Neural networks have emerged as powerful surrogates for solving partial differential equations (PDEs), offering significant computational speedups over traditional methods. However, these models suffer from a critical limitation: error accumulation during long-term rollouts, where small inaccuracies compound exponentially, eventually causing complete divergence from physically valid solutions. We present PhysicsCorrect, a training-free correction framework that enforces PDE consistency at each prediction step by formulating correction as a linearized inverse problem based on PDE residuals. Our key innovation is an efficient caching strategy that precomputes the Jacobian and its pseudoinverse during an offline warm-up phase, reducing computational overhead by two orders of magnitude compared to standard correction approaches. Across three representative PDE systems -- Navier-Stokes fluid dynamics, wave equations, and the chaotic Kuramoto-Sivashinsky equation -- PhysicsCorrect reduces prediction errors by up to 100x while adding negligible inference time (under 5\%). The framework integrates seamlessly with diverse architectures including Fourier Neural Operators, UNets, and Vision Transformers, effectively transforming unstable neural surrogates into reliable simulation tools that bridge the gap between deep learning's computational efficiency and the physical fidelity demanded by practical scientific applications.
View on arXiv@article{huang2025_2507.02227,
title={ PhysicsCorrect: A Training-Free Approach for Stable Neural PDE Simulations },
author={ Xinquan Huang and Paris Perdikaris },
journal={arXiv preprint arXiv:2507.02227},
year={ 2025 }
}