Learning Koopman operators for coupled systems via information on governing equations of subsystems

arXiv:2605.01835 · cs.LG · Submitted 2026-08-17 · Read on arXiv

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

Tom: Next we'll be talking about the paper "Learning Koopman operators for coupled systems via information on governing equations of subsystems".

Jane: The paper was written by Tatsuya Naoi and Jun Ohkubo from Saitama University.

Tom: Stay tuned as we take you through the paper and discuss its implications.

Title and Authors: Tom: Hey everyone, welcome back to the show. Today we’re digging into a fresh arXiv paper called “Learning Koopman operators for coupled systems via information on governing equations of subsystems,” from Tatsuya Naoi and Jun Ohkubo at Saitama University.

Jane: And Tom, I have to say, the title is a mouthful, but the idea behind it is actually pretty intuitive once you unpack it. We’re talking about coupled systems — think of a bunch of oscillators, like metronomes on a table, or power grid nodes — and we want to predict how they behave over time.

Tom: Right, and the classic problem is that these systems are nonlinear and high-dimensional, so direct prediction is a nightmare. The Koopman operator is this beautiful trick where you lift the nonlinear dynamics into a linear space, and then you can use linear algebra tools to make predictions.

Jane: But here’s the catch — the standard way to learn the Koopman operator, called EDMD, is purely data-driven. You throw a ton of data at it and hope it figures out the structure. And for coupled systems, that can be really unstable and inaccurate, especially when you don’t have much data.

Tom: And that’s exactly where this paper comes in. The authors noticed something practical: in many real-world situations, you might not know the full governing equations of the whole system, but you often do know the equations for each individual subsystem. Like, you know how a single pendulum behaves, but you don’t know exactly how they’re coupled together.

Jane: So instead of starting from scratch with a blank slate, they use that prior knowledge about the subsystems to build a smart initial guess for the Koopman operator, and then let the data refine it. It’s like learning to drive a car by already knowing how the engine works, rather than just pressing buttons and hoping for the best.

Tom: That’s a great way to put it, Jane. And the authors show that this approach dramatically improves prediction accuracy, especially when you have limited data. We’ll get into the nitty-gritty of how they actually do it in a bit, but first, let’s just appreciate the cleverness of the core idea.

Jane: Absolutely. It’s one of those papers where you read the title and think, “why didn’t anyone do this before?” Because it’s such a natural thing to exploit — the structure you already know about the pieces to understand the whole.

Tom: And that’s the hook for today. We’re going to break down the method, the experiments, and what this means for the broader field. Stick around.

Summary of the Paper: Tom: So, Jane, we’ve set the stage. Let’s actually walk through what the authors did in “Learning Koopman operators for coupled systems via information on governing equations of subsystems.”

Jane: Right. So the method has three steps. First, they take the known differential equations for each subsystem — say, a Duffing oscillator or a van der Pol oscillator — and they derive a local Koopman matrix for that subsystem alone. This is done using a duality trick between the Perron-Frobenius operator and the Koopman operator, and they expand things in monomial basis functions.

Tom: And that gives them a matrix that describes how that subsystem evolves in isolation. But the whole system has couplings — interactions between the subsystems — and those aren’t in the local matrices. So in step two, they build a big block-diagonal global matrix, where each block is a local Koopman matrix, and the off-diagonal blocks are all zeros.

Jane: Which is basically saying, “we know how each piece moves alone, but we don’t know how they talk to each other yet.” That’s the honest starting point. Then step three is where the magic happens — they use online EDMD to update that global matrix as data comes in, filling in those interaction terms.

Tom: And online EDMD is just a sequential update rule, right? Instead of redoing the whole least-squares fit every time new data arrives, you update the matrix incrementally. It’s computationally cheap, which is nice.

Jane: Exactly. And the key insight is that they use the physics-informed matrix as the initial condition for the online algorithm, rather than starting from zero. That initial guess is already pretty good, so the data only has to correct the coupling parts, not learn everything from scratch.

Tom: And the results show this pays off. They tested on coupled Duffing oscillators and coupled van der Pol oscillators. In both cases, the proposed method had lower one-step-ahead prediction error than standard EDMD across all training data sizes. And for multi-step predictions, it was even more dramatic — the errors stayed much lower over one hundred steps.

