Learning Neural Koopman Operators with Dissipativity Guarantees
Yuezhu Xu, S. Sivaranjani, Vijay Gupta
TL;DR
This work develops a data-driven framework to learn neural Koopman operators for nonlinear dissipative systems, guaranteeing that the learned lifted dynamics preserve dissipativity and that these guarantees extend to the original system. The method uses a two-stage process: first train an unconstrained neural Koopman model to fit the data, then apply minimal LMIs-based perturbations to enforce strict dissipativity in the lifted space. A theoretical link, with explicit error terms from noisy data, operator truncation, and generalization, ensures that dissipativity guarantees on the learned model transfer to the true nonlinear system, enabling reliable closed-loop performance. The approach is validated on a Duffing oscillator, showing that dissipativity can be enforced with negligible perturbation while maintaining close agreement with the true dynamics away from equilibrium.
Abstract
We address the problem of learning a neural Koopman operator model that provides dissipativity guarantees for an unknown nonlinear dynamical system that is known to be dissipative. We propose a two-stage approach. First, we learn an unconstrained neural Koopman model that closely approximates the system dynamics. Then, we minimally perturb the parameters to enforce strict dissipativity. Crucially, we establish theoretical guarantees that extend the dissipativity properties of the learned model back to the original nonlinear system. We realize this by deriving an exact relationship between the dissipativity of the learned model and the true system through careful characterization of the identification errors from the noisy data, Koopman operator truncation, and generalization to unseen data. We demonstrate our approach through simulation on a Duffing oscillator model.