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

arXiv: https://arxiv.org/abs/2407.18057 · HTML: https://mlsanity.com/paper/2407.18057/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 27 references that are in the map.
- First posted on arXiv 25 Jul 2024.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Next Generation Reservoir Computing](https://mlsanity.com/paper/2106.07688/) (2021)
- [Hybrid Forecasting of Chaotic Processes: Using Machine Learning in Conjunction with a Knowledge-Based Model](https://mlsanity.com/paper/1803.04779/) (2018)
- [Automatic-differentiated Physics-Informed Echo State Network (API-ESN)](https://mlsanity.com/paper/2101.00002/) (2020)
- [Physics-Informed Echo State Networks for Chaotic Systems Forecasting](https://mlsanity.com/paper/1906.11122/) (2019)
- [Learning Hidden States in a Chaotic System: A Physics-Informed Echo State Network Approach](https://mlsanity.com/paper/2001.02982/) (2020)

## Built on by (most cited first, 3 of 3)

- [Parallel Scan Recurrent Neural Quantum States for Scalable Variational Monte Carlo](https://mlsanity.com/paper/2605.13807/) (2026)
- [Unified Spatiotemporal Physics-Informed Learning (USPIL): A Framework for Modeling Complex Predator-Prey Dynamics](https://mlsanity.com/paper/2509.13425/) (2025)
- [Flow map learning in nonlinear vector autoregressive models: influence of the feature-library structure on the training error](https://mlsanity.com/paper/2605.31438/) (2026)

## Abstract (arXiv)

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.