# Constrained Neural Ordinary Differential Equations with Stability Guarantees

arXiv: https://arxiv.org/abs/2004.10883 · HTML: https://mlsanity.com/paper/2004.10883/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Apr 2020.

## Built on by (most cited first, 13 of 13)

- [Physics-Informed Machine Learning for Modeling and Control of Dynamical Systems](https://mlsanity.com/paper/2306.13867/) (2023)
- [Neural Ordinary Differential Equations for Nonlinear System Identification](https://mlsanity.com/paper/2203.00120/) (2022)
- [Forecasting the outcome of spintronic experiments with Neural Ordinary Differential Equations](https://mlsanity.com/paper/2108.02318/) (2021)
- [Deep Learning Explicit Differentiable Predictive Control Laws for Buildings](https://mlsanity.com/paper/2107.11843/) (2021)
- [Bounded nonlinear forecasts of partially observed geophysical systems with physics-constrained deep learning](https://mlsanity.com/paper/2202.05750/) (2022)
- [Unifying physical systems' inductive biases in neural ODE using dynamics constraints](https://mlsanity.com/paper/2208.02632/) (2022)
- [DeNOTS: Stable Deep Neural ODEs for Time Series](https://mlsanity.com/paper/2408.08055/) (2024)
- [On the Stochastic Stability of Deep Markov Models](https://mlsanity.com/paper/2111.04601/) (2021)
- [A Simultaneous Approach for Training Neural Differential-Algebraic Systems of Equations](https://mlsanity.com/paper/2504.04665/) (2025)
- [Eigen-informed NeuralODEs: Dealing with stability and convergence issues of NeuralODEs](https://mlsanity.com/paper/2302.10892/) (2023)
- [Deep Neural Networks Inspired by Differential Equations](https://mlsanity.com/paper/2510.09685/) (2025)
- [A Self-Adaptive Penalty Method for Integrating Prior Knowledge Constraints into Neural ODEs](https://mlsanity.com/paper/2307.14940/) (2023)
- [A Two-Stage Training Method for Modeling Constrained Systems With Neural Networks](https://mlsanity.com/paper/2403.02730/) (2024)

## Abstract (arXiv)

Differential equations are frequently used in engineering domains, such as modeling and control of industrial systems, where safety and performance guarantees are of paramount importance. Traditional physics-based modeling approaches require domain expertise and are often difficult to tune or adapt to new systems. In this paper, we show how to model discrete ordinary differential equations (ODE) with algebraic nonlinearities as deep neural networks with varying degrees of prior knowledge. We derive the stability guarantees of the network layers based on the implicit constraints imposed on the weight's eigenvalues. Moreover, we show how to use barrier methods to generically handle additional inequality constraints. We demonstrate the prediction accuracy of learned neural ODEs evaluated on open-loop simulations compared to ground truth dynamics with bi-linear terms.