Problem
Research questions and friction points this paper is trying to address.
Dynamic Mode Decomposition
Infinite-dimensional systems
Functional data
Koopman operator
Perron-Frobenius operator
Functional dynamic mode decomposition: Learning infinite-dimensional systems from data
🤖 AI Summary
Traditional dynamic mode decomposition (DMD) relies on discretization, making it difficult to directly handle infinite-dimensional systems. This work extends projection-based and exact DMD to infinite-dimensional spaces by integrating functional data analysis with Koopman and Perron–Frobenius operator theory, enabling the direct learning of finite-rank operators from functional data. The core contribution lies in overcoming finite-dimensional limitations by demonstrating that conventional DMD is merely a special case of functional DMD, thereby achieving a unified theoretical framework. The proposed algorithm has been successfully applied to graph-theoretic systems, ordinary differential equations, and stochastic differential equations, validating its effectiveness across diverse domains.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Dynamic mode decomposition
Infinite-dimensional systems
Functional data
Finite-rank operators
Koopman operator
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