A Mesh-Free, Physics-Constrained Approach to solve Partial Differential Equations with a Deep Neural Network

Abstract

In this work, we utilized different approaches for solving partial differential equations with a deep neural network. The network respects the given physical laws of the equations by incorporating these constraints in the training process or in the network architecture. Specifically, a deep, feed-forward, and fully-connected neural network is used to approximate the partial differential equation, where the initial and boundary conditions are either hard or soft assigned. The resulting physics-informed surrogate model learns to satisfy the differential operator and the initial and boundary conditions and can be differentiated with respect to all input variables. The accuracy of the methods is demonstrated on multiple equations of different types and compared to either the exact or a finite element solution.