Adaptive multilayer neural network for solving elliptic partial differential equations with different boundary conditions

Abstract

In recent years, elliptic partial differential equations (PDEs) have been challenging in numerical mathematics, engineering, and physics. In this paper, we proposed a novel approach called Multilayer Neural Network for Partial Differential Equations (MLNPDE) based on deep learning algorithms by exploring the elliptic PDE family under certain conditions such as initial conditions and boundary conditions to govern approximation solutions of PDEs. In the proposed model, we include a multilayer neural network using a densely connected network. Moreover, the approximation solution underlying PDEs can be expressed in two terms: the first term satisfies the boundary conditions, and the second term is a function of the unknown parameters that were estimated by our proposed model. Furthermore, we derive a normal form of the boundary conditions on the first term of the approximation solution for two-dimensional elliptic partial differential equation problems with an arbitrary domain. The numerical results show that our method achieved tremendous state-of-the-art performance in accuracy and efficiency compared to the classical methods.