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

Stefan Klus; Eirini Ioannou

math.DSSep 24, 2026

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

Authors: Stefan Klus, Eirini Ioannou

Organizations: School of Mathematical & Computer Sciences, Heriot–Watt University, Edinburgh, UK · Maxwell Institute for Mathematical Sciences, University of Edinburgh and Heriot–Watt University, Edinburgh, UK

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.

Figures & tables

Figure 1 : (a) Functions

u_{i}

and

v_{i}

given by solutions of the heat equation at different times

t_{i}

and

t_{i}+\tau

, where

t_{i}=(i-1)\hskip 1.00006pt\tau

. (b) Comparison of the numerically computed eigenvalues (in blue) and the analytically computed eigenvalues

e^{-\ell^{2}\pi^{2}\tau}

(in red). (c) First four numerically computed eigenfunctions. The dotted lines represent the true eigenfunctions

\sin(\ell\hskip 1.00006pt\pi\hskip 1.00006ptx)

Figure 2 : (a) Eigenfunctions

\widetilde{\varphi}_{\ell}

of the heat equation computed using exact functional DMD. The dotted lines represent the analytically computed eigenfunctions. (b) Prediction of the dynamics using exact functional DMD, where

t_{i}=(i-1)\tau

. The dotted lines in the same color represent the true solution and the gray dotted lines the projected DMD prediction. (c) Forecasting for an oscillatory temperature profile that cannot be faithfully represented by the functions

u_{i}

v_{i}

. Although the initial approximation is inaccurate, high frequencies are smoothed out quickly and the prediction improves over time.

Figure 3 : (a) Symmetric graphon $w$ with three peaks at $0.2$ , $0.5$ , and $0.8$ . The peak in the middle is less metastable than the other two. (b) Corresponding transition density function $p$ . (c) Evolution of a probability density $\rho$ in time. The initial density, a Gaussian with bandwidth $\sigma=0.2$ centered at $x=\frac{1}{2}$ , spreads to the other clusters and converges to the invariant density, represented by the dotted blue line.

Figure 4 : (a) Dominant eigenvalues $\mu_{\ell}$ of $\mathcal{P}$ . The red crosses, shown for comparison, are the eigenvalues estimated from one long discrete-time random walk. (b) Eigenfunctions of $\mathcal{P}$ , where denotes the first, the second, and the third eigenfunction. The black dots represent the true invariant density $\pi$ . (c) Rank-3 reconstruction of $w$ using the estimated eigenfunctions. The resulting graphon is virtually indistinguishable from the true graphon shown in Figure 3 (a). The dotted gray lines separate the identified clusters.

Figure 5 : (a) Visualization of the Himmelblau potential comprising two separate wells in the left half plane and two partially merged wells in the right half plane. The blue line represents a single non-equilibrated trajectory. (b) Initial density given by a kernel density estimate computed from 5000 initial conditions sampled from a Gaussian distribution with randomly generated center and bandwidth. (c) Estimate of the density at time $\tau$ . The initial density spreads to the wells of the Himmelblau potential, but has clearly not reached the stationary distribution yet.

Figure 6 : (a) Estimated invariant density. The dotted lines mark the three identified metastable sets. (b) The second eigenfunction separates the well in the lower left corner from the others. (c) The third eigenfunction distinguishes between the well in the upper left corner and the other wells. Combining this information allows us to extract the three metastable sets shown in (a).

Figure 7 : (a) Analytically computed eigenfunctions of the Koopman–von Neumann generator associated with the linear system. The tuples $(\ell_{1},\ell_{2},\ell_{3})$ are the corresponding “quantum numbers”. (b) Numerically (blue) and analytically (black) computed eigenvalues. A darker blue implies a higher multiplicity. Since the eigenspaces are not one-dimensional, eigenvectors and hence eigenfunctions are not uniquely determined. That is, the numerically computed eigenfunctions can be superpositions of the analytically computed eigenfunctions associated with the same eigenvalue.

Figure 8 : (a) Top row: Solutions of the Kuramoto–Sivashinsky equation at different times $t$ . Bottom row: A few select numerically computed eigenfunctions of the estimated linear operator. (b) Spectrum of the operator. The eigenvalues do not lie on the unit circle in this case.

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Wavelet-Based Observables for Koopman Analysis: An Extended Dynamic Mode Decomposition Framework

Cankat Tilki, Serkan Gugercin

Department of Mathematics, Virginia Tech, Blacksburg, VA - 24061.

Dynamic Mode Decomposition Koopman Operator Wavelets Spectral Decomposition

Open paper AskPDF arXiv

We present an in-depth analysis of the Koopman semigroup via wavelet transform. Towards this goal, we start by introducing the wavelet-based observables and show that they are eigenfunctions of the Koopman semigroup when this semigroup is considered over the Banach space of continuous functions on a compact forward-invariant set endowed with the supremum norm. We then construct closed-form expressions of the action of the Koopman semigroup and its resolvent in terms of these observables. To approximate the action of Koopman semigroup numerically, we combine Extended Dynamic Mode Decomposition (EDMD) with the proposed wavelet-based observables leading to the Wavelet Dynamic Mode Decomposition via Continuous Wavelet Transform (cWDMD) algorithm. We validate our theoretical results on two numerical examples.

Abstract

Explore similar work

Wavelet-Based Observables for Koopman Analysis: An Extended Dynamic Mode Decomposition Framework

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