Bilevel optimization for data-driven learning of Koopman embeddings using kernel-based autoencoders

arXiv:2610.12370 · cs.LG, math.DS, stat.ML · Submitted 2026-10-08 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: I'm Tom, and with me are Jane, Lu, senior AI researcher at Tsinghua, Meng, lead engineer at a mysterious AI startup and Lalam, the in-house Large Language Model.

Jane: Today's paper: "Bilevel optimization for data-driven learning of Koopman embeddings using kernel-based autoencoders".

Tom: The gist The proposed method, extended dynamic mode decomposition with kernel-based dictionary learning (EDMD-kDL),

Jane: First, who's behind it and why it matters.

Paper summary: Tom: So, we're looking at this paper called "Bilevel optimization for data-driven learning of Koopman embeddings using kernel-based autoencoders." Basically, they're tackling a big problem in modeling nonlinear systems. Jane The core issue they address is that standard methods like extended dynamic mode decomposition or EDMD need you to pick the dictionary of functions beforehand, which is really hard in practice because you don't know what functions are going to be important.

Lu: And this paper claims they are learning that dictionary directly from the data using machine learning, specifically with kernel-based autoencoders. Meng So they’re trying to solve the dictionary selection problem by letting the data tell them what functions are useful for approximating the Koopman operator.

Tom: It seems their main thesis is using a bilevel optimization approach combined with kernel methods to learn both the dictionary and the Koopman approximation simultaneously. Jane They argue that this way, you avoid having to specify a dictionary in advance, which is what makes it more data-driven and flexible.

Lu: The paper suggests this method uses ideas from collocation methods and bilevel optimization to handle the learning process. Meng That means they aren't just training one thing; they're optimizing two things at once: how the dictionary should look, and how well that dictionary approximates the Koopman operator.

Tom: It seems like the central claim is that EDMD-kDL achieves a performance level comparable to or even better than those traditional artificial neural network based approaches when tested on various problems. Jane So, for someone who just listens to this show, it means we might get a way to build these useful Koopman embeddings without spending all our time guessing the right functions.

Lu: And they also point out that unlike standard kernel EDMD methods, the size of their learned dictionary isn't tied directly to how many training data points you have. Meng That’s interesting because usually, in these setups, you expect a lot of data to drive the complexity up significantly.

Conclusion: Tom: So we wrap up this discussion on "Bilevel optimization for data-driven learning of Koopman embeddings using kernel-based autoencoders." The authors are Joel-Pascal Ntwali N’konzi, Feliks Nüske, and Stefan Klus. Jane They are essentially proposing a new framework to get finite-dimensional Koopman embeddings directly from data without needing a pre-specified dictionary.

Lu: The paper shows that by combining kernel methods with bilevel optimization, they can learn the dictionary and the operator approximation together in a way that is more flexible than previous work. Meng For someone listening who might be working on this kind of modeling, it suggests a pathway to make these data-driven models less reliant on prior assumptions about what functions are relevant.

Tom: The implication for the field is that it offers a way to bypass the need for extensive manual dictionary design in complex nonlinear systems. Jane It moves us closer to having tools that can learn the necessary structure of dynamics from raw observations, which is huge for system identification and forecasting.

Lu: And they also note that this approach is scalable because the required kernel matrices depend on the number of collocation points, not necessarily how many training data points you feed into it. Meng That makes a difference for engineers because it suggests a more manageable setup when you have very large datasets to work with.

Tom: So, in short, EDMD-kDL is presented as a kernel-based method that learns the embeddings directly from data using this optimization trick. Jane It’s about making these powerful Koopman tools accessible to people who don't want to do all the heavy lifting on dictionary selection first.

Maxwell Institute for Mathematical Sciences, The University of Edinburgh · Max Planck Institute for Dynamics of Complex Technical Systems · LAAS-CNRS

cs.LG, math.DS, stat.ML

Submitted: 2026-10-08

Updated: 2026-10-08

Code: https://github.com/erichson/koopmanAE

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 76/100

The gist: The gist The proposed method, extended dynamic mode decomposition with kernel-based dictionary learning (EDMD-kDL), is a novel kernel-based approach for learning finite-dimensional Koopman embeddings

Key concepts

Koopman Operator Theory: This theory provides a linear mathematical framework to analyze complex, nonlinear dynamical systems by transforming them into an equivalent linear system in a higher-dimensional space. This linearity allows researchers to use spectral decomposition to uncover patterns and time scales within the original nonlinear dynamics.
Extended Dynamic Mode Decomposition (EDMD): EDMD is a data-driven technique used to find a finite-dimensional approximation of the Koopman operator from trajectory data. It works by estimating an operator matrix using linear regression, aiming to map the system's state space into a lower-dimensional space where the dynamics appear linear.
Kernel EDMD: Kernel EDMD uses kernel functions to implicitly define a potentially infinite-dimensional dictionary for the Koopman embedding. This allows the method to capture complex relationships without needing an explicit, pre-defined dictionary, relying instead on inner products defined by a chosen kernel.

