Modal Analysis of Spatiotemporal Data via Multi-fidelity Multi-variate Gaussian Processes

This paper focuses on the challenges associated with the existing dynamic mode decomposition (DMD) techniques for the modal analysis of spatiotemporal data, such as spectral pollution, noisy measurements, missing data, and multi-fidelity datasets. A methodology based on Multi-fidelity multi-variate Gaussian process regression (M2GPR ) is employed to address these challenges. The M2GPR method leverages the connection between Gaussian processes and the spectral representations of linear systems, and further extends to the analysis of nonlinear systems via the Koopman formalism. The capability of M2GPR is endowed by its judiciously designed kernel structure for correlation function, that emulates the linear dynamics in the state space or in the Koopman space; furthermore, the learning of correlation function does not require uniform sampling of time steps and thus can handle sparse datasets with relative ease. The single-fidelity portion of M2GPR method is demonstrated on a range of examples, with benchmarks against the DMD method. It manifests itself as a promising alternative to conventional modal analysis methods, esp in the limit of sparse and noisy datasets. In the case of noisy measurement, DMD always includes noise in every identified mode, whereas M2GPR captures a clear image for each mode. Additionally, M2GPR outperforms DMD in learning the frequencies and modes from sparse dataset, requiring as little as 20% of the dataset. Lastly, some potential extensions of the M2GPR method for modal analysis are discussed.