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

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 is 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 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.