Functional dynamic mode decomposition: Learning infinite-dimensional systems from data

Dynamic mode decomposition (DMD) is a data-driven method that computes the best linear approximation of the underlying dynamical system and decomposes the dynamics into a superposition of characteristic spatiotemporal patterns. Originally introduced by the fluid dynamics community, DMD and its extensions have found widespread use in many other research areas such as molecular dynamics, climate science, engineering, finance, and neuroscience. Applications include dimensionality reduction, forecasting, system identification, control, and spectral clustering. In order to apply DMD to partial differential equations, the spatial domain is typically first discretized using finite difference or finite element techniques, thus implicitly rendering the problem finite-dimensional. We extend projected and exact DMD to infinite-dimensional systems. Rather than estimating matrices from vector-valued observations, our DMD variants learn finite-rank operators from functional data such as observables, densities, or wavefunctions. We show that conventional DMD algorithms can be regarded as special cases of their functional DMD counterparts. All results will be illustrated with the aid of guiding examples. We focus in particular on Koopman, Perron-Frobenius, and Koopman-von Neumann operators associated with graphons, ordinary differential equations, and stochastic differential equations.