Physics-informed nonlinear vector autoregressive models for the prediction of dynamical systems
This work addresses the challenge of accurate long-term forecasting in scientific machine learning for researchers and engineers dealing with ODE-based systems, representing an incremental improvement by integrating physics constraints into an existing NVAR framework.
The authors tackled the problem of predicting dynamical systems governed by ordinary differential equations (ODEs) by developing a physics-informed nonlinear vector autoregressive (piNVAR) model, which enforces the underlying differential equation during training and shows improved prediction accuracy on systems like the chaotic Lorenz system, with concrete metrics reported in the evaluation.
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.