Spatially Aware Dictionary-Free Eigenfunction Identification for Modeling and Control of Nonlinear Dynamical Systems

arXiv:2511.22648 · cs.LG · Submitted 2026-08-21 · 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 "Spatially Aware Dictionary-Free Eigenfunction Identification for Modeling and Control of Nonlinear Dynamical Systems".

Jane: The paper was written by David Grasev from Department of Aviation Technology, University of Defence and University of Defence, Kounicova 65, Brno, 100190, Czech Republic.

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

Paper discussion segment 2: Tom: To recap, this methodology offers a powerful, self-correcting framework that allows us to map the complex hidden rules governing dynamic systems without needing explicit physical equations beforehand.

Jane: Exactly. And I think the most revolutionary part of this paper is how they use a reference trajectory to kick off the entire process—it establishes a baseline set of Koopman modes, which is key to understanding where we are starting from.

Meng: The system takes that reference point and then transforms all other observed trajectories by projecting them onto a new basis formed by those known exponential functions and the established modes. It's like finding a common language for every trajectory based on the reference point.

Lu: That transformation is brilliant because it turns what into a complex, nonlinear state space into something linear in time. We are essentially mapping the chaotic flow into predictable, linear evolution within an observable space.

Lalam: I see this as creating a unified view of the system's dynamics; regardless of how messy the input data looks, we are translating it all back to these fundamental, clean patterns defined by those eigenfunctions.

Tom: It sounds like we’re not just finding a snapshot of dynamics; we are building a physics-respecting roadmap for the the entire functional envelope by defining these initial conditions.

Jane: And this is vital because it suggests that the goal of AI in engineering isn't just predictive accuracy, but *structural* understanding—understanding the ruleset itself rather than just memorizing data points.

Meng: It’s about creating a framework where we can input any trajectory, and it understands its initial starting conditions relative to that fixed reference point. That capability is massive for systems where the state changes unpredictably throughout its life cycle.

Lu: This approach eliminates the guesswork about where the system started; by using ridge regression on that new basis, we are calculating the precise initial state values of those fundamental patterns, zero.

Lalam: The implication here is that we are making the start of a trajectory as important as its end, giving us a full picture from initial state to final evolution.

Tom: That leads us naturally to how they manage and validate these powerful patterns—the specific improvements in "Spatially Aware Dictionary-Free Eigenfunction Identification for Modeling and Control of Nonlinear Dynamical Systems."

Jane: Absolutely. Let's move into Segment four and explore the closed optimization loop that makes this possible.

Paper discussion segment 3: Tom: We’ve seen how this method works, but why is it so much more robust than prior techniques? The key lies in the closed optimization loop and the self-correction mechanism they introduced.

Jane: The major leap here is that they didn't just stop at finding initial values; they put those eigenvalues into an iterative optimization loop where the time-based fit constantly refines those parameters. This is a continuous quality check, ensuring we aren't just finding a quick approximation that fails later on in the process.

Meng: This isn's coupled with some really smart penalties, too. They utilize an upper-triangular distance matrix D to actively discourage any solutions where two different modes are too close to each other, which prevents bad or spurious eigenfunctions from ruining the entire model.

Lu: That’s a very elegant way of imposing physical constraints on the math. We are telling the AI that, mathematically speaking, we don't want two distinct physical patterns that look almost identical; we want distinct modes of behavior.

Jane: And after enforcing that diversity, they have the spatial integrity check—enforcing the Koopman Partial Differential Equation or KPDE through a specific cost function. This ensures the discovered eigenfunctions satisfy fundamental laws of physics, making them physically consistent with their spatial arrangement.

Tom: It’s essentially building an intelligent feedback loop into the core of the idea to make sure our results are reliable and durable for long-term use.

Meng: If we can ensure the AI respects those physical constraints, we're much closer to having a model that is reliable for predicting real-world behavior rather than just looking good on paper in simulation. The physical consistency is what matters most to us as system designers.

Lu: This forces the discovered eigenfunctions to satisfy fundamental laws of physics, making them physically consistent with their spatial arrangement. It’s a deep layer of structural validation that goes beyond simple regression analysis.

Lalam: I think we are moving toward an era where AI isn't just predicting outcomes, but discovering the physical laws themselves governing those outcomes. This is a genuine shift in how we approach data-driven science.

Tom: We’ve seen how this method works and why its refinement loop is so much stronger than prior techniques; now, let's explore what these powerful improvements mean for practical applications across various industries.

Jane: Absolutely. Let's move into Segment five and conclude our discussion of "Spatially Aware Dictionary-Free Eigenfunction Identification for Modeling and Control of Nonlinear Dynamical Systems."

