arXiv:2607. 01819v1 Announce Type: cross Abstract: The Koopman operator has gained considerable attention due to its ability to provide a global linear representation of highly complex dynamical systems.
By Igor Mezi\'c, Jorge Cort\'es, Karl Worthmann, Mircea Lazar, Armin Lederer
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
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:2608. 11831v1 Announce Type: new Abstract: Learning mappings between infinite-dimensional objects is a central challenge in scientific machine learning.
By Adrien Weihs, Chunyang Liao, Jingmin Sun, Hayden Schaeffer
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:2504.18184v5 Announce Type: replace
Abstract: We consider a class of statistical inverse problems involving the estimation of a regression operator from a Polish space to a separable Hilbert sp...
By Jia-Qi Yang, Lei Shi