Artificial neural networks for solving ordinary and partial differential equations

TLDR

The advent of neuroprocessors and digital signal processors makes this method particularly attractive because of expected execution speed gains. The paper proposes an artificial neural network approach to solve initial and boundary value problems and demonstrates it on various model problems with comparisons to the Galerkin finite element method. The method constructs a trial solution as the sum of a non‑parameterized part that satisfies the boundary conditions and a feedforward neural network part that is trained to satisfy the differential equation, ensuring the conditions are met by design. The approach is applicable to single ODEs, coupled ODE systems, and PDEs, and the authors show it performs comparably to the Galerkin finite element method on several test cases.

Abstract

We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the initial/boundary conditions and contains no adjustable parameters. The second part is constructed so as not to affect the initial/boundary conditions. This part involves a feedforward neural network containing adjustable parameters (the weights). Hence by construction the initial/boundary conditions are satisfied and the network is trained to satisfy the differential equation. The applicability of this approach ranges from single ordinary differential equations (ODE's), to systems of coupled ODE's and also to partial differential equations (PDE's). In this article, we illustrate the method by solving a variety of model problems and present comparisons with solutions obtained using the Galekrkin finite element method for several cases of partial differential equations. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.