Conclusion: Tom: So, we’ve covered so much ground today, from how this method finds hidden patterns to its impressive performance in controlling complex systems.

Jane: It’s truly exciting that this whole approach isn't just a theoretical exercise but a powerful tool ready to be applied in the real world for designing robust equipment.

Meng: I can't wait to see how this applies in an industrial setting where we need these highly reliable models, especially given its success with the two-spool turbojet engine.

Lu: From a theoretical standpoint, it's the ability to discover structure without pre-defining the solution that is so powerful. It allows the AI to truly learn the dynamics rather than just fitting them.

Lalam: This work shows a cultural shift towards trusting AI not just as a pattern-matcher, but as a genuine system designer capable of revealing the inherent truths of complex nonlinear dynamics.

Tom: Thank you all for sharing your insights on this fascinating research. It's clear that we're looking at a very promising foundation for what's next in data-driven modeling with "Spatially Aware Dictionary-Free Eigenfunction Identification for Modeling and Control of Nonlinear Dynamical Systems."

Jane: Indeed, let’s take a quick break and look forward to the next paper on our list while we recharge.

Conclusion: Tom: So, we've covered so much ground today, from how this method finds hidden patterns to its impressive performance in complex engine control systems, Jane.

Jane: It’s truly exciting that this whole approach isn't just a theoretical exercise but a powerful tool ready to be applied in the real world for designing robust equipment.

Tom: And before we wrap up and head off, I want to make sure everyone gets one last chance to share their final thought on "Spatially Aware Dictionary-Free Eigenfunction Identification for Modeling and Control of Nonlinear Dynamical Systems."

Lu: From a theoretical standpoint, it's the ability to discover structure without pre-defining the ultimate solution that is so powerful. It allows the AI to truly learn the dynamics rather than just fit them.

Meng: I can't wait to see how this applies in an industrial setting where we need these highly reliable models, but I think we’ve seen that practically, it's also very robust against gaps in data or unexpected operational shifts.

Lalam: This work shows the cultural shift towards trusting AI not just as a pattern-matcher, but as a genuine system designer capable of revealing the inherent truths of complex dynamics.

Tom: Thank you all for sharing your insights on this fascinating research. It's clear that we're looking at a very promising foundation for what's next in data-driven modeling.

Jane: Indeed, I think the overarching takeaway is that we are moving beyond simple prediction and into true structural understanding of complex physical systems.

Tom: A model that doesn’t just tell us *what* will happen, but *why* it has to happen based on fundamental physics—that's the ultimate goal here.

Jane: It solidifies this technique, "Spatially Aware Dictionary-Free Eigenfunction Identification for Modeling and Control of Nonlinear Dynamical Systems," as a landmark piece of work.

Tom: Well, with that summary in mind, let’s take a quick break and look forward to the next paper on our list while we recharge.

David Grasev

Department of Aviation Technology, University of Defence · University of Defence, Kounicova 65, Brno, 100190, Czech Republic

cs.LG

Submitted: 2026-08-21

Updated: 2026-08-24

Importance score: 80/100

The gist: A new approach to data-driven discovery of Koopman eigenfunctions without a pre-defined set of basis functions is proposed.

Key concepts

Koopman Modes/Eigenfunctions: These are fundamental, clean patterns identified by the method. They serve as a new basis for the system's dynamics, allowing researchers to understand the hidden rules of complex systems without needing explicit physical equations beforehand.
Linearization of State Space: The technique transforms highly complex and nonlinear state spaces into a linear form over time. This allows chaotic system flows to be mapped into predictable, observable patterns defined by these fundamental eigenfunctions.
Closed Optimization Loop: This is an iterative refinement process where eigenvalues are continuously checked against the time-based fit. It acts as a continuous quality check, ensuring the model's parameters are precise and reliable for long-term use.
Physical Constraints (Spatial Integrity Check): The method enforces fundamental laws of physics through specific cost functions. This ensures the discovered patterns are physically consistent with their spatial arrangement, preventing mathematically valid but unreal solutions.

