Data-driven Simulation Of A Nonlinear Dynamical System With Unknown Parameters

Abstract

We propose a deep learning approach for model-free simulations of a dynamical system with unknown parameters without prior knowledge. The deep learning model aims to jointly learn the nonlinear time marching operator and the effects of the unknown parameters from a data set, which consists of an ensemble of trajectories for a range of the parameters. The learning task is formulated as a statistical inference problem by considering the unknown parameters as random variables, and a variational inference method is employed to train a recurrent neural network jointly with a feedforward neural network for an approximately posterior distribution. In numerical experiments, it is found that the proposed deep learning model is capable of correctly identifying the dimensions of the random parameters and learning a representation of complex time series data.