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Functional dynamic mode decomposition: Learning infinite-dimensional systems from data
Stefan Klus, Eirini Ioannou
TL;DR
DMD is commonly applied to PDEs after discretization, making the problem finite-dimensional. This paper extends projected and exact DMD to learn operators from functional data; its examples demonstrate accurate dominant-eigenfunction estimates from few observations, while discretization recovers classical DMD as a special case.
Problem
Applying DMD to partial differential equations typically requires spatial discretization, while the paper targets learning from functional data for infinite-dimensional systems.
Source: §1 · p. 2Method
The paper extends projected and exact DMD to learn finite-rank operators from functional data rather than estimating matrices from vector-valued observations.
Source: §1 · p. 2, §4 · p. 21Results
The examples demonstrate accurate estimates of dominant eigenfunctions from a few observations, and discretized functional data recover classical DMD as a special case.
Source: §1 · p. 2, §4 · p. 22Takeaways & Limitations
Functional DMD is presented as a way to predict dynamics and analyze global system properties, including metastable sets and implied timescales.
Source: §1 · p. 2Recovering eigenfunctions requires initial conditions that excite them; using multiple initial conditions can mitigate this limitation.
Source: §2.2.2 · p. 5
Abstract
from arXiv · showhide
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.
1 Introduction
The paper extends DMD to learn best-fit operators and spectral information directly from functional data for infinite-dimensional systems, while relating the extensions to conventional DMD and generalized EDMD.
1 Introduction
- Functional data analysis replaces vectors with functions and matrices with compact operators, placing observations in an abstract infinite-dimensional Hilbert space.Source: §1 · p. 1
- DMD approximates transfer operators to compute spectral properties, with applications including metastable-set detection, forecasting, system identification, control, and spectral clustering.Source: §1 · p. 1
- Functional DMD learns a best-fit approximation of an infinite-dimensional system’s propagator and its eigenvalues and eigenfunctions directly from functional data, without necessarily discretizing continuous domains.Source: §1 · p. 2
- Prior work on infinite-dimensional functional data is limited; unlike generalized EDMD’s basis-functional dictionary, this framework represents the propagator using snapshots and focuses mainly on linear dynamics.Source: §1 · p. 2
- Projected functional DMD is a Galerkin projection, whereas exact functional DMD requires pseudoinverses of operators representing the training data.Source: §1 · p. 2
- Discretizing functional training data into vectors recovers classical DMD algorithms as special cases, and the paper also compares functional DMD with generalized EDMD.Source: §1 · p. 2
- The paper illustrates the methods with examples including random walks on graphons, Langevin dynamics, Koopman–von Neumann mechanics, and the Kuramoto–Sivashinsky equation.Source: §1 · p. 2
2 Functional DMD
This section derives two functional-data variants of DMD, analyzes their properties, and compares them with conventional DMD methods.
2.1 Problem setting and training data
The problem setting models dynamics on a separable Hilbert space and uses fixed-lag pairs of functional states as training data.
2.1 Problem setting and training data
- The state space H is a separable Hilbert space, and the system is generated by a linear operator W.Source: §2.1 · p. 3
- The generator W may be an integral or differential operator, while H may be a weighted square-integrable, Sobolev, or reproducing-kernel Hilbert space.Source: §2.1 · p. 3
- The heat equation on (0, 1) in L2 is an example, with the second spatial derivative as its linear operator.Source: §2.1 · p. 3
- Training data consist of functional states u_i and their evolved states v_i at a fixed lag time τ.Source: §2.1 · p. 3
- The state pairs can come from separate short simulations or experiments, one long trajectory, or a combination of both.Source: §2.1 · p. 3
2.2 Projected functional DMD
Projected functional DMD represents functional snapshots as a finite-dimensional subspace and computes a Galerkin operator from their Gram matrices. Its eigenfunctions and eigenvalues provide spectral estimates and forecasts for initial conditions in that subspace.
