Operator Learning in Dynamical Systems

• Operator learning in dynamical systems is the data-driven approximation of function-to-function mappings that capture the evolution of complex systems.
• It employs lifting techniques, kernel-based methods, and neural architectures to linearize nonlinear behaviors for efficient analysis.
• This approach drives advances in prediction, control, and uncertainty quantification across robotics, infrastructure, climate modeling, and quantum systems.

Operator learning in dynamical systems refers to the data-driven approximation of (often infinite-dimensional) operators that govern the evolution of dynamical systems, with emphasis on function-to-function mappings—such as the solution operator of a PDE, the Koopman or transfer operator of an ODE, or response operators in control and forecasting. In practice, these operators reveal global, potentially nonlinear, behavior through linear (or approximately linear) representations in lifted or kernel spaces, enabling advanced prediction, control, and analysis across diverse scientific and engineering domains.

1. Foundations: Operator Formulations in Dynamical Systems

A dynamical system is typically described by evolution equations, such as

$x_{t+1} = F(x_t, u_t)$

or, in continuous time, by ODEs/PDEs. Operator learning targets function spaces by representing nonlinear evolution using operators acting on observables; the canonical example is the Koopman operator $\mathcal{K}$, acting as

$[\mathcal{K}g](x) := g(F(x, u))$

for some observable $g$.

The operator perspective extends naturally to Markov, stochastic, and quantum systems, where the transfer operator, Koopman operator, or Lindblad operator act on functions or density matrices. The spectral and geometric features of these operators (eigenfunctions, singular functions, invariant sets) underpin the structure and long-term behavior of the system (Mezic, 2020, Kostic et al., 2022, Novelli, 2022, Turri et al., 24 May 2025, Jeong et al., 9 Jul 2025).

2. Methodologies: Lifting, Approximation, and Learning Strategies

Operator learning methods can be grouped into several methodological classes:

• Dictionary-based (unstructured) lifting: Using libraries of basis or dictionary functions $\{\psi_i(x)\}$, one computes finite-dimensional approximations by solving least squares problems (EDMD, DMD, kernel DMD), seeking $K : \Psi(x_{t+1}) \approx K \Psi(x_t)$ (Abraham et al., 2019, Sinha et al., 2021).
• Structured lifting via coordinate transformations: If certain system symmetries or invariances are present (such as via Lie group/immersion theory), one seeks explicit coordinate changes (Koopman eigenfunctions) to exactly linearize the dynamics in new variables (Bevanda et al., 2021).
• Kernel-based methods: Realizing the operator in a reproducing kernel Hilbert space (RKHS), the dynamics become linear in feature space, permitting efficient operator regression, spectral decomposition, and uncertainty quantification via kernel methods and Gaussian processes (Kostic et al., 2022, Li et al., 31 Oct 2024, Jorgensen et al., 15 May 2024).
• Neural operator architectures: DeepONets, Fourier Neural Operators (FNO), graph/multipole neural operators, and complex neural operators (CoNO) parametrize the mapping between infinite-dimensional function spaces, achieving mesh-invariance, robustness, and scalability (Tiwari et al., 2023, Boullé et al., 2023, Lin et al., 2022, Michałowska et al., 2023).
• Active and online operator learning: Recursive, robust, and noise-aware algorithms enable real-time operator identification and updating, particularly suited for high-frequency streaming data and real-world sensors (Sinha et al., 2019, Sinha et al., 2022, Zhang et al., 2023).
• Self-supervised and contrastive operator learning: Encoder-only learning and contrastive losses (e.g., VAMP-2, density ratio regression) are used to capture dominant dynamical modes and slow components, with emphasis on high-dimensional and scientific datasets (Turri et al., 24 May 2025, Jeong et al., 9 Jul 2025).

3. Spectral and Geometric Analysis

The spectrum of learned operators—e.g., Koopman or transfer operators—organizes and reveals key dynamical features:

• Eigenfunctions and invariant sets: Level sets of Koopman eigenfunctions correspond to invariant or almost-invariant structures (e.g., isochrons, Lyapunov submanifolds) (Mezic, 2020). Faithful representation requires learning observables that partition the state space according to the operator's spectrum and geometry.
• Singular function analysis: For stochastic systems, dominant singular subspaces (learned via low-rank approximation or VAMPnet-style losses) characterize the slowest modes, metastable sets, and spectral gaps essential for long-term prediction (Jeong et al., 9 Jul 2025).
• Memory effects and model reduction: The Mori–Zwanzig formalism projects dynamics onto resolved and unresolved components; a point spectrum enables Markovian closure, while a continuous spectrum induces memory terms, manifesting as nontrivial residuals in models learned from data (Mezic, 2020).