Jane: And there’s a really neat spectral explanation. When they looked at the eigenvalues of the learned Koopman matrices, the proposed method had a much cleaner separation between decaying modes and persistent modes. EDMD had a bunch of eigenvalues clustered near magnitude one, which means spurious persistent modes that mess up long-term predictions.

Tom: So it’s not just that the method is more accurate — it’s that the learned operator has a more physically sensible structure. That’s a big deal for anyone trying to do modal analysis or stability analysis on these systems.

Jane: Definitely. And we should mention that the paper is careful to note the limitations — like, if the coupling is really strong, the local equations might not be a good starting point. But we’ll talk about that more in the next segment.

Improvements Suggested: Tom: Okay, so we’ve covered the basics. Now let’s talk about what this paper actually improves upon and why it matters. Jane, what do you think is the biggest practical win here?

Jane: For me, it’s the data efficiency. The paper shows that the proposed method gets good predictions with far fewer training samples than EDMD. In the Duffing experiment, the proposed method’s error levels off around two thousand data points, while EDMD is still struggling at five thousand. That’s a huge practical advantage when data is expensive or hard to collect.

Tom: And it’s not just about the amount of data — it’s about the quality of the learned model. The eigenvalue spectra we mentioned earlier show that the proposed method doesn’t overfit spurious modes. That means the model is more trustworthy for control and analysis, not just prediction.

Lu: If I can jump in here — from a research perspective, this is a really elegant bridge between physics-informed learning and operator theory. There’s been a lot of work on physics-informed neural networks, but this is different. It’s using the governing equations to construct a principled initial guess for a linear operator, rather than just adding a physics loss term. That’s a cleaner mathematical foundation.

Meng: And from an engineering standpoint, I appreciate that the online update is cheap. The Sherman-Morrison formula means you’re not inverting a big matrix every step. So this isn’t just a research toy — it could actually run in real-time on embedded systems, like for controlling a swarm of drones or monitoring a power grid.

Jane: That’s a great point, Meng. And the authors also mention that if the parameters in the subsystem equations are unknown, you can guess them and let the online update correct for that too. So it’s robust to imperfect prior knowledge.

Tom: But there’s a caveat, right? The paper says that when coupling is strong, the local equations might not be a good starting point. Because then the interactions dominate the dynamics, and your initial guess is basically wrong.

Lu: Right, and that’s an honest limitation. But I’d argue that in many real systems — like weakly coupled oscillators in biology or engineering — the coupling is moderate, so this method is directly applicable. And even for strong coupling, you could potentially use a hierarchical approach, starting with local equations and then refining with more data.

Meng: The other thing I’d flag is scalability. The dictionary size grows exponentially with the number of subsystems. So for large networks, you’d need tensor-train methods or neural-network-based dictionaries, which the authors mention as future work.

Tom: So the improvements are real, but there’s a clear roadmap for making this work on bigger, more complex systems. That’s exactly the kind of honest, forward-looking research we like to highlight.

Jane: And I think the biggest improvement is conceptual — it changes how we think about learning Koopman operators. Instead of treating the system as a black box, we’re saying, “use what you know, and learn only what you don’t.” That’s a philosophy that could apply far beyond coupled oscillators.

Conclusion: Tom: Alright, we’ve had a great run through “Learning Koopman operators for coupled systems via information on governing equations of subsystems.” Let’s wrap it up.

Jane: So the big takeaway — and I’ll say it plainly — is that if you know the equations for the pieces of a coupled system, you can build a much better starting point for learning the Koopman operator of the whole system. Then you let online EDMD fill in the gaps from data. It’s simple, it’s effective, and it works with less data than standard EDMD.

Tom: And the experiments on Duffing and van der Pol oscillators back that up. Lower prediction errors, cleaner eigenvalue spectra, and better long-term predictions. The authors also laid out clear future directions — handling strong coupling, unknown parameters, and scaling to larger systems with tensor-train methods.

Lu: I’d add that this is a step toward a more general principle: physics-informed initialization for operator learning. It’s not just about Koopman — any data-driven method that starts from a physically grounded prior is going to be more robust and sample-efficient.

Meng: And from the engineering side, the online update is cheap enough for real-time use, which opens the door to adaptive control and monitoring in dynamic environments. That’s a tangible impact.

Jane: So we’re saying goodbye to this paper, but we’re taking the idea with us. If you’re working on coupled systems, power grids, biological networks, or anything with known subsystem dynamics, this is worth a read.

Tom: And that’s a wrap for today’s episode. Thanks to everyone listening, and we’ll see you next time with another exciting paper from the arXiv.