2.2.1 Functional data vectors
- Functional data vectors U and V collect the training functions as row-vectors, with the functions in U spanning a subspace of dimension at most m.Their Gram matrices support Galerkin projections and operator pseudoinverses.Source: §2.2.1 · pp. 3–4
2.2.2 Galerkin projection
- Projected functional DMD forms A=Cuu^-1Cuv as the Galerkin projection onto U, using the pseudoinverse of Cuu when it is not invertible.The eigenvalues and eigenvectors of A determine the projected eigenfunctions.Source: §2.2.2 · p. 4
- 1.00, 2.00, 3.01, and 4.02 are accurate estimates of the first four heat-equation frequencies from five pairs of functions.Figure 1 compares numerical and analytical eigenvalues and shows the first four computed eigenfunctions against the true eigenfunctions.Source: §2.2.2 · p. 5, Figure 1 · p. 5
- Functional snapshots can use kernel density estimates, spectral expansions, grids, or finite-element discretizations, and some representations avoid numerical integration for Gram matrices.For gridded data, more accurate numerical quadrature can replace the Euclidean inner product when approximating the Gram matrices.Source: §2.2.2 · p. 6
2.2.3 Best approximation
- Minimizing training-data prediction error yields A=Cuu^-1Cuv when Cuu is invertible.The derivation optimizes the sum of squared differences between A applied to each training function and its paired target.Source: §2.2.3 · p. 7
2.2.4 Spectral decomposition and forecasting
- The operator's eigenvalues and eigenfunctions express forecasts at multiples of the lag time, but only for initial conditions contained in U.The forecast is obtained by applying powers of A to the initial condition and expressing its evolution through eigenvalues, eigenfunctions, and modes.Source: §2.2.4 · p. 8
2.3 Exact functional DMD
Exact functional DMD extends DMD to functional data by learning a best-fit operator and its eigenfunctions; discretizing the functions recovers conventional exact DMD. Its forecasts are faithful in the heat-equation example, and its eigenvalues match those of projected functional DMD.
2.3.1 Functional data operators
- Functional data vectors can be treated as operators from coefficient vectors to functions, enabling operator singular-value decompositions and pseudoinverses through finite-dimensional Gram matrices.The operator U+ inverts U on functions in its range, while UU+ projects functions onto that range.Source: §2.3.1 · p. 8, §2.3.2 · p. 10
2.3.2 Singular value decomposition and pseudoinverse
- The singular-value decomposition of U is obtained from the eigendecomposition of U*U, whose eigenvalues are those of the Gram matrix Cuu.The singular values are σℓ, the right singular functions are θ(ℓ), and the corresponding left singular functions are computed from Uθ(ℓ).Source: §2.3.2 · p. 9
2.3.3 Best-fit operator and forecasting
- Exact functional DMD defines the best-fit operator as VU+, mapping each training function ui to its paired function vi when the ui are linearly independent.The algorithm computes eigenvalues and eigenvectors of a finite-dimensional matrix, then forms eigenfunctions of the learned operator; the theorem establishes that these are indeed eigenfunctions.Source: §2.3.3 · p. 10
- Both functional DMD variants produce faithful heat-equation forecasts, with exact functional DMD slightly more accurate as high-frequency errors are damped over time.For a noisy initial temperature profile poorly represented by the training functions, the initial approximation is inaccurate but the prediction improves over time.Source: §2.3.3 · p. 10, Figure 2 · p. 11
2.3.4 Comparison with projected functional DMD
- Projected and exact functional DMD have identical eigenvalues, and conventional projected and exact DMD arise as special cases when the function space is R^n.Projected eigenfunctions are written in terms of ui, whereas exact eigenfunctions are defined in terms of vi.Source: §2.3.4 · pp. 11–12
2.4 Relationships between functional DMD and generalized EDMD
Functional DMD and generalized EDMD are closely related for linear operators, but they differ in how they represent data and approximate Koopman operators. The paper also notes a basis limitation and leaves nonlinear functional DMD extensions for future work.
