Deep learning algorithm for data-driven simulation of noisy dynamical system

TL;DR

This paper introduces DE-LSTM, a deep learning model that simulates noisy nonlinear dynamical systems by approximating their probability distributions, enabling multi-step predictions without assuming specific distributional forms.

Contribution

The paper presents a novel deep learning approach using LSTM networks to directly model the probability density functions of stochastic processes, incorporating a penalized likelihood and high-dimensional integration for dynamic simulation.

Findings

DE-LSTM accurately predicts probability distributions in noisy systems.

The model captures dynamic uncertainty growth and stabilization in long-term forecasts.

It performs well on systems like Ornstein-Uhlenbeck, Mackey-Glass, and Van der Pol oscillator.

Abstract

We present a deep learning model, DE-LSTM, for the simulation of a stochastic process with an underlying nonlinear dynamics. The deep learning model aims to approximate the probability density function of a stochastic process via numerical discretization and the underlying nonlinear dynamics is modeled by the Long Short-Term Memory (LSTM) network. It is shown that, when the numerical discretization is used, the function estimation problem can be solved by a multi-label classification problem. A penalized maximum log likelihood method is proposed to impose a smoothness condition in the prediction of the probability distribution. We show that the time evolution of the probability distribution can be computed by a high-dimensional integration of the transition probability of the LSTM internal states. A Monte Carlo algorithm to approximate the high-dimensional integration is outlined. The behavior of DE-LSTM is thoroughly investigated by using the Ornstein-Uhlenbeck process and noisy observations of nonlinear dynamical systems; Mackey-Glass time series and forced Van der Pol oscillator. It is shown that DE-LSTM makes a good prediction of the probability distribution without assuming any distributional properties of the stochastic process. For a multiple-step forecast of the Mackey-Glass time series, the prediction uncertainty, denoted by the 95\% confidence interval, first grows, then dynamically adjusts following the evolution of the system, while in the simulation of the forced Van der Pol oscillator, the prediction uncertainty does not grow in time even for a 3,000-step forecast.