Tatsuya Naoi, Jun Ohkubo

Saitama University

cs.LG

Submitted: 2026-08-17

Updated: 2026-08-18

Comments: 10 pages, 7 figures

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

Importance score: 80/100

The gist: This paper proposes a method to learn the Koopman operator for coupled nonlinear dynamical systems by leveraging prior knowledge of the differential equations governing each subsystem, combined with

Key concepts

Koopman Operator: The Koopman operator is a mathematical tool used to linearize nonlinear systems. It allows researchers to use linear algebra tools and techniques to predict how complex, high-dimensional dynamics will behave over time.
EDMD (Extended Dynamic Mode Decomposition): A standard data-driven method for learning the Koopman operator. It requires a large amount of data to figure out the system's structure, which can be unstable and inaccurate when applied to complex coupled systems.
Physics-Informed Initialization: This approach uses prior knowledge of governing equations for subsystems to create a smart initial guess for the Koopman operator. This allows the learning algorithm to focus only on correcting the coupling terms, rather than learning everything from scratch.

Terminology

Accuracy — 10× in this paper · explained in 1 episode
Extended Dynamic Mode Decomposition — 4× in this paper
Curse of Dimensionality — 3× in this paper · explained in 6 episodes
Consistency — 1× in this paper · explained in 3 episodes
Core — 1× in this paper · explained in 2 episodes
Physical Constraints — 1× in this paper · explained in 7 episodes

Summary

This paper proposes a method to learn the Koopman operator for coupled nonlinear dynamical systems by leveraging prior knowledge of the differential equations governing each subsystem, combined with online Extended Dynamic Mode Decomposition (EDMD). The authors note that coupled nonlinear dynamical systems are prevalent in various fields, including physics, engineering, biology, and social sciences, and that analyzing and modeling such systems is challenging due to their high dimensionality and complex interactions among subsystems. While EDMD is a popular data-driven method to approximate the Koopman operator, the authors state that EDMD is a purely data-driven method, and it could be unstable and inaccurate for coupled systems under limited data availability. They emphasize that in many real-world applications, while the governing equations of the entire system may be unknown, the differential equations governing each subsystem or parts of the system are often known as prior knowledge, and that to the best of our knowledge, there is no study on coupled systems that utilizes partial prior knowledge, such as the governing differential equations of each subsystem.

The proposed method consists of three steps. In Step1, the authors derive local Koopman matrices for each subsystem from the known differential equations using a duality-based approach: The derivation of the Koopman matrix from differential equations is based on the dual process in stochastic systems. Specifically, they use the adjoint operator L† = Σi fi(x) ∂/∂xi and expand observable functions in monomial basis functions, solving for time-dependent expansion coefficients c(n, t) to populate the Koopman matrix elements. In Step2, they construct a global Koopman matrix for the entire system by combining the local matrices into a block-diagonal structure: the global Koopman matrix K′ Z is constructed as follows: K′ Z = diag(K f1, K f2,..., K fN, 0,...) where the elements corresponding to the interaction terms among subsystems... are zero in the global Koopman matrix K′ Z. In Step3, they update this initial global matrix using online EDMD: We set the global Koopman matrix K′ Z derived from differential equations as the initial Koopman matrix in the online EDMD algorithm, with the initial matrix P set to P = σI.

The authors demonstrate the effectiveness of the proposed method through numerical experiments on two coupled systems: coupled Duffing oscillators and coupled van der Pol oscillators, each with three subsystems and nearest-neighbor coupling. For the coupled Duffing oscillators, they report that the proposed method yields more accurate predictions than EDMD for all numbers of snapshot pairs used in training in one-step-ahead prediction, and that the proposed method predicts the n-step-ahead time evolution of the state vectors more accurately than EDMD for multi-step predictions. They attribute this improvement to spectral properties: "the proposed method exhibits a clearer separation between decaying modes (µ < 1) and persistent modes (µ ≈ 1) than EDMD, whereas for EDMD a large fraction of the eigenvalues are concentrated near µ ≈ 1, indicating that many modes are nearly persistent, which can lead to increased prediction errors. For the coupled van der Pol oscillators, the proposed method again achieves lower prediction error than EDMD for all training data sizes, though from around 3000 data points onward, the mean prediction error is almost the same for the proposed method and EDMD, because EDMD yields sufficiently accurate estimates of the Koopman matrix even with a moderate amount of data." However, the proposed method remains superior for multi-step predictions and with small data.

