Evaluating the accuracy of the dynamic mode decomposition

Dynamic mode decomposition (DMD) gives a practical means of extracting dynamic information from data, in the form of spatial modes and their associated frequencies and growth/decay rates. DMD can be considered as a numerical approximation to the Koopman operator, an infinite-dimensional linear operator defined for (nonlinear) dynamical systems. This work proposes a new criterion to estimate the accuracy of DMD on a mode-by-mode basis, by estimating how closely each individual DMD eigenfunction approximates the corresponding Koopman eigenfunction. This approach does not require any prior knowledge of the system dynamics or the true Koopman spectral decomposition. The method may be applied to extensions of DMD (i.e., extended/kernel DMD), which are applicable to a wider range of problems. The accuracy criterion is first validated against the true error with a synthetic system for which the true Koopman spectral decomposition is known. We next demonstrate how this proposed accuracy criterion can be used to assess the performance of various choices of kernel when using the kernel method for extended DMD. Finally, we show that our proposed method successfully identifies modes of high accuracy when applying DMD to data from experiments in fluids, in particular particle image velocimetry of a cylinder wake and a canonical separated boundary layer.