analiza aiqbench
This research introduces a neural power iteration algorithm to approximate dominant Koopman modes for nonlinear dynamical systems, offering a data-driven approach that avoids explicit operator projection. By leveraging neural networks within a lifted space, the method simplifies complex dynamics into linear forms to facilitate control and analysis while providing theoretical convergence guarantees. The technique effectively addresses dimensionality challenges without requiring traditional anti-collapse mechanisms, making it a significant advancement for machine learning applications in systems control.
Data-driven Koopman mode approximation: A neural power iteration algorithm
arXiv:2608.26943v1 Announce Type: cross Abstract: This paper proposes a novel data-driven algorithm to approximate the dominant eigenfunctions (aka.~modes) of the Koopman operator of nonlinear dynamical systems using neural networks. The relevance of learning the dominant Koopman modes is to approximate nonlinear dynamics by linear ones in a lifted space, thereby enabling simplified control and analysis. To fight the curse of dimensionality arising from using expressive templates (here neural networks) for the mode approximation, the proposed method leverages a power-iteration scheme that directly learns the dominant Koopman modes without explicitly constructing the projection of the Koopman operator on the template of functions. Our approach connects to other approaches in the literature that avoid the curse of dimensionality by learning small dictionaries of functions, but differs from them in that we do not require ``anti-collapse mechanisms'' to ensure that the learned dictionary is expressive enough to approximate the Koopman operator since our power-iteration scheme is designed to converge toward the dominant modes of the projected Koopman operator. The approach is fully data-driven, requiring only sampled state transitions. Theoretical guarantees are provided, showing convergence under increasing sample size and network width (in connection with the neural tangent kernel theorem). Numerical experiments demonstrate that the method achieves accurate and smooth approximations of dominant modes while avoiding the limitations of traditional techniques such as extended dynamic mode decomposition.
Abstract:This paper proposes a novel data-driven algorithm to approximate the dominant eigenfunctions (aka.~modes) of the Koopman operator of nonlinear dynamical systems using neural networks. The relevance of learning the dominant Koopman modes is to approximate nonlinear dynamics by linear ones in a lifted space, thereby enabling simplified control and analysis. To fight the curse of dimensionality arising from using expressive templates (here neural networks) for the mode approximation, the proposed method leverages a power-iteration scheme that directly learns the dominant Koopman modes without explicitly constructing the projection of the Koopman operator on the template of functions. Our approach connects to other approaches in the literature that avoid the curse of dimensionality by learning small dictionaries of functions, but differs from them in that we do not require ``anti-collapse mechanisms'' to ensure that the learned dictionary is expressive enough to approximate the Koopman operator since our power-iteration scheme is designed to converge toward the dominant modes of the projected Koopman operator. The approach is fully data-driven, requiring only sampled state transitions. Theoretical guarantees are provided, showing convergence under increasing sample size and network width (in connection with the neural tangent kernel theorem). Numerical experiments demonstrate that the method achieves accurate and smooth approximations of dominant modes while avoiding the limitations of traditional techniques such as extended dynamic mode decomposition.
Systems and Control (eess.SY); Machine Learning (cs.LG); Optimization and Control (math.OC)
arXiv:2608.26943 [eess.SY]
(or arXiv:2608.26943v1 [eess.SY] for this version)
https://doi.org/10.48550/arXiv.2608.26943 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
Text integral sub licența deschisă a publicației · 320 cuvinte
Subiecte conexe
[AINews] Reality Checks on AI News (Yegge shuts down Gas Town, Databricks’ +60% Astra cost)
Latent Space · 17 sept. 2026
Synthesis Superintelligence: from Semiconductors to Superconductors — Periodic Labs’ Liam Fedus and Ekin Dogus Cubuk
Latent Space · 8 oct. 2026
Koopman Theory and Metaethics
LessWrong · 19 sept. 2026
Bilevel optimization for data-driven learning of Koopman embeddings using kernel-based autoencoders
arXiv · 9 oct. 2026