Speaker
Description
Koopman operators and transfer operators represent nonlinear dynamics in state space through its induced action on linear spaces of observables and measures, respectively. This framework enables the use of linear operator theory for analysis and modeling of nonlinear dynamical systems, and has received considerable interest over the years from mathematical, computational, and domain-scientific disciplines. In this talk, we present data-driven techniques for spectral approximation of Koopman and transfer operators of measure-preserving ergodic flows that are based on operators with compact resolvent. Our approach performs a bounded transformation of the Koopman generator (an operator implementing directional derivatives of observables along the dynamical flow), followed by smoothing by a Markov semigroup of kernel integral operators. This results in a skew-adjoint, compact operator with compact resolvent whose eigendecomposition is expressible as a variational generalized eigenvalue problem amenable to approximation using Galerkin methods. A key aspect of these methods is that they are physics-informed, in the sense of making direct use of dynamical vector field information through automatic differentiation of kernel functions. Solutions of the eigenvalue problem reconstruct evolution operators that preserve unitarity of the underlying Koopman group while spectrally converging to it in a suitable limit. In addition, the computed eigenfunctions have representatives in a reproducing kernel Hilbert space, enabling out-of-sample evaluation of learned dynamical features. We illustrate this method with numerical experiments on integrable and chaotic systems.