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Data-driven approximation of the Koopman generator: Model reduction, system identification, and control
Stefan Klus, Feliks Nüske, Sebastian Peitz, Jan-Hendrik Niemann, Cecilia Clementi, Christof Schütte
TL;DR
The paper addresses how to approximate Koopman generators and use them for analysis and learning of deterministic and stochastic dynamical systems. It introduces gEDMD, an EDMD extension that computes generator spectral quantities and supports system identification, coarse-graining, and control. The resulting framework recovers governing equations and provides model-reduction and predictive-control applications, subject to basis, derivative, and sampling constraints.
Problem
The study seeks a general data-driven framework for approximating Koopman generators and applying them to deterministic and stochastic systems.
Method
gEDMD reformulates EDMD to estimate Koopman-generator matrices and their eigenvalues, eigenfunctions, and modes from data, supporting system identification, coarse-graining, and control.
Results
The methods recover deterministic and stochastic governing-equation components and support coarse-grained models and efficient model predictive control strategies.
Takeaways & Limitations
gEDMD unifies generator-based spectral analysis with system identification, model reduction, and control, while deterministic equation recovery reduces to SINDy.
Takeaways & Limitations
Learned models depend on dictionary expressivity and accurate derivative estimates; matrix-logarithm approaches additionally require sufficiently small sampling times.
Abstract
from arXiv · showhide
We derive a data-driven method for the approximation of the Koopman generator called gEDMD, which can be regarded as a straightforward extension of EDMD (extended dynamic mode decomposition). This approach is applicable to deterministic and stochastic dynamical systems. It can be used for computing eigenvalues, eigenfunctions, and modes of the generator and for system identification. In addition to learning the governing equations of deterministic systems, which then reduces to SINDy (sparse identification of nonlinear dynamics), it is possible to identify the drift and diffusion terms of stochastic differential equations from data. Moreover, we apply gEDMD to derive coarse-grained models of high-dimensional systems, and also to determine efficient model predictive control strategies. We highlight relationships with other methods and demonstrate the efficacy of the proposed methods using several guiding examples and prototypical molecular dynamics problems.
1 Introduction
The paper introduces gEDMD, a data-driven extension of EDMD for approximating Koopman generators in deterministic and stochastic systems. It develops applications in system identification, model reduction, and control.
- The study presents a general framework for computing matrix approximations of Koopman generators for deterministic and stochastic systems.
- gEDMD extracts generator eigenvalues, eigenfunctions, and modes directly from data without trajectory integration.
- Empirical generator estimates can recover governing equations for deterministic and stochastic dynamical systems and connect gEDMD with SINDy and related methods.
- The method supports coarse-grained modeling from full-system data and flexible, efficient model predictive control strategies.
- The proposed methods are demonstrated using guiding examples and illustrative benchmark problems.
2 The Koopman operator and its generator
This section defines Koopman operators and their infinitesimal generators for deterministic and stochastic dynamics, then describes Galerkin approximations from basis functions and data.
- Deterministic dynamical systems: For deterministic dynamics, the Koopman semigroup evolves observables through the system flow, while its infinitesimal generator governs their time evolution.
- Non-deterministic dynamical systems: For stochastic differential equations, the generator includes drift and diffusion contributions and yields the Kolmogorov backward equation.
- Non-deterministic dynamical systems: The adjoint generator produces the Fokker–Planck or Kolmogorov forward equation for stochastic systems.
- Galerkin approximation: Galerkin approximation represents observables in a finite basis and estimates projected generator matrices from integrals or data.
- Generator eigenvalues can represent decay time scales through their reciprocals, called implied time scales.
- Non-compactness and continuous spectra are outside the paper’s scope because their numerical treatment is often challenging and may require regularization.
3 Infinitesimal generator EDMD
The paper reformulates EDMD to approximate infinitesimal generators directly from data, naming the resulting method gEDMD.
- EDMD can be reformulated to compute Koopman or Perron–Frobenius infinitesimal generators from data; the resulting method is called gEDMD.
