Physics-informed nonlinear vector autoregressive models for the prediction of dynamical systems

TL;DR

This paper introduces physics-informed nonlinear vector autoregressive models (piNVAR) that incorporate differential equation knowledge into machine learning models to improve the prediction of various dynamical systems.

Contribution

The paper develops a physics-informed NVAR framework that enforces differential equations and jointly trains data-driven and physics-based models for improved system prediction.

Findings

piNVAR accurately predicts solutions of ODE systems

Joint training enhances model robustness and accuracy

Effective for nonlinear and chaotic dynamical systems

Abstract

Machine learning techniques have recently been of great interest for solving differential equations. Training these models is classically a data-fitting task, but knowledge of the expression of the differential equation can be used to supplement the training objective, leading to the development of physics-informed scientific machine learning. In this article, we focus on one class of models called nonlinear vector autoregression (NVAR) to solve ordinary differential equations (ODEs). Motivated by connections to numerical integration and physics-informed neural networks, we explicitly derive the physics-informed NVAR (piNVAR) which enforces the right-hand side of the underlying differential equation regardless of NVAR construction. Because NVAR and piNVAR completely share their learned parameters, we propose an augmented procedure to jointly train the two models. Then, using both data-driven and ODE-driven metrics, we evaluate the ability of the piNVAR model to predict solutions to various ODE systems, such as the undamped spring, a Lotka-Volterra predator-prey nonlinear model, and the chaotic Lorenz system.