Solving flows of dynamical systems by deep neural networks and a novel deep learning algorithm

Machine learning becomes popular and is used for a wide range of problems in various areas of applied sciences. In dynamical systems, machine learning methods are applied to solve differential equations. In this paper, we develop an artificial network to solve systems of ordinary differential equations. For the network, we use a multilayer perceptron networks, which is a fully connected feedforward network to predict the flow of a specific system. In order to improve the long time estimation of the method, we introduced an upper bound control strategy. To deal with stiff ODE systems(such as slow-fast coupled system), a new algorithm, Finite Neural Element method, is introduced. By numerical simulation, the novel algorithm is proved to have better efficiency and accuracy than direct machine learning method.