Learning Under Laws: A Constraint-Projected Neural PDE Solver that Eliminates Hallucinations

TL;DR

This paper introduces a physics-constrained neural PDE solver, CPL, that enforces physical laws during training to eliminate hallucinations and violations, ensuring stable and lawful solutions.

Contribution

The paper proposes Constraint-Projected Learning (CPL), a novel framework that integrates physical constraints directly into neural PDE solvers, improving their stability and physical fidelity.

Findings

Conservation holds at machine precision.

Total-variation growth is suppressed.

Solutions remain stable and physically lawful.

Abstract

Neural networks can approximate solutions to partial differential equations, but they often break the very laws they are meant to model-creating mass from nowhere, drifting shocks, or violating conservation and entropy. We address this by training within the laws of physics rather than beside them. Our framework, called Constraint-Projected Learning (CPL), keeps every update physically admissible by projecting network outputs onto the intersection of constraint sets defined by conservation, Rankine-Hugoniot balance, entropy, and positivity. The projection is differentiable and adds only about 10% computational overhead, making it fully compatible with back-propagation. We further stabilize training with total-variation damping (TVD) to suppress small oscillations and a rollout curriculum that enforces consistency over long prediction horizons. Together, these mechanisms eliminate both hard and soft violations: conservation holds at machine precision, total-variation growth vanishes, and entropy and error remain bounded. On Burgers and Euler systems, CPL produces stable, physically lawful solutions without loss of accuracy. Instead of hoping neural solvers will respect physics, CPL makes that behavior an intrinsic property of the learning process.