Physics-Informed Nonlinear Vector Autoregressive Models for the Prediction of 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. This yields a statistical framework to capture and predict dynamic relationships between multiple time series variables with the extra knowledge that the observed data are from ODEs. 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. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/samuelhocking/pinvar-sisc-2024 and in the supplementary materials ( pinvar-sisc-2024-main.zip [6.77MB]). [Formula: see text]