Recursive Koopman Learning

• Recursive Koopman Learning is a data-driven framework that iteratively updates finite-dimensional approximations of the Koopman operator to model and control nonlinear systems.
• By lifting state trajectories into an observable space, RKL converts nonlinear dynamics into approximately linear evolution, enhancing prediction and stability analysis.
• Integration with machine learning enables continuous parameter updates, adaptive control synthesis, and efficient detection of regime shifts and basin boundaries.

Recursive Koopman Learning (RKL) is a data-driven framework for modeling, analysis, prediction, and control of nonlinear dynamical systems by iteratively or continuously updating finite-dimensional approximations of the Koopman operator as new system data becomes available. By “lifting” state trajectories to an observable space in which their evolution is (approximately) linear, RKL enables adaptive modeling, control synthesis, and improved sample efficiency in rapidly changing and uncertain environments. The design and effectiveness of RKL are fundamentally governed by the mathematical properties of Koopman eigenfunctions, spectra, and the interaction between stability, continuity, and control as revealed by operator-theoretic analysis and machine learning approximation theory.

1. Spectral Foundations and Stability in Recursive Operators

Recursive Koopman Learning capitalizes on the key property that, for a nonlinear system with flow $F^t(x)$, Koopman eigenfunctions $\phi$ and eigenvalues $\lambda$ satisfy the fundamental scaling law:

$\phi(F^t(x)) = e^{\lambda t} \phi(x)$

This property holds even under minimal regularity assumptions (such as non-continuous eigenfunctions). In RKL, this exponential law is embedded as an explicit constraint or as a regularization penalty during recursive updates (e.g., using minimization objectives of the form $$\|\phi(F^{\Delta t}(x)) - e^{\lambda \Delta t}\phi(x)\|^2$$).

Ensuring that recursively updated eigenfunctions encode this scaling is critical for capturing stability characteristics:

• If $\text{Re}(\lambda)<0$, and $\phi$ vanishes along a trajectory, the corresponding basin is locally stable.
• Theorem 2.1 asserts that for $\text{Re}(\lambda)>0$, boundedness of $\phi$ within a region implies all trajectories will eventually exit that region.

Hence, recursive update rules are typically designed to detect and correct deviations from exponential scaling, ensuring the learned Koopman representation informs about local and global stability with each data assimilation cycle.

2. Continuity, Discontinuity, and Representation Implications

A salient analytical result is that Koopman eigenfunctions may be discontinuous at basin boundaries in multi-attractor systems: Theorem 3.1 establishes that if $\phi$ (with eigenvalue $\lambda = 0$) varies across isolated fixed points $x_A$, $x_B$, then $\phi$ cannot be continuous globally. For machine learning approximators—whether fixed dictionaries (EDMD) or neural network-based—this has deep consequences:

• Smooth function approximators may fail to capture discontinuities, leading to model bias or "averaging out" basin boundaries.
• Recursive learning strategies must either employ basis expansions capable of modeling piecewise or discontinuous structure (e.g., hard activations, adaptive local bases) or integrate explicit mechanisms for domain decomposition and basin detection.

Anomalous jumps in recursively estimated eigenfunctions near certain regions can serve as indicators to trigger localized learning, dictionary augmentation, or regularization that respects physical domain separation.

3. Controllability Constraints and Locality of Recursive Control

Theorem 4.1 demonstrates that when null eigenfunctions (with $\lambda=0$) are continuous but differ among basins, no bounded control input can connect states across basins. This has direct implications for RKL-enabled control synthesis:

• Recursive Koopman-based controllers will inherently be locally optimal (only within the current basin of attraction).
• Control policies learned through RKL with bounded $\psi_{xu}(x,u)$ observables must accept the impossibility of “instantaneous jumps” in eigenfunction values and therefore the inability to transition across discontinuous domain boundaries with bounded actions.
• Recursive update laws for control, such as

$\dot{\phi}_x = Q^{-1} L_{xu} \psi_{xu}(x,u)$

must monitor and enforce boundedness of $\psi_{xu}(x,u)$ to avoid non-physical transitions.

