Learning the Koopman Eigendecomposition: A Diffeomorphic Approach

Petar Bevanda, Johannes Kirmayr, Stefan Sosnowski, Sandra Hirche

Abstract: We present a novel data-driven approach for learning linear representations of a class of stable nonlinear systems using Koopman eigenfunctions. Utilizing the spectral equivalence of topologically conjugate systems, we construct Koopman eigenfunctions corresponding to the nonlinear system to form linear predictors of nonlinear systems. The conjugacy map between a nonlinear system and its Jacobian linearization is learned via a diffeomorphic neural network. The latter allows for a well-defined, supervised learning problem formulation. Given the learner is diffeomorphic per construction, our learned model is asymptotically stable regardless of the representation accuracy. The universality of the diffeomorphic learner leads to the universal approximation ability for Koopman eigenfunctions - admitting suitable expressivity. The efficacy of our approach is demonstrated in simulations.