Learning interpretable and stable dynamical models via mixed-integer Lyapunov-constrained optimization

In this paper, we consider the data-driven discovery of stable dynamical models with a single equilibrium. The proposed approach uses a basis-function parameterization of the differential equations and the associated Lyapunov function. This modeling approach enables the discovery of both the dynamical model and a Lyapunov function in an interpretable form. The Lyapunov conditions for stability are enforced as constraints on the training data. The resulting learning task is a mixed-integer quadratically constrained optimization problem that can be solved to optimality using current state-of-the-art global optimization solvers. Application to two case studies shows that the proposed approach can discover the true model of the system and the associated Lyapunov function. Moreover, in the presence of noise, the model learned with the proposed approach achieves higher predictive accuracy than models learned with baselines that do not consider Lyapunov-related constraints.