Constrained Neural Ordinary Differential Equations with Stability Guarantees

Aaron Tuor, Jan Drgona, Draguna Vrabie

Keywords: Deep Learning, Ordinary Differential Equations, Physics Informed Machine Learning, Physics Informed Neural Networks, Eigenvalue Constraints

TL;DR: 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.

Abstract: 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.

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:2004.10883/code)