PI-DOSnet: A Physics-Informed Deep Operator-Splitting Network for Evolution Partial Differential Equations

Abstract

Evolution partial differential equations (PDEs) describe time-dependent physical systems governed by differential laws and arise widely across science and engineering. In recent years, operator learning has emerged as a powerful and efficient paradigm for solving evolution PDEs by learning mappings between infinite-dimensional function spaces, enabling solution prediction without explicit time-step integration. In this work, we propose PI-DOSnet, a physics-informed operator learning framework built upon DOSnet and operator splitting. Unlike purely data-driven operator learning methods, PI-DOSnet incorporates physical constraints during training, allowing it to operate even in the absence of paired input-output data. Once trained, PI-DOSnet performs long-time inference of PDE solutions through an iterative strategy. We analyze the linear stability and approximation error of PI-DOSnet and demonstrate its accuracy, efficiency, and robustness through multiple numerical experiments. Moreover, for the Allen--Cahn equation, PI-DOSnet achieves energy stable solutions even with a large time-step size.