Numerical Solution and Bifurcation Analysis of Nonlinear Partial Differential Equations with Extreme Learning Machines

TL;DR

This paper introduces a machine learning-based numerical scheme using Extreme Learning Machines to solve nonlinear PDEs and analyze bifurcations, demonstrating superior efficiency over traditional finite difference and finite element methods for larger grids.

Contribution

The paper presents a novel ELM-based numerical method for solving nonlinear PDEs and bifurcation analysis, outperforming classical methods in accuracy and efficiency for larger computational grids.

Findings

ELM method outperforms FD and FEM for medium to large grids.

ELM provides comparable results to FEM for small to medium grids.

The approach effectively computes bifurcation diagrams of nonlinear PDEs.

Abstract

We address a new numerical scheme based on a class of machine learning methods, the so-called Extreme Learning Machines with both sigmoidal and radial-basis functions, for the computation of steady-state solutions and the construction of (one dimensional) bifurcation diagrams of nonlinear partial differential equations (PDEs). For our illustrations, we considered two benchmark problems, namely (a) the one-dimensional viscous Burgers with both homogeneous (Dirichlet) and non-homogeneous mixed boundary conditions, and, (b) the one and two-dimensional Liouville-Bratu-Gelfand PDEs with homogeneous Dirichlet boundary conditions. For the one-dimensional Burgers and Bratu PDEs, exact analytical solutions are available and used for comparison purposes against the numerical derived solutions. Furthermore, the numerical efficiency (in terms of accuracy and size of the grid) of the proposed numerical machine learning scheme is compared against central finite differences (FD) and Galerkin weighted-residuals finite-element methods (FEM). We show that the proposed ELM numerical method outperforms both FD and FEM methods for medium to large sized grids, while provides equivalent results with the FEM for low to medium sized grids.