Stabilized Neural Differential Equations for Learning Dynamics with Explicit Constraints

Alistair White, Niki Kilbertus, Maximilian Gelbrecht, Niklas Boers

Keywords: neural differential equations, neural ordinary differential equations, constraints, conservation laws, stabilization, dynamical systems, dynamics, scientific machine learning, physics-informed machine learning

TL;DR: We introduce a simple yet powerful stabilization method that allows us to learn dynamics with arbitrary explicit constraints from observed solution trajectories.

Abstract: Many successful methods to learn dynamical systems from data have recently been introduced. However, ensuring that the inferred dynamics preserve known constraints, such as conservation laws or restrictions on the allowed system states, remains challenging. We propose stabilized neural differential equations (SNDEs), a method to enforce arbitrary manifold constraints for neural differential equations. Our approach is based on a stabilization term that, when added to the original dynamics, renders the constraint manifold provably asymptotically stable. Due to its simplicity, our method is compatible with all common neural differential equation (NDE) models and broadly applicable. In extensive empirical evaluations, we demonstrate that SNDEs outperform existing methods while broadening the types of constraints that can be incorporated into NDE training.

Submission Number: 11082