# Enforcing governing equation constraints in neural PDE solvers via training-free projections

arXiv: https://arxiv.org/abs/2511.17258 · HTML: https://mlsanity.com/paper/2511.17258/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 24 references that are in the map.
- First posted on arXiv 21 Nov 2025.

## Builds on (most cited first, 9 of 9)

- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [DiffusionPDE: Generative PDE-Solving Under Partial Observation](https://mlsanity.com/paper/2406.17763/) (2024)
- [InverseBench: Benchmarking Plug-and-Play Diffusion Priors for Inverse Problems in Physical Sciences](https://mlsanity.com/paper/2503.11043/) (2025)
- [CoCoGen: Physically-Consistent and Conditioned Score-based Generative Models for Forward and Inverse Problems](https://mlsanity.com/paper/2312.10527/) (2023)
- [Physical Symmetries Embedded in Neural Networks](https://mlsanity.com/paper/1904.08991/) (2019)
- [Guaranteed Conservation of Momentum for Learning Particle-based Fluid Dynamics](https://mlsanity.com/paper/2210.06036/) (2022)
- [Physics vs Distributions: Pareto Optimal Flow Matching with Physics Constraints](https://mlsanity.com/paper/2506.08604/) (2025)
- [Precipitation Nowcasting Using Physics Informed Discriminator Generative Models](https://mlsanity.com/paper/2406.10108/) (2024)
- [Generating Physical Dynamics under Priors](https://mlsanity.com/paper/2409.00730/) (2024)

## Built on by (most cited first, 4 of 4)

- [Divergence-Free Diffusion Models for Incompressible Fluid Flows](https://mlsanity.com/paper/2601.19368/) (2026)
- [Predict-Project-Renoise: Sampling Diffusion Models under Hard Constraints](https://mlsanity.com/paper/2601.21033/) (2026)
- [Enabling Structure-Only Initialization and Out-of-Distribution Generalization in GNN-based Molecular Dynamics Simulators](https://mlsanity.com/paper/2605.09495/) (2026)
- [Source Anchoring for Physical Consistency in Flow Matching Models](https://mlsanity.com/paper/2609.33510/) (2026)

## Abstract (arXiv)

Neural PDE solvers used for scientific simulation often violate governing equation constraints. While linear constraints can be projected cheaply, many constraints are nonlinear, complicating projection onto the feasible set. Dynamical PDEs are especially difficult because constraints induce long-range dependencies in time. In this work, we evaluate two training-free, post hoc projections of approximate solutions: a nonlinear optimization-based projection, and a local linearization-based projection using Jacobian-vector and vector-Jacobian products. We analyze constraints across representative PDEs and find that both projections substantially reduce violations and improve accuracy over physics-informed baselines.