Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems

TL;DR

This paper introduces a neural network-based method combining multi-step numerical schemes to identify and predict complex nonlinear and chaotic dynamical systems from data, advancing automated modeling in science and engineering.

Contribution

It presents a novel approach integrating multi-step schemes with neural networks for data-driven discovery of nonlinear dynamical systems, including chaotic and biological models.

Findings

Accurately learns complex nonlinear dynamics

Forecasts future states effectively

Identifies basins of attraction in chaotic systems

Abstract

The process of transforming observed data into predictive mathematical models of the physical world has always been paramount in science and engineering. Although data is currently being collected at an ever-increasing pace, devising meaningful models out of such observations in an automated fashion still remains an open problem. In this work, we put forth a machine learning approach for identifying nonlinear dynamical systems from data. Specifically, we blend classical tools from numerical analysis, namely the multi-step time-stepping schemes, with powerful nonlinear function approximators, namely deep neural networks, to distill the mechanisms that govern the evolution of a given data-set. We test the effectiveness of our approach for several benchmark problems involving the identification of complex, nonlinear and chaotic dynamics, and we demonstrate how this allows us to accurately learn the dynamics, forecast future states, and identify basins of attraction. In particular, we study the Lorenz system, the fluid flow behind a cylinder, the Hopf bifurcation, and the Glycoltic oscillator model as an example of complicated nonlinear dynamics typical of biological systems.