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
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
arXiv:2605. 01835v2 Announce Type: replace Abstract: Nonlinear coupled systems are ubiquitous in science and engineering.
By Tatsuya Naoi, Jun Ohkubo
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:2601. 17090v2 Announce Type: replace-cross Abstract: Partial differential equations (PDEs) govern complex systems, yet neural operators often struggle to efficiently capture the long-range, nonlocal interactions inherent in their solution maps.
By Noam Koren, Rafael Moschopoulos, Kira Radinsky, Elad Hazan
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:2607. 12570v1 Announce Type: cross Abstract: Multiscale problems are notoriously difficult to tackle using traditional numerical methods, as accurately resolving fine-scale features often requires prohibitively fine discretizations.
By Marc Haltmayer, Jaemin Seo, Yuseung Lee, Sungyeop Lee, Jaehoon Jeong, Jae Yong Lee
arXiv:2607. 29036v1 Announce Type: new Abstract: Sparse identification of nonlinear dynamics (SINDy) and PDE functional identification (PDE-FIND) recover parsimonious ordinary and partial differential equations (ODEs and PDEs) from data.
By Pongpisit Thanasutives, Yoshinobu Kawahara
The paper investigates how training shapes the geometry of recurrent network dynamics in Reservoir Computing (RC) networks used for temporal forecasting. It demonstrates that, in a linear continuous-time reservoir with infinite data, training data create an invariant subspace whose dimension matches the number of dominant modes. For a simplified diagonal linear reservoir, the study connects dominant eigenvalues and eigenvectors to the spectrum of a backward Dynamic Mode Decomposition matrix, providing a finite-dimensional approximation of the backward-time Koopman operator, and illustrates these phenomena through simulation while suggesting extensions to nonlinear RC.
By Noa Kaplan, Alberto Padoan, Anastasia Bizyaeva
arXiv:2507.07292v2 Announce Type: replace
Abstract: We develop a new and general encode-approximate-reconstruct operator learning model that leverages learned neural representations of bases for inpu...
By Jacob Hauck, Yanzhi Zhang
The paper introduces DOODL, a framework that learns a dictionary of spectral dynamics to represent related dynamical systems as points on a low‑dimensional manifold in operator space. By constraining operator estimation to this learned manifold, DOODL provides compact, interpretable embeddings and enables fast, accurate operator estimation from short, partially observed trajectories. Experiments on metastable Langevin dynamics and turbulent plasma simulations show that DOODL achieves one to two orders of magnitude lower errors than independent estimation methods, especially in low‑data regimes.
By Thibaut Germain, Sami Chemlal, R\'emi Flamary, Vladimir R. Kostic, Karim Lounici