arXiv:2606. 05131v1 Announce Type: new Abstract: Koopman theory turns nonlinear dynamics into a linear spectral problem.
By Kelan Gray, Finlay Brown, Nicolas Boull\'e, Matthew J. Colbrook
arXiv:2407. 06312v2 Announce Type: replace-cross Abstract: Many systems resist analytical modeling, making data-driven inference of dynamics important.
By Matthew J. Colbrook, Igor Mezi\'c, Alexei Stepanenko
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
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
arXiv:2609. 23529v1 Announce Type: new Abstract: Neural operators have emerged as powerful surrogates for solving partial differential equations (PDEs), yet their reliability under distribution shift remains a critical barrier to deployment.
By Hang-Cheng Dong, Pengcheng Cheng
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