Multi-Step Physics-Informed Deep Operator Neural Network for Directly Solving Partial Differential Equations

Abstract

This paper establishes a method for solving partial differential equations using a multi-step physics-informed deep operator neural network. The network is trained by embedding physics-informed constraints. Different from traditional neural networks for solving partial differential equations, the proposed method uses a deep neural operator network to indirectly construct the mapping relationship between the variable functions and solution functions. This approach makes full use of the hidden information between the variable functions and independent variables. The process whereby the model captures incredibly complex and highly nonlinear relationships is simplified, thereby making network learning easier and enhancing the extraction of information about the independent variables in partial differential systems. In terms of solving partial differential equations, we verify that the multi-step physics-informed deep operator neural network markedly improves the solution accuracy compared with a traditional physics-informed deep neural operator network, especially when the problem involves complex physical phenomena with large gradient changes.