This fundamentally limits global controllability unless the recursive scheme explicitly incorporates model switching, rare-event learning, or hybridized multi-model frameworks that are sensitive to basin transitions.

4. Integration with Machine Learning: Recursive Parameter Updates

In practice, machine learning models such as neural networks, EDMD, and adaptive basis expansions approximate Koopman eigenfunctions and the associated spectra from time-series data. RKL systems operationalize these methods via:

• Continuous parameter updates (online or in-memory batch) to accommodate new temporal data segments, ensuring that the Koopman representation adapts to changing system regimes.
• Loss function designs that incorporate explicit regularization for exponential scaling:

$$\text{Loss} = \text{Prediction Error} + \alpha \left\| \phi(x_{t+1}) - e^{\lambda \Delta t} \phi(x_t) \right\|^2$$

• Monitoring for discontinuities in the eigenfunctions, employing adaptive learning rates or multi-resolution representations to manage the challenges posed by rapid eigenfunction changes or detect new basins.
• The necessity to sometimes transition from a global model to local models (piecewise RKL), triggered by the identification of boundaries via the learned eigenfunction structure.

5. Advantages and Practical Limitations of Recursive Koopman Learning

The recursive framework allows several unique operational benefits:

• Embedding of stability, continuity, and local controllability constraints directly into the learning process.
• Real-time adaptation to regime shifts, time-varying parameters, or evolving environments by recursively updating the learned operator.
• Systematic detection of when the Koopman representation is “well-behaved”—that is, when the model remains bounded and respects the exponential dynamic—vs. when it displays failure modes (such as discontinuity smearing).

However, explicit limitations include:

• Difficulty modeling discontinuities at basin boundaries, particularly when relying on standard smooth approximators.
• Pronounced sensitivity to estimation errors near basin separatrices, leading to numerical instabilities or degraded operator accuracy in challenging regions.
• Inability to perform global controllability due to intrinsic spectral and eigenfunction structure; recursive models are necessarily localized unless augmented with advanced switching or rare-event learning.

6. Design Principles and Future Directions

The analytic insights from the referenced work dictate several design principles for the formulation of efficient and robust RKL systems:

• Recursive update criteria must directly enforce the scaling law $$\phi(F^{\Delta t}(x)) \approx e^{\lambda \Delta t} \phi(x)$$.
• Continuous monitoring of learned eigenfunction continuity to inform adaptivity in modeling architecture (choice of dictionary, neural network topology, or domain decomposition).
• Explicit integration of theoretical control limitations into the recursive learning process to prevent overextension of the global policy and respect natural system boundaries.
• Exploration of hybrid approaches that combine smooth and discontinuous basis functions, or develop multi-resolution models that adapt to the geometric structure uncovered by evolving Koopman modes.

Anticipated research avenues include recursive mechanisms for automated basin detection, adaptive basis selection informed by sharp eigenfunction transitions, and new recursive control strategies for multi-basin, multi-mode settings that explicitly respect the fundamental limitations derived from Koopman spectral analysis.

7. Summary Table: Analytical Implications for RKL Algorithm Design

Analytical Result	RKL Algorithmic Implication	Required Mechanism
Exponential scaling law	Regularization of operator learning	Loss terms, update monitoring
Eigenfunction discontinuities	Adaptive or piecewise modeling, domain splitting	Domain detection, local models
Control across basins forbidden	Local optimality, no global policy propagation	Basin-aware learning, switching
Non-local stability detection	Stability-informed recursive parameter update	Spectral-based monitoring

By grounding RKL design in these results, future systems can robustly extend Koopman operator frameworks to real-world, nonstationary, discontinuous, and locally controlled dynamical systems while offering a route for stability and control-aware learning at scale (Bakker et al., 2020).