2.4 Relationships between functional DMD and generalized EDMD
- Functional DMD represents the propagator using snapshots and focuses on linear dynamics, whereas generalized EDMD projects Koopman operators onto preselected basis functionals.For linear operators, the two approaches are closely related.Source: §2.4 · pp. 12–13
- In one example, DMD yields Koopman modes rather than Koopman eigenfunctions, while a Galerkin approximation based on inner products with ui approximates eigenfunctions and eigenfunctionals.The latter construction uses Cuu and Cuv as its data matrices.Source: §2.4 · pp. 12–13
- The functions ui may not be a good basis for approximating eigenfunctions of the adjoint operator, and nonlinear functional DMD extensions are left for future work.Products and powers of principal eigenfunctionals can construct additional eigenfunctionals.Source: §2.4 · pp. 12–13
3 Applications
The applications section introduces potential uses of the functional DMD framework and presents numerical results.
3 Applications
- The section presents numerical results to illustrate potential applications of the proposed functional DMD framework.Source: §3 · p. 13
3.1 Graphons
The graphon application uses transfer operators to characterize random walks and their metastable clusters. Functional DMD applied to evolving densities can identify clusters and reconstruct graphons without access to walker positions.
3.1.1 Transfer operators for discrete-time random walks
- A graphon is a measurable function encoding edge weights or probabilities between continuum vertices; the paper considers connected symmetric graphons, whose random walks are reversible.For discrete-time walks, the Koopman and Perron–Frobenius operators describe the associated transfer dynamics.Source: §3.1 · p. 13, §3.1.1 · pp. 13–14
- The graphon can be reconstructed only up to a multiplicative factor because scaling it does not change transition probabilities.Source: §3.1.1 · p. 14
3.1.2 Transfer operators for continuous-time random walks
- For continuous-time random walks with exponentially distributed waiting times, Q and its adjoint govern observable and density evolution, while their negatives define the graphon Laplacians.In the three-cluster example, metastability makes convergence to the invariant density slow and is associated with eigenvalues close to one.Source: §3.1.2 · pp. 14–15, Figure 3 · p. 15
3.1.3 Learning graphons from functional data
- Using only evolving densities, functional DMD identifies graphon clusters and reconstructs the graphon up to a multiplicative constant; the eigenfunctions also support forecasting.The example extracts the invariant density before the simulation has nearly reached it.Source: §3.1.3 · pp. 15–16
- Three dominant eigenvalues followed by a spectral gap indicate three metastable sets; k-means on the corresponding eigenfunctions identifies clusters, and those functions reconstruct the graphon.The example also compares estimated eigenvalues with values from discrete-time random walks.Source: §3.1.3 · p. 16
3.2 Stochastic differential equations
The section applies functional DMD to Langevin dynamics, using evolving densities to estimate transfer-operator spectra and identify metastable structure. In the Himmelblau example, the analysis identifies three metastable sets and estimates their associated eigenvalues.
3.2.1 Transfer operators for Langevin dynamics
- For overdamped Langevin dynamics, the Kolmogorov backward and Fokker–Planck equations govern observables and densities, with Koopman and Perron–Frobenius propagators.The invariant Gibbs density satisfies L*π = 0, equivalently Pτπ = π.Source: §3.2.1 · p. 16
- Langevin systems can exhibit metastability; in the Himmelblau example, the close right-hand wells are treated as one, yielding an expectation of three metastable sets.The illustrated density spreads toward the potential wells by time τ but has not reached stationarity.Source: §3.2.1 · p. 16, Figure 5 · p. 17
3.2.2 Detecting invariant densities and metastable states
- Functional DMD estimates dominant Perron–Frobenius eigenfunctions from density estimates, which can reveal the invariant density, metastable sets, potential, timescales, and number of sets.The approach uses evolving functions rather than requiring individually tracked particle trajectories.Source: §3.2.2 · pp. 17–18
- Standard EDMD gave λ1≈1, λ2≈0.854, and λ3≈0.699, close to projected functional-DMD estimates; the analysis identified three metastable sets.The sets are extracted from the estimated eigenfunctions using SEBA.Source: Figure 6 · p. 17, §3.2.2 · p. 18
3.3 Koopman–von Neumann mechanics
The section applies functional DMD to Koopman–von Neumann wavefunctions for ordinary differential equations, including a linear system with analytically available spectral structure. The computed spectrum lies on the unit circle and the resulting spectral properties approximate analytic results and support forecasting.
