchmark dataset consisting of numerical simulations for both time-dependent and time-independent PDEs across a wide range of physical behaviors and parameters. Our dataset includes evaluation splits that assess performance in both in-distribution and various out-of-distribution settings. Using our benchmark dataset, we comprehensively explore the design space of neural networks for PDE inverse problems along three key dimensions: (1) optimization procedures, analyzing the role of supervised, self-supervised, and test-time training objectives on performance, (2) problem representations, where we study the value of architectural choices with different inductive biases and various conditioning strategies, and (3) scaling, which we perform with respect to both model and data size. Our experiments reveal several practical insights: 1) neural networks perform best with a two-stage training procedure: initial supervision with PDE parameters followed by test-time fine-tuning using the PDE residual, 2) incorporating PDE derivatives as input features consistently improves accuracy, and 3) increasing the diversity of initial conditions in the training data yields greater performance gains than expanding the range of PDE parameters. We make our dataset and codebase publicly available.
Key facts. Cited by 2 papers in the map (data through 14 Oct 2026). Builds on 35 of its 67 references that are in the map. First posted on arXiv 25 May 2026.
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35 of 67 references.
References · 35
- 2015Deep Residual Learning for Image Recognitioncited by 52,096
- 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
- 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
- 2021Swin Transformer V2: Scaling Up Capacity and Resolutioncited by 1,228
- 2021Characterizing possible failure modes in physics-informed neural networkscited by 643
- 2017Deep Learning Scaling is Predictable, Empiricallycited by 582
- 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 527
- 2021Machine learning accelerated computational fluid dynamicscited by 471
- 2022PDEBENCH: An Extensive Benchmark for Scientific Machine Learningcited by 413
- 2021Explaining Neural Scaling Lawscited by 363
- 2021Choose a Transformer: Fourier or Galerkincited by 343
- 2023GNOT: A General Neural Operator Transformer for Operator Learningcited by 336
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- 2024Transolver: A Fast Transformer Solver for PDEs on General Geometriescited by 326
- 2021Factorized Fourier Neural Operatorscited by 249
- 2022U-NO: U-shaped Neural Operatorscited by 230
- 2020Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solverscited by 205
- 2024Poseidon: Efficient Foundation Models for PDEscited by 200
- 2022Towards Multi-spatiotemporal-scale Generalized PDE Modelingcited by 197
- 2023PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEscited by 87
- 2024Latent Neural Operator for Solving Forward and Inverse PDE Problemscited by 80
- 2022Lie Point Symmetry Data Augmentation for Neural PDE Solverscited by 77
- 2023Benchmarking Autoregressive Conditional Diffusion Models for Turbulent Flow Simulationcited by 71
- 2025InverseBench: Benchmarking Plug-and-Play Diffusion Priors for Inverse Problems in Physical Sciencescited by 68
- 2023Neural Inverse Operators for Solving PDE Inverse Problemscited by 65
- 2022Generalizing to New Physical Systems via Context-Informed Dynamics Modelcited by 50
- 2022Learning to correct spectral methods for simulating turbulent flowscited by 49
- 2023LagrangeBench: A Lagrangian Fluid Mechanics Benchmarking Suitecited by 27
- 2024Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspectivecited by 25
- 2023BubbleML: A Multi-Physics Dataset and Benchmarks for Machine Learningcited by 17
- 2023Neural Spectral Methods: Self-supervised learning in the spectral domaincited by 14
- 2023Understanding the Expressivity and Trainability of Fourier Neural Operator: A Mean-Field Perspectivecited by 11
- 2023Interpretable Meta-Learning of Physical Systemscited by 11
- 2024Physics-Informed Deep Inverse Operator Networks for Solving PDE Inverse Problemscited by 11
- 2024PDE Control Gym: A Benchmark for Data-Driven Boundary Control of Partial Differential Equationscited by 10
- 2025EddyFormer: Accelerated Neural Simulations of Three-Dimensional Turbulence at Scalecited by 6
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