Functional dynamic mode decomposition learns infinite dimensional systems from data
Functional dynamic mode decomposition: Learning infinite-dimensional systems from data
Machine Learning
Summary
Dynamic mode decomposition (DMD) is a technique to simplify and understand complex changing systems by breaking them into patterns. The authors extend DMD to work directly with infinite-dimensional data, like continuous functions, rather than limiting it to simpler finite data. This lets them analyze systems modeled by equations involving continuous spaces without approximating them as finite sets. Their approach connects to important mathematical operators used to describe dynamics in areas like networks and random processes, helping to generalize DMD for broader scientific use.
What this means in practice
- •For climate modelers: Analyze continuous spatial climate data without discretizing the domain for better system understanding and prediction.
- •For financial engineers: Model and forecast complex infinite-dimensional financial systems using functional data instead of finite discretizations.
Tested on simulated data.
Authors
Stefan Klus, Eirini Ioannou
Abstract
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.