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

arXiv: https://arxiv.org/abs/2507.02227 · HTML: https://mlsanity.com/paper/2507.02227/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 34 references that are in the map.
- First posted on arXiv 3 Jul 2025.

## Builds on (most cited first, 24 of 24)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Group Normalization](https://mlsanity.com/paper/1803.08494/) (2018)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2403.03542/) (2024)
- [Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks](https://mlsanity.com/paper/1906.05747/) (2019)
- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)
- [CViT: Continuous Vision Transformer for Operator Learning](https://mlsanity.com/paper/2405.13998/) (2024)
- [Learning to correct spectral methods for simulating turbulent flows](https://mlsanity.com/paper/2207.00556/) (2022)
- [Applications of physics informed neural operators](https://mlsanity.com/paper/2203.12634/) (2022)
- [Teaching the Incompressible Navier-Stokes Equations to Fast Neural Surrogate Models in 3D](https://mlsanity.com/paper/2012.11893/) (2020)
- [Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2210.03008/) (2022)
- [Generative Learning of the Solution of Parametric Partial Differential Equations Using Guided Diffusion Models and Virtual Observations](https://mlsanity.com/paper/2408.00157/) (2024)
- [NeuralStagger: Accelerating Physics-constrained Neural PDE Solver with Spatial-temporal Decomposition](https://mlsanity.com/paper/2302.10255/) (2023)
- [LordNet: An Efficient Neural Network for Learning to Solve Parametric Partial Differential Equations without Simulated Data](https://mlsanity.com/paper/2206.09418/) (2022)
- [Residual-Based Error Corrector Operator to Enhance Accuracy and Reliability of Neural Operator Surrogates of Nonlinear Variational Boundary-Value Problems](https://mlsanity.com/paper/2306.12047/) (2023)
- [Goal-Oriented A-Posteriori Estimation of Model Error as an Aid to Parameter Estimation](https://mlsanity.com/paper/2205.15131/) (2022)

## Built on by (most cited first, 8 of 8)

- [Neural-Initialized Newton: Accelerating Nonlinear Finite Elements via Operator Learning](https://mlsanity.com/paper/2511.06802/) (2025)
- [Towards Robust Surrogate Models: Benchmarking Machine Learning Approaches to Expediting Phase Field Simulations of Brittle Fracture](https://mlsanity.com/paper/2507.07237/) (2025)
- [Error-Conditioned Neural Solvers](https://mlsanity.com/paper/2606.27354/) (2026)
- [Learning Heat-based Equations in Self-similar variables](https://mlsanity.com/paper/2602.00872/) (2026)
- [ARC-STAR: Auditable Post-Hoc Correction for PDE Foundation Models](https://mlsanity.com/paper/2605.22222/) (2026)
- [Dynamics-Aware Weighting for Deep Learning Forecasts of Chaotic Systems](https://mlsanity.com/paper/2608.22277/) (2026)
- [DU-NO: A Parameter-Efficient Double U-Shaped Neural Operator for Phase-Resolving Wave Modeling](https://mlsanity.com/paper/2609.12115/) (2026)
- [Physics-Refined Spatiotemporal Forecasting on Open-Boundary Hydrologic Graphs](https://mlsanity.com/paper/2610.01765/) (2026)

## Abstract (arXiv)

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.