PhysicsCorrect: A Training-Free Approach for Stable Neural PDE Simulations

TL;DR

Neural PDE surrogates often suffer from error accumulation during long-term rollouts, limiting practical use. PhysicsCorrect introduces a training-free, physics-informed correction that projects each predicted state onto the PDE-consistent manifold by solving a linearized inverse problem using the PDE residual and a precomputed Jacobian pseudoinverse, achievable via offline caching. Across 2D Navier–Stokes, 2D wave, and Kuramoto–Sivashinsky benchmarks and across FNO, UNet, and ViT architectures, it yields 1–2 orders of magnitude reductions in error with negligible inference overhead. This approach robustly bridges the speed of neural solvers with the fidelity required for real-world scientific simulations, while outlining clear avenues for scaling and future refinement.

Abstract

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, including 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.

Figures (15)

• Figure 1: PhysicsCorrect stabilizes neural PDE solver rollouts by projecting erroneous predictions back onto the manifold of physically consistent solutions.
• Figure 2: Long-term rollout accuracy comparison for the 2D Navier-Stokes benchmark. The baseline neural operator (brown) exhibits error accumulation, while our predictor-corrector approach (blue) maintains stability throughout the simulation, closely matching the performance of idealized one-step rollouts (yellow).
• Figure 3: PhysicsCorrect's caching strategy efficiency on 2D Navier-Stokes. Left: Relative $L_2$ error vs. reference solutions over 200 time steps. Right: PDE residual magnitude per step. While the baseline (brown) shows increasing error and residual, both uncached (yellow) and Jacobian-cached (blue) corrections maintain low values. Pseudoinverse caching (red) preserves performance while reducing computational cost by 163x.
• Figure 4: Performance comparison of our physics-informed correction approach across different PDE systems and neural architectures. Left axis (bars): Relative L2 error of baseline models (lighter colors) versus corrected models (darker colors) for Navier-Stokes (NS), wave equation, and Kuramoto-Sivashinsky (KS) equations at the final state after long rollouts (1000 time step rollout for NS and KS equations; 100 time steps for wave equation). Error bars represent the standard deviation for 5 seeds. Right axis (line): Relative computational cost of the corrected approach compared to baseline. The correction framework consistently reduces error across all PDEs and architectures with minimal computational overhead. Detailed rollout histories and comparison with baseline models are provided in the Appendix.
• Figure 5: One-step correction on the 2D Navier-Stokes equation. From left to right: ground truth solution from numerical simulation, prediction error of baseline FNO, prediction error after our correction, PDE residual of baseline prediction, and PDE residual after correction. Note the significant reduction in both error magnitude (10× improvement) and PDE residual (100× improvement), demonstrating that our correction effectively projects predictions onto the manifold of physically consistent solutions.