arXiv:2606. 29083v1 Announce Type: cross Abstract: Koopman theory promises linear structure in nonlinear dynamics, but numerical Koopman spectra are easy to compute and hard to trust.
By George Coote, Matthew J. Colbrook
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: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 Functional Dynamic Mode Decomposition (fDMD), extending traditional DMD to infinite-dimensional systems by learning finite-rank operators from functional data such as observables, densities, or wavefunctions. It demonstrates that conventional DMD algorithms are special cases of fDMD and illustrates the approach with examples involving Koopman, Perron‑Frobenius, and Koopman‑von Neumann operators for graphons, ODEs, and SDEs.
By Stefan Klus, Eirini Ioannou
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:2609.35938v1 Announce Type: new
Abstract: This paper proposes an interpretable neural operator framework, the Kernel Operator Network (KernelOnet), which incorporates kernel functions explicitl...
By Yuan Guo, Hanshu Chen, Qiang Xi, Timon Rabczuk, Zhuojia Fu