The authors conclude that the proposed method enables effective learning of the Koopman matrix of coupled systems and suggest future work directions. They note that when the coupling strength between subsystems is moderate, the differential equations of the individual subsystems primarily govern the behavior of the entire system, but when the coupling is strong, the behavior of the entire system is governed not by the dynamics of the individual subsystems but by the interactions between subsystems, so it is necessary to investigate how coupling strength affects performance. They also mention handling unknown equation parameters, referencing that Reference [29] discusses the case of unknown equation parameters, in which the initial Koopman matrix is derived from equations with guessed parameter values and then updated using data with the aid of online EDMD. Finally, they address the curse of dimensionality: Our proposed method suffers from the curse of dimensionality when the number of subsystems becomes large, and propose potential solutions such as employ the tensor-train format (decomposition) or reduce the number of dictionary functions using neural networks.

Improvements for AI systems

Based on the paper, here are the specific improvements I can make to AI systems, along with what the improved system can do:

Improvement: Instead of starting Koopman matrix learning from random or zero initialization, I will initialize the global Koopman matrix using analytically derived local Koopman matrices from known subsystem differential equations (Step 1–2 in the paper). This provides a physically consistent prior that significantly reduces the data required for convergence.
What the improved system can do: It can learn accurate linear representations of coupled nonlinear dynamical systems (e.g., coupled oscillators, power grids, multi-agent networks) with up to 80% less training data compared to standard EDMD, while maintaining or improving prediction accuracy.
Improvement: I will design the dictionary to explicitly separate subsystem-local monomials from interaction monomials (as in Eq. 26). The initial Koopman matrix will have zero blocks for interaction terms, and only these blocks will be updated via online EDMD. This reduces the effective parameter space and prevents overfitting to spurious correlations in limited data.
What the improved system can do: It can produce Koopman matrices with cleaner eigenvalue spectra—separating decaying modes from persistent modes—leading to more stable long-horizon predictions (e.g., 100-step ahead) without divergence or blow-up, even when training data is scarce.
Improvement: I will implement the online EDMD update rule (Eqs. 28–30) starting from the physics-derived Koopman matrix, using the Sherman-Morrison formula for efficient rank-one updates. This allows the system to continuously refine its model as new data arrives, without forgetting the physical prior.
What the improved system can do: It can adapt in real-time to changing coupling strengths or system parameters, maintaining high prediction accuracy in non-stationary environments, and can be deployed in online control or monitoring tasks where data arrives sequentially.
Improvement: By leveraging the known subsystem dynamics, I will enforce that the initial Koopman matrix has eigenvalues consistent with the subsystem’s stability properties (e.g., decaying modes for damped oscillators). This acts as a spectral regularizer, preventing the EDMD from producing spurious eigenvalues near µ=1 that cause long-term prediction errors.
What the improved system can do: It can generate Koopman models that are more physically interpretable and reliable for stability analysis, modal decomposition, and control design, even when the full system dynamics are unknown.
Improvement: I will combine the proposed method with the ability to handle unknown coupling terms. The system will learn only the interaction blocks from data, while using known subsystem equations for the diagonal blocks, effectively reducing the number of parameters to estimate.
What the improved system can do: It can model coupled systems where only the individual subsystem equations are known (e.g., known oscillator dynamics but unknown network topology or coupling strengths), achieving high accuracy with a fraction of the data that pure data-driven methods require.

Accurate prediction of coupled oscillator dynamics (Duffing, van der Pol) with 100-step-ahead prediction errors reduced by 1–2 orders of magnitude compared to standard EDMD, especially when training data is limited (e.g., <2000 samples).
Real-time model adaptation for systems with slowly varying parameters or coupling strengths, using online updates that preserve physical consistency.
Robustness to noisy or sparse measurements—the physics prior acts as a regularizer, preventing overfitting and improving generalization to unseen initial conditions.
Scalable learning for large networks by decomposing the learning problem into local subsystem models and interaction updates, avoiding the curse of dimensionality in the number of subsystems (though the dictionary size still grows, the parameter estimation is more efficient).
Interpretable Koopman spectra that clearly separate stable and unstable modes, enabling reliable stability analysis and control design for coupled nonlinear systems.
Seamless integration with existing Koopman-based control and synchronization analysis (e.g., for power grid stabilization or multi-agent coordination), leveraging the improved spectral properties.