基于方程驱动神经网络的非线性参数系统分岔学习
Learn bifurcations of nonlinear parametric systems via equation-driven neural networks
摘要：
非线性参数系统广泛应用于科学和工程中的非线性动力学建模。这些非线性系统在参数空间上的分岔分析通常用来研究解的结构，如解的个数和稳定性。在本文中，我们开发了一种新的机器学习方法，通过所谓的方程驱动神经网络（EDNN）来计算分叉。EDNN由两步优化组成：第一步是通过训练经验解数据来近似参数的解函数;第二步是通过使用第一步中获得的近似神经网络来计算分叉。理论上的收敛性分析和数值实现的几个例子已经证明了所提出的方法的可行性。
英文摘要：
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.
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