Learn bifurcations of nonlinear parametric systems via equation-driven neural networks

TL;DR

This paper introduces a novel machine learning method called equation-driven neural networks (EDNNs) to efficiently compute bifurcations in nonlinear parametric systems, combining data approximation and bifurcation analysis.

Contribution

The paper develops a new two-step EDNN approach for bifurcation analysis, integrating neural network approximation with bifurcation computation, supported by theoretical and numerical validation.

Findings

EDNNs accurately approximate solution functions of nonlinear systems.

The method successfully identifies bifurcation points in example systems.

Theoretical convergence of EDNNs is established.

Abstract

Nonlinear parametric systems have been widely used in modeling nonlinear dynamics in science and engineering. Bifurcation analysis of these nonlinear systems on the parameter space are usually used to study the solution structure such as the number of solutions and the stability. In this paper, we develop a new machine learning approach to compute the bifurcations via so-called equation-driven neural networks (EDNNs). The EDNNs consist of a two-step optimization: the first step is to approximate the solution function of the parameter by training empirical solution data; the second step is to compute bifurcations by using the approximated neural network obtained in the first step. Both theoretical convergence analysis and numerical implementation on several examples have been performed to demonstrate the feasibility of the proposed method.