arXiv:2605. 01835v2 Announce Type: replace Abstract: Nonlinear coupled systems are ubiquitous in science and engineering.
By Tatsuya Naoi, Jun Ohkubo
This paper introduces a data‑driven neural power‑iteration algorithm for approximating the dominant eigenfunctions (modes) of the Koopman operator in nonlinear dynamical systems. By avoiding explicit construction of the operator’s projection, the method sidesteps the curse of dimensionality that plagues expressive neural templates. The authors provide theoretical convergence guarantees tied to sample size and network width, and demonstrate through numerical experiments that the approach yields accurate, smooth mode approximations without the drawbacks of traditional techniques such as extended dynamic mode decomposition.
By Guillaume O. Berger, Rapha\"el M. Jungers
arXiv:2604. 03086v2 Announce Type: replace-cross Abstract: This work establishes a rigorous bridge between infinite-dimensional delay dynamics and finite-dimensional Koopman learning, with explicit and interpretable error guarantees.
By Santosh Mohan Rajkumar, Dibyasri Barman, Kumar Vikram Singh, Debdipta Goswami
arXiv:2405. 07312v5 Announce Type: replace-cross Abstract: This paper presents a novel Koopman composition operator representation framework for control systems in reproducing kernel Hilbert spaces (RKHSs) that is free of explicit dictionary or input parametrizations.
By Petar Bevanda, Bas Driessen, Lucian Cristian Iacob, Stefan Sosnowski, Roland T\'oth, Sandra Hirche
The paper introduces K$^2$SVD, a method that learns the leading singular functions of the Koopman operator by optimizing a Hilbert-Schmidt objective, producing a low‑rank, interpretable Koopman representation with a compact latent space. In this space, temporal evolution is modeled with a linear Gaussian state‑space model and inference is performed via Kalman filtering to reduce noise accumulation in multi‑step predictions. Experiments demonstrate that K$^2$SVD outperforms state‑of‑the‑art methods on multiple datasets, achieving faster prediction speeds and lower computational cost.
By Ruiquan Li, Yuheng Bu
The paper introduces EDMD-kDL, a kernel-based method that learns finite-dimensional Koopman embeddings directly from data using bilevel optimization and collocation techniques. It combines kernel dictionary learning with extended dynamic mode decomposition to produce Koopman approximations without pre-specifying a dictionary. Experiments on sea-surface-temperature forecasting and video data show that EDMD‑kDL matches or surpasses ANN-based approaches while remaining scalable to large datasets.
By Joel-Pascal Ntwali N'konzi, Feliks N\"{u}ske, Stefan Klus