Learning the Koopman Eigendecomposition: A Diffeomorphic Approach
This work addresses the challenge of constructing linear predictors for nonlinear systems in control and dynamical systems, representing an incremental advance by integrating operator, system, and learning theories.
The authors tackled the problem of learning linear representations for stable nonlinear systems by introducing a diffeomorphic approach using Normalizing Flows to guarantee stability and universality in approximating Koopman eigenfunctions, with efficacy demonstrated through simulation examples.
We present a novel data-driven approach for learning linear representations of a class of stable nonlinear systems using Koopman eigenfunctions. By learning the conjugacy map between a nonlinear system and its Jacobian linearization through a Normalizing Flow one can guarantee the learned function is a diffeomorphism. Using this diffeomorphism, we construct eigenfunctions of the nonlinear system via the spectral equivalence of conjugate systems - allowing the construction of linear predictors for nonlinear systems. The universality of the diffeomorphism learner leads to the universal approximation of the nonlinear system's Koopman eigenfunctions. The developed method is also safe as it guarantees the model is asymptotically stable regardless of the representation accuracy. To our best knowledge, this is the first work to close the gap between the operator, system and learning theories. The efficacy of our approach is shown through simulation examples.