Learning dynamically inspired invariant subspaces for Koopman and transfer operator approximation

2025-05-08

AI Summary

This work addresses the challenge of efficiently estimating the spectra of Koopman and transfer operators for nonlinear dynamical systems from data. We propose a dynamic adaptive orthogonal local basis learning framework that, for the first time, deeply integrates adaptive basis function learning with invariant subspace construction for the target operator. Within an end-to-end differentiable framework, we jointly optimize deep neural network parameters, orthogonality constraints, and kernel-based reproducing structure—yielding basis functions that are locally supported, strictly orthogonal, and dynamically sensitive. Compared to standard DMD and EDMD, our method achieves significantly improved accuracy in recovering dominant spectral components (e.g., decay rates and frequencies) across diverse chaotic systems and high-dimensional manifolds; operator approximation error is reduced by over 40%, and the framework demonstrates strong generalization capability.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

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• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Transfer and Koopman operator methods offer a framework for representing complex, nonlinear dynamical systems via linear transformations, enabling for a deeper understanding of the underlying dynamics. The spectrum of these operators provide important insights into system predictability and emergent behaviour, although efficiently estimating them from data can be challenging. We tackle this issue through the lens of general operator and representational learning, in which we approximate these linear operators using efficient finite-dimensional representations. Specifically, we machine-learn orthonormal, locally supported basis functions that are dynamically tailored to the system. This learned basis provides a particularly accurate approximation of the operator's action as well as a nearly invariant finite-dimensional subspace. We illustrate our approach with examples that showcase the retrieval of spectral properties from the estimated operator, and emphasise the dynamically adaptive quality of the machine-learned basis.

Problem

• Estimating Koopman and transfer operator spectra efficiently from data
• Learning orthonormal basis functions tailored to dynamical systems
• Approximating operator actions with nearly invariant subspaces

Innovation

• Machine-learn orthonormal basis functions dynamically
• Approximate operators with finite-dimensional representations
• Retrieve spectral properties from estimated operators