3.3.1 The Koopman–von Neumann generator
- The Koopman–von Neumann generator describes wavefunction evolution; its skew-adjointness makes the fixed-lag propagator unitary and its finite-dimensional projection a unitary matrix.The framework could potentially support simulation of classical dynamical systems on quantum computers.Source: §3.3 · p. 18, §3.3.1 · pp. 18–19
3.3.2 Linear systems and invariant subspaces
- For a linear-system benchmark, a conservation law defines invariant subspaces where eigenvalues and eigenfunctions can be computed analytically.Products and powers of principal eigenfunctions generate further eigenfunctions; combinations can share an eigenvalue.Source: §3.3.2 · p. 19
3.3.3 Spectral decomposition and forecasting
- Exact functional DMD estimates the propagator’s eigenvalues and eigenfunctions from functional time-series data for the Koopman–von Neumann equation.The example uses a 20-dimensional invariant subspace and 15 snapshot pairs.Source: §3.3.3 · p. 19
- Computed eigenvalues lay on the unit circle, with reported multiplicities 3, 3, 2, and 1; the estimated spectral properties approximated analytic results and supported prediction.The eigenfunctions represent non-decaying periodic patterns with different frequencies.Source: §3.3.3 · p. 20, Figure 7 · p. 20
3.4 Kuramoto–Sivashinsky equation
The section tests exact functional DMD on the two-dimensional Kuramoto–Sivashinsky equation, treating its nonlinear right-hand side through an estimated linear operator. The resulting spectrum is not confined to the unit circle, and functional DMD is presented as a way to project infinite-dimensional data onto slowly evolving dynamics.
3.4 Kuramoto–Sivashinsky equation
- Because the nonlinear right-hand side is approximated by a linear operator, the eigenfunctions are less clearly interpretable, though the operator may retain information about global dynamics.The example uses a periodic domain and a 128×128 Fourier basis.Source: §3.4 · p. 20
- For the Kuramoto–Sivashinsky example, exact functional DMD estimates eigenfunctions and a spectrum whose eigenvalues do not lie on the unit circle.The analysis uses 200 snapshots with lag time τ = 1.Source: Figure 8 · p. 21, §3.4 · p. 21
- Functional DMD can project infinite-dimensional data onto slowly evolving dynamics, where Markov state models or manifold-learning methods can then be constructed.A few modes or time-lagged components can capture characteristic behavior such as metastability in complex multiscale systems.Source: §3.4 · p. 21
4 Conclusion
The conclusion presents projected and exact functional DMD as finite-rank operator-learning methods for infinite-dimensional systems, with classical DMD recovered through spatial discretization. The approach works directly with functions and is illustrated across several dynamical-system settings, while the authors identify open questions around dimensionality, limits, and spurious eigenvalues.
4 Conclusion
- Projected and exact functional DMD learn finite-rank operators from functional data, and conventional DMD algorithms arise as special cases after discretizing the domain and evaluating functions on grid points.The methods are applied to transfer operators associated with ODEs, SDEs, and graphons.Source: §4 · p. 21
- Working with observables, densities, or wavefunctions removes the need to track particles, which is advantageous when only density estimates are available or particles cannot be distinguished.Source: §4 · p. 21
- Open issues include the infinite-snapshot limit, representing functions in high-dimensional state spaces, extending the algorithms to nonlinear infinite-dimensional systems, and functional DMD’s potential for spurious eigenvalues.The authors also suggest that incorporating domain knowledge could improve accuracy and efficiency.Source: §4 · p. 22