Solving Poisson's equation using a physics-informed natural gradient descent neural network with Dirichlet distribution

Abstract

Solving Poisson's equation using a physics-informed natural gradient descent neural network with Dirichlet distribution

Abdulkadirov R.I., Lyakhov P.A., Nagornov N.N.

Incoming article date: 25.09.2023

In this paper, a physics-informed neural network containing natural gradient descent is proposed to solve the boundary value problem of the Poisson equation. Machine learning methods used in solving partial differential equations are an alternative to the finite element method. Traditional numerical methods for solving differential equations are not capable of solving arbitrary problems of mathematical physics with equivalent efficiency, unlike machine learning methods. The loss function of the neural network is responsible for the accuracy of solving initial and boundary value problems of partial differential equations. The more efficiently the loss function is minimized, the more accurate the resulting solution is. The most traditional optimization algorithm is adaptive moment estimation, which is still used in deep learning today. However, this approach does not guarantee achieving a global minimum of the loss function. We propose to use natural gradient descent with the Dirichlet distribution which increase the accuracy of solving the Poisson equation.

Keywords: natural gradient descent, Poisson equation, Fisher matrix, finite element method, neural networks