Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds
Authors: Gustav Conradie, Nicolas Boullé, Jean-Christophe Loiseau, Steven L. Brunton, Matthew J. Colbrook
Organizations: Centre for Mathematical Sciences, University of Cambridge, UK · Department of Mathematics, Imperial College London, UK · Laboratoire DynFluid, Arts et Métiers Institute of Technology, France · Department of Mechanical Engineering, University of Washington, USA
Koopman operator theory provides a global linear representation of nonlinear dynamics and underpins many data-driven methods. In practice, however, finite-dimensional feature spaces induced by a user-chosen dictionary are rarely invariant, so closure failures and projection errors lead to spurious eigenvalues, misleading Koopman modes, and overconfident forecasts. This paper addresses a central validation problem in data-driven Koopman methods: how to quantify invariance and projection errors for an arbitrary feature space using only snapshot data, and how to use these diagnostics to produce actionable guarantees and guide dictionary refinement? A unified a posteriori methodology is developed for certifying when a Koopman approximation is trustworthy and improving it when it is not. Koopman invariance is quantified using principal angles between a subspace and its Koopman image, yielding principal observables and a principal angle decomposition (PAD), a dynamics-informed alternative to SVD truncation with significantly improved performance. Multi-step error bounds are derived for Koopman and Perron--Frobenius mode decompositions, including RKHS-based pointwise guarantees, and are complemented by Gaussian process expected error surrogates. The resulting toolbox enables validated spectral analysis, certified forecasting, and principled dictionary and kernel learning, demonstrated on chaotic and high-dimensional benchmarks and real-world datasets, including cavity flow and the Pluto--Charon system.
Figures & tables
Figure 1: Workflow for computing error bounds on Koopman operators and learning a dictionary.
Figure 2: A summary of the numerical examples of this paper and some of the results of the experiments applied to them. Principal observables are discussed in section 3 , and error bounds and expected errors are discussed in sections 4 and 5 , respectively.
Computed quantity
Pseudocode
Theoretical results
Principal angles and observables
algorithm 1
section 3.2
Principal angle decomposition
algorithm 2
section 3.3
Error bounds on KMD/PFMD
algorithms 3 and 4
section 4.2
Expected errors of KMD/PFMD
algorithms 5 and 6
section 5.2
Table 1: The key algorithms of this paper.
JV=(GVAV∗AVLV)∈C2n×2n,
Algorithm 1 Principal angles and observables between V and KV .
Figure 3: Left: The Duffing attractor. Middle: Principal angles on the Duffing oscillator for a finite subspace of Chebyshev polynomials. Right: One-step prediction errors for the corresponding principal observables. The solid line denotes a piecewise-linear best fit with a single breakpoint.
Figure 4: Left: Forecast errors for the full Koopman matrix on the Duffing oscillator versus the Koopman operator truncated using a SVD or PAD. Right: One-step prediction errors for Koopman operators truncated via PAD and SVD, shown as a function of the number of modes.
Figure 5: Left: Two trajectories on the Lorenz attractor. Middle: Exact prediction errors for the full Lorenz system and for truncated Koopman operators using PAD and SVD. Right: The dependence of one-step prediction error on the number of modes for both PAD and SVD.
Figure 6: Sharpness of bounds. The exact errors and error bound formula to first-order (minimum of eqs. 20 and 21 ) and full-order (minimum of eqs. 15 and 16 ) for a system with Lebesgue spectrum.
Figure 7: Exact L2 errors and error bounds for the Duffing oscillator (left) and Lorenz system (right) using a truncated SVD and PAD with the same number of modes. The PAD method outperforms the SVD for small numbers of time steps, in terms of both exact errors and error bounds.
Figure 8: Exact RKHS errors and error bounds eq. 24 for the Duffing oscillator (left) and Lorenz system (right) using a Matérn kernel.
Figure 9: Left: A snapshot of the cavity flow system. Right: Exact L2 -errors and error bounds for the cavity flow system. The error bounds overestimate the true error by a factor of over 10 .
Figure 10: L2 (top) and RKHS (bottom) norm forecast errors, error bounds and expected bounds for the Duffing oscillator (left) and Lorenz system (right), along with variance error bars including three standard deviations.
Figure 11: Exact error, error bounds, expected error and three standard deviation error bars for the cavity flow system. The expected error tracks the error much more closely than the strict bound.
Figure 12: Left: Orbits of moons in the Pluto–Charon system with Pluto at the origin. Right: RKHS-norm errors, strict error bounds, expected errors and three standard deviation error bars for the Pluto–Charon system. While the strict bounds overestimate the true error, the expected errors track it closely.
Figure 13: First row: Exact and predicted trajectories with kEDMD for the position of Styx (smallest body in the Pluto–Charon system) relative to Pluto. Second row: Exact relative pointwise errors together with the corresponding error bounds and expected errors. Third and fourth rows: As in the first and second rows, but for the velocities of Styx relative to Pluto.
Figure 14: Left: Averaged error bounds, exact L2 errors, and pointwise errors for the Pluto–Charon system over 500 iterations of the Adam stochastic optimization algorithm. Right: Comparison of exact L2 forecast errors and error bounds over 20 time steps for the initial parameter choice (marked by a cross) and the best parameters found by Adam (marked by a star).
Koopman theory promises linear structure in nonlinear dynamics, but numerical Koopman spectra are easy to compute and hard to trust. A finite EDMD matrix always has eigenvalues; the problem is that many of them may have nothing to do with the infinite-dimensional operator. In this paper we make spectral reliability the objective of dictionary learning. We train neural-network dictionaries not merely to predict the next snapshot, but to minimize Residual Dynamic Mode Decomposition residuals: operator-level a posteriori errors that test whether computed eigenvalues and modes are genuine Koopman spectral objects. To keep the learned observables from collapsing into an unstable coordinate system, the loss also penalizes the condition number of the lifted data matrix. Thus the method couples two requirements that should not be separated: small Koopman residuals and a well-conditioned representation. The result is a learned dictionary that is expressive, numerically stable, and spectrally disciplined. Across conservative and dissipative benchmark systems, the method sharply reduces spectral pollution, improves residual pseudospectral inclusion, and lowers forecast error relative to standard fixed dictionaries. On sea-surface temperature data, it gives cleaner Koopman diagnostics and substantially better one-step forecasts from noisy observations with no governing equations. The message is simple: neural Koopman learning should be judged not by prediction alone, but by whether its spectral claims can be certified. Residuals provide the certificate; conditioning makes it computable.
George Coote, Matthew J. Colbrook
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, CB3 0WA, United Kingdom
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.
The Koopman operator has gained considerable attention due to its ability to provide a global linear representation of highly complex dynamical systems. The operator describes nonlinear dynamics in a linear way through the lens of real- or complex-valued observable functions. Data-driven techniques, like extended dynamic mode decomposition (EDMD), kernel EDMD, and machine-learning methods, can be used to generate finite-dimensional approximations accompanied by finite-data error bounds. In this tutorial paper, we provide a concise introduction into Koopman operator theory and its use in systems and control. A particular focus is put on data-driven surrogate models, their extension to systems with inputs, and controller design using Koopman operator theory. Moreover, we demonstrate the key techniques, i.e., EDMD and Koopman MPC. To this end, we provide simulation studies including source code on GitHub to enable the interested reader to experience the Koopman operator in systems and control step by step.
Igor Mezić, Jorge Cortés, Karl Worthmann +2
Department of Mechanical Engineering, University of California, Santa Barbara, USA · Department of Mechanical and Aerospace Engineering, University of California, San Diego, USA · Optimization-based Control Group, TU Ilmenau, Germany +2