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

TL;DR

The paper addresses automatic discovery of nonlinear dynamical systems from time-series data. It introduces a hybrid framework that couples multistep time-stepping schemes with neural networks to learn the governing vector field f(x) from data without requiring derivatives. The approach is demonstrated on diverse benchmarks, including the Lorenz system, cylinder wake flow, Hopf bifurcation, and glycolytic oscillator, showing accurate dynamics learning, forecasting, and basin identification; it also analyzes the relative performance of Adams-Bashforth, Adams-Moulton, and BDF schemes. The work presents a memory-enabled, scalable method for data-driven dynamical systems with potential extensions to irregular sampling and physics-informed priors.

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.

Figures (7)

• Figure 1: Harmonic Oscillator: Trajectories of the two-dimensional damped harmonic oscillator with cubic dynamics are depicted in the left panel while the corresponding phase portrait is plotted in the right panel. Solid colored lines represent the exact dynamics while the dashed black lines demonstrate the learned dynamics. The identified system correctly captures the form of the dynamics and accurately reproduces the phase portrait.
• Figure 2: Lorenz System: The exact phase portrait of the Lorenz system (left panel) is compared to the corresponding phase portrait of the learned dynamics (right panel).
• Figure 3: Lorenz System: The exact trajectories of the Lorenz systems is compared to the corresponding trajectories of the learned dynamics. Solid blue lines represent the exact dynamics while the dashed black lines demonstrate the learned dynamics.
• Figure 4: Flow past a cylinder: A snapshot of the vorticity field of a solution to the Navier-Stokes equations for the fluid flow past a cylinder.
• Figure 5: Flow past a cylinder: The exact phase portrait of the cylinder wake trajectory in reduced coordinates (left panel) is compared to the corresponding phase portrait of the learned dynamics (right panel).