4. Algorithmic Implementations and Computational Considerations

Practical implementation involves multiple choices and trade-offs:

• Real-time, online, and recursive estimation: Algorithms such as recursive EDMD and robust online Koopman operator learning update operator estimates as new data arrives—critical for large-scale or safety-critical systems (e.g., power networks, adaptive robot control), with computational efficiency underpinned by the use of the matrix inversion lemma and careful initialization (Sinha et al., 2019, Sinha et al., 2022).
• Data enrichment and robust optimization: For sparse or noisy data, enrichment via local synthetic samples and robust least-squares formulations (min–max optimizations) control ill-posedness and improve the fidelity of the learned operator (Sinha et al., 2021).
• Uncertainty quantification: Operator-valued kernels and kernel-based Bayesian filters model uncertainties in both operator and state prediction, yielding probabilistic priors/posteriors and supporting minimum-variance recursive estimation—especially important in settings with incomplete or noisy data (Jorgensen et al., 15 May 2024, Li et al., 31 Oct 2024, Subedi et al., 4 Apr 2025).
• Neural operator design: Architectures leverage low-rank decompositions (DeepONet), translation invariance (FNO, CoNO), and local/hierarchical structure (graph/multipole operators), enabling mesh-invariance, improved generalization, and computational scalability. Efficient integration into deep learning pipelines is emphasized in recent work by eliminating unstable numerical operations from the objective (Boullé et al., 2023, Jeong et al., 9 Jul 2025).

5. Applications and Empirical Validation

Operator learning underpins advances across scientific modeling, engineering control, and computational prediction:

• Model-based robotic control: Koopman operator models, coupled with active learning and information-theoretic measures (e.g., Fisher information), enable rapid learning and improved stabilization/tracking performance in robotic systems under uncertainty (Abraham et al., 2019).
• Real-time monitoring and smart infrastructure: Online operator learning supports the dynamic analysis and control of power grids, HVAC systems, and similar infrastructure by furnishing up-to-date, computationally tractable operator representations from streaming sensor data (Sinha et al., 2019, Sinha et al., 2022).
• High-dimensional scientific modeling: Recent encoder-only and self-supervised approaches scale to systems with terabyte-scale datasets for applications such as protein folding, small molecule binding, and global climate dynamics, producing interpretable dynamical modes (e.g., capturing slow folding dynamics or ENSO events) and outperforming traditional approaches on high-fidelity metrics (Turri et al., 24 May 2025).
• Quantum systems and non-commutative probability: Operator learning extends to open quantum systems, enabling the modeling and spectral analysis of observables and the extraction of symmetries via the Koopman operator's spectrum (Novelli, 2022, Jorgensen et al., 15 May 2024).
• Stochastic dynamics and uncertainty modeling: Hybrid deterministic–stochastic operator learning (e.g., stochastic flow map learning employing ResNets and GANs) captures both averaged trajectories and fluctuating paths in noisy environments, relevant for climate, finance, and biology (Chen et al., 2023).

6. Mathematical Theory, Limitations, and Future Directions

• Connections between facets of learning: Approximation of the primary transformation law, invariant set, Koopman operator, and finite-state Markov approximations each reside in different mathematical spaces. Direct approximation of the map $f$ need not guarantee faithful recovery of all operator-theoretic or topological properties (e.g., spectral gaps, invariant measures) (Berry et al., 20 Sep 2024).
• Limits of predictability: The top Lyapunov exponent and other spectral features determine the regime over which forecasts remain accurate. Direct (non-iterated) and iterative forecasting methods display distinct error growth rates, with instability and embedding limitations presenting fundamental obstructions (Berry et al., 20 Sep 2024).
• Operator-valued kernels and non-commutative extensions: Universal factorization results and new non-commutative Radon–Nikodym theorems inform the design of kernels for machine learning in both classical and quantum dynamical systems, enabling the transfer of tools from functional analysis into the learning context (Jorgensen et al., 15 May 2024).
• Active data collection and optimal experiment design: Adaptive sampling, critical sample selection, and information-gain-driven experimentation can substantially reduce the data required for robust operator learning, especially in high-dimensional or resource-limited settings (Zhang et al., 2023, Subedi et al., 4 Apr 2025).
• Uncertainty quantification and robust deployment: Bayesian frameworks, ensemble and conformal methods, and kernel-based uncertainty estimation are identified as essential for reliable application of learned operators in scientific computing, control, and decision-making (Subedi et al., 4 Apr 2025, Li et al., 31 Oct 2024).

Operator learning thus provides a unified framework linking nonlinear dynamical systems, linear operator theory, data-driven modeling, and advanced machine learning. The field is evolving rapidly, integrating advances in computational efficiency, representation theory, uncertainty quantification, and active data acquisition to address foundational and practical challenges across science and engineering.