koopman operator theory

Koopman operator learning for nonlinear dynamical systems traditionally relies on pre-specified, finite-dimensional observable spaces, imposing restrictive closure assumptions. Method: This paper introduces an online sparse learning paradigm within a reproducing kernel Hilbert space (RKHS), establishing— for the first time—theoretical equivalence between the Koopman operator and the conditional mean embedding operator. The proposed framework is fully nonparametric and does not assume observables’ closure. It integrates conditional mean embeddings, stochastic operator approximation, sparse kernel methods, and trajectory-driven optimization. Contribution/Results: We provide the first finite-time last-iterate convergence guarantee under trajectory sampling. Numerical experiments demonstrate substantial improvements in dynamic prediction accuracy and generalization performance, while maintaining representation sparsity and robust convergence—empirically validating the theoretical guarantees.

This work addresses modeling bias in Koopman operator learning for nonlinear dynamical systems under non-closed function spaces. Methodologically, it introduces a nonparametric sparse online learning framework, formulating the Koopman operator action as a conditional mean embedding (CME) in a reproducing kernel Hilbert space (RKHS) and devising a trajectory-sampling-based online sparse stochastic approximation algorithm. Theoretical contributions include the first convergence analysis for online Koopman learning in non-closed, nonparametric settings, yielding a finite-time last-iterate error bound and extending beyond classical finite-dimensional stochastic approximation frameworks. Experiments demonstrate substantial improvements in generalization performance and computational efficiency over existing approaches.

The Koopman operator provides a data-driven linearization framework for nonlinear dynamical systems, but its infinite-dimensionality impedes spectral estimation convergence and undermines reliability in analyzing continuous spectra and systems lacking spectral gaps. Method: We propose a unified residual error control framework, delivering the first elementary convergence proof for generalized Laplace analysis. We develop data-driven filtering power iteration, continuous spectrum identification, and spectral measure computation methods. Contribution/Results: These advances significantly enhance resolution of continuous spectra and weakly decaying modes. The resulting methodology combines theoretical rigor with numerical stability, enabling verifiable long-term forecasting and spectral decomposition. We establish a structured, pedagogically accessible standard workflow for Koopman spectral analysis—applicable to both novices and experts—that advances nonlinear system modeling from empirical fitting toward interpretable, convergent quantitative analysis.

High-dimensional spatiotemporal chaotic systems are often dominated by continuous spectra, yet existing data-driven approaches frequently suffer from instability, limited interpretability, and poor scalability. This work proposes KoopGen—a generator-based neural Koopman framework that explicitly decomposes dynamics into conservative (skew-adjoint) and dissipative (self-adjoint) components via a state-dependent Koopman generator, while rigorously embedding operator-theoretic constraints. Notably, KoopGen achieves the first explicit separation of self-adjoint and skew-adjoint parts within the generator without relying on finite-dimensional assumptions or explicit spectral parameterizations. Experiments ranging from nonlinear oscillators to high-dimensional chaotic systems demonstrate that KoopGen substantially improves long-term prediction accuracy and stability, uncovering learnable and interpretable structural components underlying continuous-spectrum dynamics.

In dynamical systems modeling, the selection of basis function dictionaries often relies on problem-specific prior knowledge and lacks adaptability. To address this bottleneck, we propose a gradient-based learnable dictionary optimization framework: for the first time, the basis function dictionary is parameterized as trainable weights and jointly optimized with model parameters via end-to-end backpropagation. This unified approach enhances three prominent data-driven modeling paradigms—Extended Dynamic Mode Decomposition (EDMD), Sparse Identification of Nonlinear Dynamics (SINDy), and PDE-FIND—while preserving basis interpretability and significantly improving generalization and robustness. Extensive experiments on diverse benchmarks—including the Ornstein–Uhlenbeck process, Chua’s circuit, a nonlinear heat equation, and protein folding dynamics—demonstrate consistent improvements in modeling accuracy. The results validate the framework’s broad applicability across heterogeneous dynamical systems and modeling tasks.

This study addresses the challenges of parameter identifiability, long-term prediction instability, and limited interpretability in nonlinear epidemic models by proposing a novel integration of Koopman operator theory with physics-informed neural networks (PINNs). The approach embeds epidemic dynamics as hard constraints via automatic differentiation and leverages structure-preserving nonstandard finite difference schemes to generate high-fidelity training data. By lifting the system into a latent observable space where evolution is approximately linear, the method substantially enhances both parameter identifiability and long-term predictive stability while improving model interpretability. Comprehensive experiments on synthetic mpox data and real-world COVID-19 datasets from Germany, Morocco, and Switzerland demonstrate that the proposed framework consistently outperforms conventional PINNs and Koopman-EDMD methods in parameter estimation, trajectory reconstruction, and long-range forecasting.

This work proposes a learning and control framework based on Koopman operator regression for nonlinear switched systems with unknown dynamics. By leveraging finite data, the system dynamics are learned within a reproducing kernel Hilbert space to construct a linear switched predictive model, which is then integrated with model predictive control (MPC) to solve an infinite-horizon optimal control problem. The approach provides, for the first time, a theoretically guaranteed closed-loop control strategy for switched nonlinear systems, with rigorous derivation of both the learning rate for Koopman dynamic approximation and a suboptimality bound on the closed-loop MPC performance. Numerical experiments on the Duffing oscillator demonstrate the effectiveness of the proposed method.

This study addresses the challenge of global linear modeling and control for highly nonlinear dynamical systems by leveraging Koopman operator theory. By introducing observable functions, the nonlinear dynamics are lifted into a higher-dimensional space where they admit an approximately linear representation. A data-driven surrogate model is constructed through a synergistic integration of Extended Dynamic Mode Decomposition (EDMD), kernelized EDMD, and machine learning techniques. The work innovatively extends the Koopman framework to input-affine systems, proposing a unified modeling approach and a corresponding Koopman-based Model Predictive Control (MPC) design methodology. Numerical simulations demonstrate that the proposed method achieves high-fidelity modeling accuracy and effective closed-loop control performance. Full reproducibility is supported by the accompanying open-source implementation.

This work addresses the performance degradation of machine learning models in non-stationary environments caused by temporal domain drift by proposing a model-agnostic, zero-retraining adaptive framework. The approach models the sequence of model parameters as a trajectory of a nonlinear dynamical system and identifies its linear Koopman operator using extended dynamic mode decomposition (EDMD) with a Fourier-augmented observation dictionary. Leveraging a warm-start training protocol, the framework autonomously predicts future parameter trajectories without requiring future labels, enabling efficient adaptation. Moreover, it uncovers an interpretable dynamical structure underlying decision boundary drift. Evaluated across six datasets, the method achieves average accuracies between 0.981 and 1.000 over 100 future timesteps, demonstrating robustness and effectiveness under diverse distribution shift scenarios.

This work addresses the challenge of simultaneously achieving stability, interpretability, and generalization in time series forecasting by proposing a novel architecture that integrates learnable Koopman operators with Transformer-based backbones such as PatchTST, Informer, and Autoformer. By designing four variants of the Koopman operator, the method enables explicit control over the spectral properties, stability, and rank of the linear transition operator within deep forecasting models for the first time, allowing flexible interpolation between strictly stable and unconstrained dynamics. Experiments demonstrate that the approach significantly improves the bias-variance trade-off, numerical conditioning, and interpretability of latent dynamics across multi-horizon forecasting tasks, effectively combining theoretical guarantees with data-driven flexibility.