Koopman spectral analysis to capture underlying dynamic features in random pressure fields

Signal processing and system characterization tasks are often addressed utilizing state-space modeling methods where the evolution operator of a dynamical system is defined in phase space. The identification of this transient mapping by definition requires prior knowledge of dynamical mechanisms governing the underlying physics, which reversely poses challenges to dynamical systems analysis, especially when the dynamic equilibrium is not available or partially known. Fortunately, unprecedented data availability in this information explosion era facilitates effective learning of hidden dynamical mechanisms directly from data. This paper provides an equation-free learning method that describes a nonlinear/chaotic dynamical system from a functional space rather than the conventional phase space via the introduction of the Koopman operator. Spectral analysis of the Koopman operator provides a physical characterization of transient signals in terms of a set of dynamically coherent structures. The resolution scheme is designed in an eigendecomposition form where identified eigen-tuples can not only capture dominant spatial structures but also isolate each structure with a specific frequency and a corresponding growth/decay rate. To demonstrate the unique feature of this approach, we consider learning the evolution dynamics of random wind pressures fields over a tall building model using limited experimental data. The learned results indicate the proposed method can accurately identify a wide range of dynamical mechanisms, each of which is closely connected to a physical phenomenon.