Two-stage projection physics-informed neural networks for preserving conservation laws

Abstract

Physics-informed neural networks (PINNs) can attain small residuals for partial differential equations while drifting from global invariants. We separate residual-based training from invariant enforcement: a standard PINN is trained first, and its output is then projected onto a selected invariant manifold at inference. The correction requires neither retraining nor a conservation penalty in the training objective. We derive explicit projection maps for the Korteweg–de Vries and nonlinear Schrödinger equations and for a mean-conserving Kuramoto–Sivashinsky benchmark. Spatial field evaluation and quadrature are accelerated on a graphics processing unit, while one scalar update is computed for each evaluation time. Across these three one-dimensional benchmarks, projection reduces the selected-invariant violation by a factor of approximately 1.1×103– 1.3×104 relative to the unprojected PINN, while prediction errors against the supplied numerical comparison trajectories change little. These results establish a benchmark-level post-processing correction over a unit time horizon; they do not establish long-horizon stability or generality beyond equations with a tractable scalar invariant.