Learning Under Laws: A Constraint-Projected Neural PDE Solver that Eliminates Hallucinations
This addresses the issue of hallucinations in neural PDE solvers for physics and engineering applications, representing a novel method rather than an incremental improvement.
The paper tackled the problem of neural networks violating physical laws when solving partial differential equations, and introduced Constraint-Projected Learning (CPL) to enforce constraints like conservation and entropy, achieving stable solutions with machine-precision conservation and bounded error on Burgers and Euler systems.
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.