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

James H. Adler, Samuel Hocking, Xiaozhe Hu, Shafiqul Islam

TL;DR

The authors address predicting dynamical systems governed by ODEs by augmenting nonlinear vector autoregression (NVAR) with physics-informed training, yielding piNVAR, a model whose parameters are shared between the data-driven update and an explicit ODE-consistency term. By deriving the time derivative of the NVAR output via the chain rule, piNVAR enforces the governing equation, enabling simultaneous, linear-time training and improved adherence to the dynamics. Across undamped springs, Lotka-Volterra, and Lorenz systems, piNVAR improves both data-driven metrics (valid time) and physics-driven metrics (discrete energy) relative to purely data-driven NVAR, especially under appropriate ODE-weighting and regularization. The method offers a transparent, efficient framework for physics-constrained forecasting with potential applications to a broad class of ODE-based 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.

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

TL;DR

The authors address predicting dynamical systems governed by ODEs by augmenting nonlinear vector autoregression (NVAR) with physics-informed training, yielding piNVAR, a model whose parameters are shared between the data-driven update and an explicit ODE-consistency term. By deriving the time derivative of the NVAR output via the chain rule, piNVAR enforces the governing equation, enabling simultaneous, linear-time training and improved adherence to the dynamics. Across undamped springs, Lotka-Volterra, and Lorenz systems, piNVAR improves both data-driven metrics (valid time) and physics-driven metrics (discrete energy) relative to purely data-driven NVAR, especially under appropriate ODE-weighting and regularization. The method offers a transparent, efficient framework for physics-constrained forecasting with potential applications to a broad class of ODE-based 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.
Paper Structure (19 sections, 63 equations, 2 figures, 7 tables)

This paper contains 19 sections, 63 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Nonlinear vector autoregression (NVAR) models
  3. Physics-informed NVAR
  4. PINNs
  5. NVAR differentiation
  6. Physics-informed training
  7. Illustrative example
  8. Numerical experiments
  9. Test problems
  10. Undamped spring
  11. Lotka-Volterra
  12. Lorenz
  13. Cross-validation routine
  14. Evaluation metrics
  15. Valid time
  16. ...and 4 more sections

Figures (2)

  • Figure 1: Recursive prediction of system state $\mathbf{x}=[x_1\ x_2\ x_3]^\top$ for the Lorenz system. Blue: model prediction, orange: target, dashed line: valid time for $M=0.01$. Here, $t_{valid}=5,005$
  • Figure 2: Support coefficient functions. Left: smooth support for $m=1.0$, $t=0.0$, and sharpness $\xi=1,\ldots,5$, right: non-smooth support for $a=-1$, $b=-0.85$, $c=0.85$, $d=1.0$