3.1 Deterministic dynamical systems
For deterministic systems, gEDMD fits generator action on dictionary functions using state and derivative data, enabling spectral analysis and recovery of governing equations. Noise may require thresholding or other sparsification techniques.
- Generator approximation: gEDMD uses measured states, corresponding time derivatives, and a chosen dictionary of basis functions to approximate generator action.
- Generator approximation: The generator matrix is estimated by solving a least-squares problem matching differentiated dictionary evaluations to their basis-function evaluations.
- Generator approximation: The generator may be sparse even when the Koopman operator for a finite time map is not.
- System identification: Diagonalizing the learned generator provides eigenvalues and eigenfunctions, while full-state observables enable reconstruction of the governing equations and Koopman modes.
- System identification: For the guiding deterministic example, both the Koopman-mode representation and the direct basis-function representation identify the governing equations correctly.
- System identification: When derivative estimates are noisy or inaccurate, denoising, total-variation regularization, or iterative hard thresholding may be needed to remove spurious nonzero entries.
- Conservation laws: Conserved quantities correspond to generator eigenfunctions with eigenvalue λ = 0, allowing gEDMD to seek non-trivial invariants from data.
3.2 Non-deterministic dynamical systems
For stochastic differential equations, gEDMD approximates the generator from data and supports eigenfunction computation, system identification, and diffusion recovery. Its accuracy depends on basis selection, derivative estimates, and data quality.
- The stochastic formulation requires first and second derivatives of basis functions, which can be precomputed analytically.
- gEDMD converges in the infinite-data limit to a Galerkin approximation of the stochastic generator.
- Generator approximation: Using 100 test points, gEDMD produced Ornstein–Uhlenbeck Koopman eigenfunctions virtually indistinguishable from the analytical solution.Replacing monomials with Gaussian basis functions considerably improved the Perron–Frobenius eigenfunctions.
- Generator approximation: gEDMD recovered dominant Perron–Frobenius eigenfunctions for a metastable double-well system using 30000 test points and 300 radial basis functions.The second eigenfunction clearly separates the two wells.
- System identification: The method identifies drift and diffusion terms, recovering a=σσ⊤ rather than σ itself.For the Ornstein–Uhlenbeck process, drift is recovered correctly, while monomials only through order three recover drift but not diffusion.
- System identification: Learned stochastic models depend crucially on whether drift and diffusion lie in the dictionary and whether derivatives are estimated accurately.Noisy data can produce nonsparse solutions, while noise may cause the diffusion term to be overestimated.
3.3 Relationships with other methods
The paper relates gEDMD to SINDy, Koopman lifting, and KRONIC. These methods differ in data requirements, generator construction, and assumptions about derivatives, sampling, or known eigenvalues.
- For deterministic systems, SINDy is a special case of gEDMD because both use states and corresponding time derivatives.
- Koopman lifting estimates the Koopman operator from time-lagged data and obtains the generator through a matrix logarithm.
- Koopman lifting avoids time derivatives and basis-function partial derivatives but requires sufficiently small lag times to control matrix-logarithm ambiguity.Its spectral accuracy is constrained by the sampling rate.
- With exact training derivatives, gEDMD is generally more accurate than Koopman lifting; when derivatives are approximated, accuracy depends on finite-difference order and step size.
- KRONIC discovers Koopman eigenfunctions for control and conservation-law detection, while assuming an eigenvalue is known or learned simultaneously.Despite different derivations, it is strongly related to gEDMD for deterministic systems.
4 Further applications
The Koopman generator has applications beyond identifying fast and slow modes, governing equations, and conservation laws. These further applications are briefly demonstrated.
- The section extends beyond identifying fast and slow modes.
- It also considers applications beyond identifying governing equations.
- Conservation-law identification is likewise not the section’s only focus.
4.1 Coarse-grained dynamics and gEDMD
gEDMD provides a data-driven Galerkin approximation of coarse-grained Koopman generators, enabling reduced-model identification from full-system data. The framework supports direct and separate identification, with tests showing accurate effective parameters and implied time scales.
- 4.1.1 Galerkin approximation: gEDMD converges to a Galerkin approximation of the coarse-grained generator using basis functions defined on the reduced space.Applying gEDMD to lifted reduced functions yields the corresponding Galerkin projection.
