Koopman for stochastic dynamics: error bounds for kernel extended dynamic mode decomposition
Provides rigorous theoretical error bounds for a widely used method (kEDMD) in stochastic dynamics, addressing a gap in understanding its accuracy.
The paper proves L∞-error bounds for kernel extended dynamic mode decomposition (kEDMD) approximants of the Koopman operator for stochastic dynamical systems, splitting the error into deterministic (fill distance) and probabilistic (Monte Carlo sampling) components, and illustrates bounds on Langevin-type SDEs.
We prove $L^\infty$-error bounds for kernel extended dynamic mode decomposition (kEDMD) approximants of the Koopman operator for stochastic dynamical systems. To this end, we establish Koopman invariance of suitably chosen reproducing kernel Hilbert spaces and provide an in-depth analysis of the pointwise error in terms of the data points. The latter is split into two parts by showing that kEDMD for stochastic systems involves a kernel regression step leading to a deterministic error in the fill distance as well as Monte Carlo sampling to approximate unknown expected values yielding a probabilistic error in terms of the number of samples. We illustrate the derived bounds by means of Langevin-type stochastic differential equations involving a nonlinear double-well potential.