Terminology

Accuracy — 12× in this paper · explained in 1 episode
Core — 6× in this paper · explained in 2 episodes
Benchmark — 5× in this paper · explained in 2 episodes
Adam — 3× in this paper · explained in 2 episodes
Limit Cycles — 3× in this paper · explained in 2 episodes
Regularization — 3× in this paper · explained in 3 episodes
Sparsity — 3× in this paper · explained in 4 episodes
Embeddings — 2× in this paper · explained in 2 episodes
Hilbert Space — 2× in this paper · explained in 2 episodes
PID Controller — 2× in this paper · explained in 2 episodes
Physical Constraints — 2× in this paper · explained in 7 episodes
Spatially Aware Dictionary-Free Eigenfunction Discovery — 2× in this paper
Abundance — 1× in this paper · explained in 1 episode
Annual American Control Conference — 1× in this paper
Closed-Loop Optimization — 1× in this paper · explained in 2 episodes
Closed-Loop System — 1× in this paper · explained in 4 episodes
Consistency — 1× in this paper · explained in 3 episodes
Decoupling — 1× in this paper · explained in 2 episodes
Fine-Tuning — 1× in this paper · explained in 5 episodes
Hysteresis — 1× in this paper · explained in 3 episodes
Interpretability — 1× in this paper · explained in 15 episodes
Model Predictive Control (MPC) — 1× in this paper · explained in 5 episodes
Objective Function — 1× in this paper · explained in 2 episodes
Quantization — 1× in this paper · explained in 12 episodes
System Identification — 1× in this paper · explained in 3 episodes

Summary

A new approach to data-driven discovery of Koopman eigenfunctions without a pre-defined set of basis functions is proposed. This method, referred to as Spatially Aware Dictionary-Free Eigenfunction Discovery (SADFED), aims to simplify the process and find samples of eigenfunctions using only the knowledge of the eigenvalues, expressing the problem solely in terms of the eigenvalues.

The core methodology involves utilizing a reference trajectory for which Koopman mode amplitudes are first identified. The Koopman mode decomposition is then transformed into a new basis, which contains fundamental functions of eigenvalues and time. The initial values of the eigenfunctions (0) are obtained by projecting trajectories onto this basis via a regularized least-squares fit (ridge regression). A global optimizer is employed to optimize the eigenvalues.

Crucially, mapping initial-state values to eigenfunction values reveals their spatial structure, enabling the numerical computation of their gradients. This spatial structure integrity is enforced through a Koopman partial differential equation (KPDE) cost function (J KPDE), where deviations from the KPDE are penalized. The total optimization objective is defined as J = J temp + gamma J KPDE, with opt = arg min J.

The approach was successfully tested on several benchmark nonlinear dynamical systems:

FitzHugh-Nagumo system with inputs: The study demonstrates that the numerically computed gradient can be directly utilized for input dynamics modeling and control design.
Van der Pol and Duffing oscillators: The method reveals geometric features of the state space, such as invariant partitions (isostables) and isochrons.
A 2-spool turbojet engine with control: This complex, non-affine system was used to demonstrate the framework's applicability to model a control-affine representation of dynamics before Koopman eigenfunction identification.

The study demonstrates that incorporating principal eigenvalues and spatial structure integrity promotion significantly improves the accuracy of Koopman predictors. The approach effectively discovers Koopman spectral components even with sparse state-space sampling and reveals geometric features of the state space, such as invariant partitions. Furthermore, the numerical approximation of the eigenfunction gradient can be used for input dynamics modeling and control design.

For systems with a single fixed point not in the origin, a zero eigenvalue (lambda 0 = 0) is included to represent a constant eigenfunction (phi 0), which acts as an indicator function to tell in which invariant subset the point in the state space is located. For controlled systems, the gradient grad 0 is utilized to model input dynamics. The resulting cost function J KPDE allows for the explicit enforcement of spatial structure, and when combined with temporal projection error minimization (J temp), yields robust solutions. The final results support the practicality of this approach for use with various dynamical systems, leading to a state-dependent Riccati equation and an optimal tracking LQG controller that outperforms classical Proportional-Integral (PI) controllers in the control of the 2-spool GTE.

Improvements for AI systems

Current AI systems often struggle with systems that possess multiple invariant subsets or discontinuous state-space boundaries (e.g, separatrix regions). The SADFED framework allows for an automated partitioning of the state space based on the discovered spatial structure of the first eigenfunction (phi 0).

Improvement: Implement a dynamic Invariant Subset Detector that uses phi 0 as an indicator function. Instead of applying a single global optimization loop, the system autonomously splits the optimization into localized sub-problems for each identified invariant region.

AI Capability: The resulting AI system can accurately model and predict trajectories in complex, discontinuous systems (e.g., bifurcating or multi-fixed point dynamics) without requiring manual pre-sectorization of the state space.

The reliance on a single reference trajectory (x*) is a bottleneck for system coverage, especially when the state space is vast or highly non-linear.

Modeling input dynamics (G(x)u) in a dictionary-free manner is complex for control design, particularly in systems where G(x) is non-affine. The SADFED method provides the necessary spatial gradient (grad 0).

The current optimization uses sequential global (PSO) and local (NM) optimizers, which is computationally expensive.

The sensitivity of the KPDE cost function (J KPDE) to noise and interpolation quality is a known weakness when dealing with discontinuities.