- 4.1.1 Galerkin approximation: Data sampling the invariant distribution of the original process can identify a matrix representation of the coarse-grained generator for system identification, simulation, and control.The reduced generator matrix is used analogously to the full-system generator model.
- 4.1.2 Separate Identification: Separate identification estimates the reduced potential and diffusion independently for reversible systems, then combines them to define the coarse-grained dynamics.The approach exploits reversibility, which characterizes the dynamics through a scalar potential and diffusion field.
- 4.1.2 Separate Identification: Separate basis sets permit constraints such as diffusion positive definiteness, while the resulting dynamics are automatically reversible and their spectrum can be used for validation.The potential and diffusion can also be learned with nonlinear models, and generator spectra can be compared directly.
- 4.1.2 Separate Identification: Direct system identification is more general because local-mean-force reconstruction may be inaccurate when dynamics orthogonal to the reaction-coordinate manifold are slow.This is the stated boundary of the separate-identification approach.
- 4.1.3 Example 1: Lemon-slice potential: In the lemon-slice potential, both methods accurately represent effective parameters, while the separately learned diffusion is virtually indistinguishable from the analytical solution.The polynomial-basis diffusion is more oscillatory, whereas the learned diffusion uses a Gaussian basis.
- 4.1.3 Example 1: Lemon-slice potential: The first three implied time scales from gEDMD agree very well with a Markov state model, and the optimal-diffusion generator matches them almost perfectly.The comparison uses the three nontrivial slowest generator eigenvalues.
- 4.1.4 Example 2: Alanine dipeptide: For alanine dipeptide, gEDMD applied to one million Langevin-dynamics snapshots reproduces reference implied time scales after rescaling by an effective time unit.The same agreement holds for time scales based on the generator matrix corresponding to the optimal diffusion.
4.2 Control
The Koopman generator supports model predictive control for deterministic and stochastic systems while allowing flexible time steps and prediction of expected values without repeated sampling. Examples with Burgers dynamics and the Ornstein–Uhlenbeck process demonstrate improved tracking and accurate stochastic predictions.
- Control: MPC repeatedly solves a finite-horizon tracking problem, applies only the first control segment, and then shifts the horizon.The formulation penalizes control effort through α and is resolved while the plant runs.
- Control: Koopman-generator surrogate models enable MPC by replacing system dynamics with a linear prediction model in observable coordinates.The observable transformation z = ψ(f(x)) yields a linear system through the generator approximation.
- Control: Generator-based control removes the finite-lag restriction of Koopman-operator surrogates, permitting arbitrary and adaptive control time steps.Discrete-time operator models restrict switching points, whereas the generator also supports efficient second-order methods for switching-time optimization.
- Partial differential equations: Burgers-equation tracking improves significantly as the constant-input time step decreases, with the generator framework reducing error by two orders of magnitude.The generator permits adaptive time-step selection without new data and computations when the lag changes.
- Stochastic differential equations: For the controlled Ornstein–Uhlenbeck process, generator predictions accurately reproduce expected values, while MPC and switching-time control achieve strong tracking performance.The open-loop switching experiment uses p = 200 switches, and the generator trajectory is almost indistinguishable from the simulated expected value.
5 Conclusion
The paper extends EDMD to data-driven Koopman and Perron–Frobenius generator approximation and demonstrates applications in model reduction, system identification, and control. It also identifies unresolved questions about convergence, spectral approximation, and coarse-grained dynamics.
- Conclusion: The proposed framework approximates Koopman or Perron–Frobenius generators from data and supports decomposition into eigenvalues, eigenfunctions, and modes.The implementation is available in Python with code and examples released online.
- Conclusion: The framework connects generator approximation with system identification, with SINDy emerging as a special case for deterministic dynamics.The paper also discusses governing-equation identification for deterministic and stochastic systems.
- Open questions: Open questions concern convergence as both data and basis size grow, spectral approximation without a pure point spectrum, and coarse-graining from dominant spectral terms.These questions define boundaries for the current theoretical understanding and modeling scope.