Randomized neural network with Petrov–Galerkin methods for solving linear and nonlinear partial differential equations

Abstract

We present a new approach for solving partial differential equations (PDEs) based on randomized neural networks and the Petrov–Galerkin method, which we call the RNN-PG methods. This method uses randomized neural networks to approximate unknown functions and allows for a flexible choice of test functions, such as finite element basis functions, Legendre or Chebyshev polynomials, or neural networks. We apply the RNN-PG methods to various problems, including Poisson problems with primal or mixed formulations, and time-dependent problems with a space–time approach. This paper is adapted from the work originally posted on arXiv.com by the same authors (arXiv:2201.12995, Jan 31, 2022). The new ingredients include non-linear PDEs such as Burger’s equation and a numerical example of a high-dimensional heat equation. Numerical experiments show that the RNN-PG methods can achieve high accuracy with a small number of degrees of freedom. Moreover, RNN-PG has several advantages, such as being mesh-free, handling different boundary conditions easily, solving time-dependent problems efficiently, and solving high-dimensional problems quickly. These results demonstrate the great potential of the RNN-PG methods in the field of numerical methods for PDEs.