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

TL;DR

This paper introduces two training-free projection methods to enforce nonlinear and long-range PDE constraints in neural solvers, significantly reducing violations and enhancing accuracy without additional training.

Contribution

It presents novel, training-free post hoc projection techniques for neural PDE solvers to better enforce complex governing constraints.

Findings

Both projections reduce constraint violations substantially.

Projections improve the accuracy of neural PDE solutions.

Methods are effective across various PDE types.

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.