One-Step Evolution for Long-Time Extrapolation: An Error-Bound-Informed and Prior-Guided Neural Residual Framework for Autonomous PDEs

Abstract

Accurate simulation of the long-time evolution of systems governed by partial differential equations (PDEs) is central to scientific computing. Among existing deep learning?based approaches for solving PDEs, neural operators typically rely on extensive trajectory data, whereas physics-informed meth?ods often exhibit limited stability during long-time extrapolation. For a well-posed autonomous PDE, long-time trajectories can be generated by repeated composition of a fixed-step evolution operator; hence, long-time extrapolation depends on controlling the approximation error of this operator and the propagation of that error under recursive composition. Accordingly, we propose a numerical-prior-guided, physics-constrained method trained without ground-truth trajectory supervision: a low-cost numerical prior reduces the difficulty of approximating the one?step evolution operator, while a weak-form PDE residual provides a computable proxy for the one-step error term in the error?propagation bound. We validate the method on five benchmark cases spanning four PDE classes and compare it with ten physics?informed learning methods under a unified protocol that excludes ground-truth trajectories from training and model selection. The results indicate that, in all five cases, the proposed method reduces long-time extrapolation error relative to the numerical prior and outperforms the best competing baseline in each case, thereby improving long-time simulation accuracy across different PDEs without ground-truth trajectory supervision. The source code developed for this paper will be made publicly available upon acceptance of the manuscript.