Solving Poisson Equation by Physics-Informed Neural Network with Natural Gradient Descent with Momentum

Abstract

In this paper, we present a solution of the first boundary value problem of the Poisson equation using physics-informed neural networks, which optimize the loss function by natural gradient descent with Dirichlet distribution. This algorithm proved his ability in optimization of multidimensional functions with high accuracy, taking little number of iterations and avoiding local minimums. This fact means, that proposed algorithm will minimize loss function as well. Also, we obtain the exact and numerical solutions, implied due to separation of variable (Fourier method) and finite element method, respectively. The main goal of this paper is to rise the accuracy of solution of the Poisson equation in a short time. We will demonstrate the error estimate of solutions received by finite element method and physics-informed neural network, equipped with adaptive moment estimates and natural gradient descent with momentum and Nesterov condition.