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
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 ui and vi given by solutions of the heat equation at different times ti and ti+τ , where ti=(i−1)τ . (b) Comparison of the numerically computed eigenvalues (in blue) and the analytically computed eigenvalues e−ℓ2π2τ (in red). (c) First four numerically computed eigenfunctions. The dotted lines represent the true eigenfunctions sin(ℓπx) .
Figure 2 : (a) Eigenfunctions φℓ 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 ti=(i−1)τ . 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 ui or vi . 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 ρ in time. The initial density, a Gaussian with bandwidth σ=0.2 centered at x=21 , spreads to the other clusters and converges to the invariant density, represented by the dotted blue line.
Figure 4 : (a) Dominant eigenvalues μℓ of P . The red crosses, shown for comparison, are the eigenvalues estimated from one long discrete-time random walk. (b) Eigenfunctions of P , where denotes the first, the second, and the third eigenfunction. The black dots represent the true invariant density π . (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 τ . 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 (ℓ1,ℓ2,ℓ3) 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.
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.
Cankat Tilki, Serkan Gugercin
Department of Mathematics, Virginia Tech, Blacksburg, VA - 24061.
Koopman theory turns nonlinear dynamics into a linear spectral problem. In computation, however, everything depends on a hard finite-dimensional choice: the observables must be expressive, nearly invariant under the dynamics, and, ideally, compatible with composition. Deep Koopman methods learn flexible coordinates, whereas structure-preserving methods enforce operator identities on fixed dictionaries. We combine these ideas by introducing Deep Embedded Multiplicative Dynamic Mode Decomposition (DeepMDMD), a method that learns a latent space and a partition of it, while enforcing the Koopman product rule as an exact algebraic constraint. Training alternates between an exact multiplicative operator update and a differentiable latent-clustering step that promotes Koopman closure. The result is a finite transition map on learned latent cells. Its nonzero spectrum lies on the unit circle, its dictionary is shaped by the dynamics rather than by ambient geometry, and forecasts are made in latent coordinates before being decoded to physical space. Across Hamiltonian, chaotic, and fluid examples, DeepMDMD learns dictionaries that are far more compact and dynamically coherent than those produced by geometric MDMD partitions. It reduces spectral pollution, reveals richer continuous-spectrum structure, and gives stable forecasts under severe noise. In high-dimensional flows, including a 158,624-dimensional cylinder wake and a noisy Re=20,000 lid-driven cavity, it preserves coherent structures and long-time spectral statistics where state-space MDMD fails. These results suggest a practical rule for Koopman learning: learn the coordinates, constrain the algebra.
Kelan Gray, Finlay Brown, Nicolas Boullé +1
Department of Mathematics, Imperial College London, London, SW7 2AZ, UK. · Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK.
Nonlinear coupled systems are ubiquitous in science and engineering. The analysis and modeling of such systems is challenging due to their high dimensionality and complex interactions among subsystems. In recent years, operator-theoretic methods based on the Koopman operator have attracted attention as a powerful tool for analyzing and modeling nonlinear dynamical systems. Extended dynamic mode decomposition (EDMD) is one of the most popular methods to approximate the Koopman operator. However, EDMD is a purely data-driven method, and it could be unstable and inaccurate for coupled systems under limited data availability. In this paper, we propose a method to learn the Koopman operator for coupled systems using the differential equations governing each subsystem. We also demonstrate its effectiveness through numerical experiments on coupled oscillator systems.
Tatsuya Naoi, Jun Ohkubo
Graduate School of Science and Engineering, Saitama University, Sakura, Saitama, 338-8570, Japan