Terminology

Embeddings — 9× in this paper · explained in 2 episodes
Hilbert Space — 7× in this paper · explained in 2 episodes
Regularization — 6× in this paper · explained in 3 episodes
Objective Function — 3× in this paper · explained in 2 episodes
Accuracy — 2× in this paper · explained in 1 episode
Benchmark — 2× in this paper · explained in 2 episodes
Conference on Decision and Control — 2× in this paper
Consistency — 2× in this paper · explained in 3 episodes
Loss Function — 2× in this paper · explained in 2 episodes
Core — 1× in this paper · explained in 2 episodes
Interpretability — 1× in this paper · explained in 15 episodes
Inverse Problem — 1× in this paper · explained in 2 episodes
Robustness — 1× in this paper · explained in 5 episodes
System Identification — 1× in this paper · explained in 3 episodes
Trace — 1× in this paper · explained in 2 episodes

Summary

The gist The proposed method, extended dynamic mode decomposition with kernel-based dictionary learning (EDMD-kDL), is a novel kernel-based approach for learning finite-dimensional Koopman embeddings directly from data by combining ideas from collocation methods and bilevel optimization<ref:2610.12370#pg6>.

Background on Koopman Learning

The Koopman operator theory provides a linear framework for analyzing nonlinear dynamical systems and has become a major tool for data-driven modeling<ref:2610.12370#pg5> The Koopman operator is linear, even when the state-space dynamics are nonlinear<ref:2610.12370#pg4> This linearity makes it amenable to spectral decomposition, which encodes important properties of the dynamical system such as dominant time scales and coherent patterns<ref:2610.12370#pg5> The main interest from a data-driven perspective is to obtain a finite dimensional approximation of the Koopman operator from trajectory data<ref:2610.12370#pg6>

Extended Dynamic Mode Decomposition (EDMD)

The EDMD algorithm works by considering a dictionary of functions defining a so-called Koopman embedding, which is a mapping from the state space to a typically high-dimensional space such that the embedded dynamical system is, at least approximately, linear<ref:2610.12370#pg6> Given training data, the matrix representing the operator is estimated by solving a linear regression problem<ref:2610.12370#pg6> The minimum-norm solution of this problem is given by Kb = ΨY Ψ†X = ΨY Ψ⊤XΨXΨ⊤X†<ref:2610.12370#pg6>

Kernel EDMD and Bilevel Optimization

The choice of dictionary has a significant impact on the performance of the EDMD algorithm because it is assumed that the dictionary spans, at least approximately, a Koopman-invariant subspace<ref:2610.12370#pg6> Kernel EDMD leverages the well-known kernel trick to perform EDMD with a possibly infinite-dimensional dictionary implicitly defined by a chosen kernel<ref:2610.12370#pg6> The proposed method, EDMD-kDL, is less parametrized than ANN-based counterparts and relies on numerically solving a nonlinear optimization problem involving the values taken by the Koopman embedding map at a finite set of collocation points<ref:2610.12370#pg6> This formulation is derived by leveraging ideas from collocation methods and bilevel optimization<ref:2610.12370#pg6>

EDMD-kDL Algorithm

The derivation of EDMD-kDL involves solving a bilevel optimization problem where the outer problem seeks the optimal matrix K and dictionary U, while the inner problem optimizes over the complementary dictionary phi by treating pseudo-values as noisy observations<ref:2610.12370#pg6> The inner optimization problem is solved by finding qbl = (G + γ mc I)−1 ul, provided that G is invertible<ref:2610.12370#pg6> The final optimization problem over U and K can be formulated as a linear combination of the standard EDMD loss, an autoencoder reconstruction loss, and an RKHS regularization term<ref:2610.12370#pg6> Algorithm 1 details the steps to assemble the Gram matrix G, perform SVD, solve for Ub using L-BFGS or a preferred optimizer, and then iteratively find qbl to assemble the full dictionary ψb and EDMD matrix Kb<ref:2610.12370#pg6>

Performance and Applications

Across all tested settings, EDMD-kDL achieves performance comparable to or better than the ANN-based methods<ref:2610.12370#pg6> In the two-dimensional nonlinear pendulum model, EDMD-kDL consistently outperforms the neural network approaches in terms of prediction error and phase portrait preservation, especially when noise levels are substantial<ref:2610.12370#pg6> For global sea-surface temperature forecasting, EDMD-kDL and SINDy-SHRED produce similar forecasts across the board, with EDMD-kDL maintaining a low error level over the entire forecast horizon compared to KAE and cKAE<ref:2610.12370#pg6> Furthermore, EDMD-kDL is able to accurately predict the dynamics of a pendulum directly from video data, a setting in which ANN-based approaches drastically failed<ref:2610.12370#pg6> The method is scalable to large datasets by design since the size of the required kernel matrices depends on the number of collocation points rather than the size of the training dataset<ref:2610.12370#pg6>

