Numerical solution and bifurcation analysis of nonlinear partial differential equations with extreme learning machines

Gianluca Fabiani,Lucia Russo,Constantinos Siettos,Francesco Calabrò

doi:10.1007/s10915-021-01650-5

Abstract

We address a new numerical method based on a class of machine learning methods, the so-called Extreme Learning Machines (ELM) 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 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 numerical accuracy, size of the grid and execution times) of the proposed numerical machine-learning method is compared against central finite differences (FD) and Galerkin weighted-residuals finite-element (FEM) methods. We show that the proposed numerical machine learning method outperforms in terms of numerical accuracy both FD and FEM methods for medium to large sized grids, while provides equivalent results with the FEM for low to medium sized grids; both methods (ELM and FEM) outperform the FD scheme. Furthermore, the computational times required with the proposed machine learning scheme were comparable and in particular slightly smaller than the ones required with FEM.

Highlights

The solution of partial differential equations (PDEs) with the aid of machine learning as an alternative to conventional numerical analysis methods can been traced back in the early ’90s
We present some known properties of the proposed problems and provide details on their numerical solution with Finite Differences (FD), FEM and the proposed machine learning (ELM) scheme with both logistic and Gaussian radial basis functions (RBF) transfer functions
In all the computations with FD, FEM and the proposed machine learning (ELM) scheme, the convergence criterion for Newton’s iterations was the L2 norm4 of the relative error between the solutions resulting from successive iterations; the convergence tolerance was set to 10−6

Summary

Introduction

The solution of partial differential equations (PDEs) with the aid of machine learning as an alternative to conventional numerical analysis methods can been traced back in the early ’90s. Very recently [51,52] addressed the use of numerical Gaussian Processes and Deep Neural Networks (DNNs) with collocation to solve time-dependent non-linear PDEs circumventing the need for spatial discretization of the differential operators. In [28], DNNs were used to solve high-dimensional nonlinear parabolic PDEs including the Black-Scholes, the HamiltonJacobi-Bellman and the Allen-Cahn equation. In [62], the authors used FNN to solve modified high-dimensional diffusion equations: the training of the FNN is achieved iteratively using an unsupervised universal machine-learning solver. In [21], the authors have used DNN to construct non-linear reduced-order models of time-dependent parametrized PDEs

Objectives

Results

Conclusion