  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.
