Variational inference formulation for a model-free simulation of a dynamical system with unknown parameters by a recurrent neural network

TL;DR

This paper introduces a variational inference-based recurrent neural network approach for model-free simulation of dynamical systems with unknown parameters, effectively learning from time series data without prior system knowledge.

Contribution

It develops a novel probabilistic recurrent neural network framework employing variational inference to identify unknown parameters and simulate system dynamics without explicit models.

Findings

More accurate simulation compared to standard RNNs

Correctly identifies dimensions of unknown parameters

Learns complex time series representations

Abstract

We propose a recurrent neural network for a "model-free" simulation 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 time series dataset. We assume that the time series data set 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. A latent variable is introduced to model the effects of the unknown parameters, and a variational inference method is employed to simultaneously train probabilistic models for the time marching operator and an approximate posterior distribution for the latent variable. Unlike the classical variational inference, where a factorized distribution is used to approximate the posterior, we employ a feedforward neural network supplemented by an encoder recurrent neural network to develop a more flexible probabilistic model. The approximate posterior distribution makes an inference on a trajectory to identify the effects of the unknown parameters. The time marching operator is approximated by a recurrent neural network, which takes a latent state sampled from the approximate posterior distribution as one of the input variables, to compute the time evolution of the probability distribution conditioned on the latent variable. In the numerical experiments, it is shown that the proposed variational inference model makes a more accurate simulation compared to the standard recurrent neural networks. 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.