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

2025-11-21

AI Summary

Neural network–based PDE solvers often violate physical conservation laws due to nonlinear constraints and long-range temporal dependencies. To address this, we propose a training-free post-hoc projection framework that enforces predicted solutions to reside within the feasible set defined by the governing equations. Our approach comprises two complementary components: (i) a high-accuracy nonlinear optimization scheme for constraint correction, and (ii) an efficient, stable local linearization projection leveraging Jacobian-vector and vector-Jacobian products. To our knowledge, this is the first systematic study evaluating training-free projection strategies for dynamical PDEs. Experiments on canonical equations—including Burgers’ and KdV—demonstrate substantial reductions in constraint violation, superior accuracy and physical consistency compared to baselines such as PINNs, and effective mitigation of error accumulation over long-time integration.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Search and Optimization: Non-convex Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

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.

Problem

• Neural PDE solvers often violate governing equation constraints during simulation
• Nonlinear constraints complicate projection onto feasible sets in PDE solutions
• Dynamical PDE constraints create long-range temporal dependencies in solutions

Innovation

• Training-free projections enforce governing equation constraints
• Nonlinear optimization-based projection reduces constraint violations
• Local linearization projection uses Jacobian-vector products