PDEInvBench: A Comprehensive Dataset and Design Space Exploration of Neural Networks for PDE Inverse Problems
2026-05-25 • Machine Learning
Machine LearningComputer Vision and Pattern Recognition
AI summaryⓘ
The authors study how to estimate unknown physical parameters in systems described by partial differential equations (PDEs) using neural networks. They created a new benchmark dataset called PDEInvBench containing many simulations to test different approaches for these inverse problems. Their experiments show that a two-step training method, using both supervised learning and fine-tuning, works best, and adding derivatives of the PDE as input helps accuracy. They also find that having diverse starting conditions in training data improves results more than just varying the physical parameters. The dataset and code are made publicly available to support further research.
partial differential equationsinverse problemsneural networksparameter estimationsupervised learningself-supervised learningtest-time trainingPDE residualbenchmark datasetmodel scaling
Authors
Divyam Goel, Nithin Chalapathi, Sanjeev Raja, Aditi S. Krishnapriyan
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.