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.

翻译：机器学习技术近年来在求解微分方程方面引起了广泛关注。训练这些模型传统上属于数据拟合任务，但微分方程表达式的知识可用于补充训练目标，从而推动了物理信息科学机器学习的发展。本文聚焦于一类称为非线性向量自回归（NVAR）的模型，用于求解常微分方程（ODEs）。受数值积分与物理信息神经网络之间联系的启发，我们显式推导出物理信息NVAR（piNVAR），该模型无论NVAR结构如何都能强制满足底层微分方程的右侧表达式。由于NVAR与piNVAR完全共享学习参数，我们提出了一种增强训练流程来联合训练这两个模型。随后，通过数据驱动和ODE驱动的双重评估指标，我们系统评估了piNVAR模型在预测各类ODE系统解方面的能力，包括无阻尼弹簧系统、Lotka-Volterra捕食者-猎物非线性模型以及混沌Lorenz系统。