  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.
