Improved Deep Neural Networks with Domain Decomposition in Solving Partial Differential Equations

Wei Wu, Xinlong Feng, Hui Xu

Abstract: An improved neural networks method based on domain decomposition is proposed to solve partial differential equations, which is an extension of the physics informed neural networks (PINNs). Although recent research has shown that PINNs perform effectively in solving partial differential equations, they still have difficulties in solving large-scale complex problems, due to using a single neural network and gradient pathology. In this paper, the proposed approach aims at implementing calculations on sub-domains and improving the expressiveness of neural networks to mitigate gradient pathology. By investigations, it is shown that, although the neural networks structure and the loss function are complicated, the proposed method outperforms the classical PINNs with respect to training effectiveness, computational accuracy, and computational cost.