Future Research Directions

Potential avenues for future research include elucidating convergence properties and finite-data error bounds for EDMD-kDL, potentially by combining ideas from standard kernel EDMD methods and error analysis of kernel collocation methods<ref:2610.12370#pg6> Extending the proposed method to work directly with high-dimensional data by jointly learning a nonlinear projection map akin to kernel principal component analysis is also a problem that would be investigated in the future<ref:2610.12370#pg6> Finally, computational cost can be reduced by starting with a first-order optimizer before switching to L-BFGS when training EDMD-kDL models<ref:2610.12370#pg6>

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Bilevel optimization for data-driven learning of Koopman embeddings using kernel-based autoencoders Joel-Pascal Ntwali N’konzi, Feliks Nüske, Stefan Klus (Dated: 9 October 2026)

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Koopman operator theory provides a linear framework for analyzing nonlinear dynamical systems and has become a major tool for data-driven modeling<ref:2610.12370#pg5> The Koopman operator is linear, even when the state-space dynamics are nonlinear<ref:2610.12370#pg4> This linearity makes it amenable to spectral decomposition, which encodes important properties of the dynamical system such as dominant time scales and coherent patterns<ref:2610.12370#pg5> The main interest from a data-driven perspective is to obtain a finite dimensional approximation of the Koopman operator from trajectory data<ref:2610.12370#pg6>

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II. INTRODUCTION Dynamical systems, whether deterministic or stochastic, are ubiquitous in science and engineering due to their effectiveness in modeling a wide range of natural phenomena<ref:2610.12370#pg5> In contrast to the standard geometric approach to dynamics, the Koopman perspective focuses on the temporal evolution of observables, which are scalar functions of the state<ref:2610.12370#pg5> Despite its great success, the EDMD algorithm is limited by the constraint that a suitable subspace in which the operator approximation is carried out has to be determined a priori, which is a challenging task in practice<ref:2610.12370#pg6> This has prompted the development of various dictionary learning approaches for EDMD/VAC in the literature<ref:2610.12370#pg6>

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An alternative approach for tackling the dictionary selection problem is kernel EDMD, which has been derived in two separate ways<ref:2610.12370#pg6> The distinguishing feature of this approach is that the embedding map can, in theory, be infinite-dimensional since only inner products between embedded vectors – which can be obtained by evaluating a symmetric positive semi-definite kernel – are required<ref:2610.12370#pg6>

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Learning Koopman dictionaries with neural networks Kernel EDMD relies on the assumption that the induced RKHS is Koopman-invariant<ref:2610.12370#pg6> An alternative approach is to simultaneously learn an approximately Koopman-invariant dictionary and the corresponding EDMD matrix directly from data<ref:2610.12370#pg6> This idea was first proposed by the authors of Ref. 8, who considered hybrid dictionaries of the form x → ψθ(x) = ρ(x)⊤, ϕθ(x)⊤⊤ ∈ R N, where ρ: omega → R o is a fixed component, typically the full-state observable<ref:2610.12370#pg6>

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We propose a novel approach, called extended dynamic mode decomposition with kernel-based dictionary learning (EDMD-kDL), for learning finite-dimensional Koopman embeddings of nonlinear dynamics from data<ref:2610.12370#pg6> In contrast to standard methods that rely on ANNs, EDMD-kDL optimizes EDMD dictionaries over an RKHS associated with a user-specified kernel<ref:2610.12370#pg6> Furthermore, our approach is fundamentally different from standard kernel EDMD in that the dimensionality of learned dictionary is independent of the number of training data points<ref:2610.

Improvements for AI systems

Bold header: Bilevel Optimization for Dictionary Learning

The system can learn a dictionary of functions directly from data by solving a bilevel optimization problem to simultaneously learn a kernel dictionary and the corresponding Koopman approximation, as stated in the abstract. This allows for learning embeddings directly from data without relying on pre-specified dictionaries.

Bold header: Scalable Kernel Dictionary Learning

The proposed method is scalable to large datasets by design since the size of the required kernel matrices depends on the number of collocation points rather than the size of the training dataset, addressing a major limitation in traditional methods. This enables applying EDMD-kDL to massive, real-world datasets like global SST or video data effectively.

Bold header: Robust Non-smooth Prediction for Nonlinear Systems

The system can achieve comparably or better performance than the ANN-based approaches on challenging nonlinear problems like the undamped pendulum and Kármán vortex shedding, even under significant noise (EDMD-kDL outperforms all the other methods in terms of prediction errors when the noise level is substantial).

Bold header: Accurate Phase Preservation Under Noise

For nonlinear dynamics, EDMD-kDL properly balances the preservation of phase and amplitude, whereas other methods like EDMD-RFF suffer from phase drift leading to high errors when noise is large, resulting in superior long-term prediction accuracy.

Bold header: Direct Video Data Interpretation

The system can be trained to predict the next frame of a video directly from raw video data, a task where EDMD-kDL produces sensible predictions over the entire test horizon, contrasting with ANN-based approaches that drastically failed in this setting.

Bold header: Long-Term Forecasting Accuracy

For complex time series like global sea-surface temperature (SST), EDMD-kDL maintains a low error level over the entire forecast horizon, achieving a maximum relative error of 6.7% compared to SINDy-SHRED's 11.8% and KAE's 20.9%.