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

2025-11-05

AI Summary

Neural networks solving partial differential equations (PDEs) often violate fundamental physical principles—such as mass conservation, entropy production, positivity, and shock dynamics. To address this, we propose the Constraint Projection Learning (CPL) framework, which encodes conservation laws, the Rankine–Hugoniot condition, entropy conditions, and positivity constraints into differentiable projection operators embedded throughout training. Integrated with total variation diminishing (TVD) regularization and rollout curriculum learning, CPL enforces strict physical compliance at every optimization step. Experiments on the Burgers and Euler equations demonstrate that solutions exhibit exact conservation, bounded total variation, no error accumulation, long-term stability, and machine-precision adherence to physical laws. This work presents the first end-to-end, differentiable, and compact integration of multiple physics-based constraints in neural PDE solvers, significantly enhancing solution reliability and generalizability.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Computer Vision: Learning & Optimization for CV
• Machine Learning: Learning on the Edge & Model Compression

Application Category

• Responsible Web: Consent frameworks and practices on the web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

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.

Problem

• Ensures neural PDE solvers strictly obey physical laws during training
• Projects network outputs onto constraint sets for conservation and entropy
• Eliminates physical violations like mass creation and entropy breaches

Innovation

• Differentiable projection enforces physical constraints during training
• Total-variation damping suppresses oscillations in neural solutions
• Rollout curriculum ensures long-term prediction consistency