PDEInvBench: A Comprehensive Dataset and Design Space Exploration of Neural Networks for PDE Inverse Problems

• URL: http://arxiv.org/abs/2605.25353v2
• Date: Tue, 26 May 2026 06:06:38 GMT
• Title: PDEInvBench: A Comprehensive Dataset and Design Space Exploration of Neural Networks for PDE Inverse Problems
• Abstract summary: Intemporal problems in partial differential equations (PDEs) involve estimating the physical parameters of a observed system from solution fields.<n>Nerve networks are well-suited for PDE parameter estimation due to their capability to model function-to-function space transformations.<n>PDEInvBench is a comprehensive benchmark dataset consisting of numerical simulations for both time-dependent and time-independent PDEs.
• Score: 11.742898472476796
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

Related papers

• Physics-based deep kernel learning for parameter estimation in high dimensional PDEs [2.5088504346370684]
  Inferring parameters of high-dimensional partial differential equations (PDEs) poses significant computational and inferential challenges.<n>This paper introduces a novel two-stage Bayesian framework that integrates training, physics-based deep kernel learning (DKL) with Hamiltonian Monte Carlo (HMC)<n> Numerical experiments on canonical and high-dimensional inverse PDE problems demonstrate that our framework accurately estimates parameters, provides reliable uncertainty estimates, and effectively addresses challenges of data sparsity and model complexity.
  arXiv  Detail & Related papers  (2025-09-17T14:56:31Z)
• High precision PINNs in unbounded domains: application to singularity formulation in PDEs [83.50980325611066]
  We study the choices of neural network ansatz, sampling strategy, and optimization algorithm.<n>For 1D Burgers equation, our framework can lead to a solution with very high precision.<n>For the 2D Boussinesq equation, we obtain a solution whose loss is $4$ digits smaller than that obtained in citewang2023asymptotic with fewer training steps.
  arXiv  Detail & Related papers  (2025-06-24T02:01:44Z)
• Paving the way for scientific foundation models: enhancing generalization and robustness in PDEs with constraint-aware pre-training [49.8035317670223]
  A scientific foundation model (SciFM) is emerging as a promising tool for learning transferable representations across diverse domains.<n>We propose incorporating PDE residuals into pre-training either as the sole learning signal or in combination with data loss to compensate for limited or infeasible training data.<n>Our results show that pre-training with PDE constraints significantly enhances generalization, outperforming models trained solely on solution data.
  arXiv  Detail & Related papers  (2025-03-24T19:12:39Z)
• Mechanistic PDE Networks for Discovery of Governing Equations [52.492158106791365]
  We present Mechanistic PDE Networks, a model for discovery of partial differential equations from data.<n>The represented PDEs are then solved and decoded for specific tasks.<n>We develop a native, GPU-capable, parallel, sparse, and differentiable multigrid solver specialized for linear partial differential equations.
  arXiv  Detail & Related papers  (2025-02-25T17:21:44Z)
• Advancing Generalization in PINNs through Latent-Space Representations [71.86401914779019]
  Physics-informed neural networks (PINNs) have made significant strides in modeling dynamical systems governed by partial differential equations (PDEs)<n>We propose PIDO, a novel physics-informed neural PDE solver designed to generalize effectively across diverse PDE configurations.<n>We validate PIDO on a range of benchmarks, including 1D combined equations and 2D Navier-Stokes equations.
  arXiv  Detail & Related papers  (2024-11-28T13:16:20Z)
• Learning a Neural Solver for Parametric PDE to Enhance Physics-Informed Methods [14.791541465418263]
  We propose learning a solver, i.e., solving partial differential equations (PDEs) using a physics-informed iterative algorithm trained on data.<n>Our method learns to condition a gradient descent algorithm that automatically adapts to each PDE instance.<n>We demonstrate the effectiveness of our approach through empirical experiments on multiple datasets.
  arXiv  Detail & Related papers  (2024-10-09T12:28:32Z)
• Parameterized Physics-informed Neural Networks for Parameterized PDEs [24.926311700375948]
  In this paper, we propose a novel extension, parameterized physics-informed neural networks (PINNs) PINNs enable modeling the solutions of parameterized partial differential equations (PDEs) via explicitly encoding a latent representation of PDE parameters. We demonstrate that P$2$INNs outperform the baselines both in accuracy and parameter efficiency on benchmark 1D and 2D parameterized PDEs.
  arXiv  Detail & Related papers  (2024-08-18T11:58:22Z)
• Unisolver: PDE-Conditional Transformers Towards Universal Neural PDE Solvers [53.79279286773326]
  We present Unisolver, a novel Transformer model trained on diverse data and conditioned on diverse PDEs.<n>Unisolver achieves consistent state-of-the-art on three challenging large-scale benchmarks, showing impressive performance and generalizability.
  arXiv  Detail & Related papers  (2024-05-27T15:34:35Z)
• Deep Equilibrium Based Neural Operators for Steady-State PDEs [100.88355782126098]
  We study the benefits of weight-tied neural network architectures for steady-state PDEs. We propose FNO-DEQ, a deep equilibrium variant of the FNO architecture that directly solves for the solution of a steady-state PDE.
  arXiv  Detail & Related papers  (2023-11-30T22:34:57Z)
• LatentPINNs: Generative physics-informed neural networks via a latent representation learning [0.0]
  We introduce latentPINN, a framework that utilizes latent representations of the PDE parameters as additional (to the coordinates) inputs into PINNs. We use a two-stage training scheme in which the first stage, we learn the latent representations for the distribution of PDE parameters. In the second stage, we train a physics-informed neural network over inputs given by randomly drawn samples from the coordinate space within the solution domain.
  arXiv  Detail & Related papers  (2023-05-11T16:54:17Z)
• Neural Partial Differential Equations with Functional Convolution [30.35306295442881]
  We present a lightweighted neural PDE representation to discover the hidden structure and predict the solution of different nonlinear PDEs. We leverage the prior of translational similarity'' of numerical PDE differential operators to drastically reduce the scale of learning model and training data.
  arXiv  Detail & Related papers  (2023-03-10T04:25:38Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.