EDMD-kDL: Bilevel Optimization for Data-Driven Learning of Koopman Embeddings with Kernel Methods

Forum topic · 小凯 · 2026-10-12

Summary

Researchers Joel-Pascal Ntwali N'konzi, Feliks Nüske, and Stefan Klus introduce EDMD-kDL, a kernel-based method for learning finite-dimensional Koopman embeddings directly from data, presented in arXiv paper 2610.12370. Koopman operator theory offers a linear framework for analyzing nonlinear dynamical systems, but methods like extended dynamic mode decomposition (EDMD) require the observation dictionary to be specified in advance. While recent machine-learning approaches learn the dictionary using artificial neural network autoencoders, kernel methods remain underexplored despite their interpretability and theoretical tractability. EDMD-kDL combines a collocation approach with bilevel optimization to simultaneously learn the kernel dictionary and the corresponding Koopman approximation. In numerical experiments including global sea surface temperature prediction and learning directly from video data, EDMD-kDL matches or exceeds state-of-the-art ANN methods across all tested scenarios. Unlike standard kernel methods, its kernel matrix size depends on the number of collocation points rather than the training set size, making the approach scalable to large datasets by design.

Paper Overview

Field: Machine Learning Authors: Joel-Pascal Ntwali N'konzi, Feliks Nüske, Stefan Klus Published: 2026-10-08 arXiv: 2610.12370

Abstract

Koopman operator theory provides a linear framework for analyzing nonlinear dynamical systems and has become a major tool for data-driven modeling. A central challenge, however, is that finite-dimensional approximations computed by methods such as extended dynamic mode decomposition (EDMD) require the dictionary to be specified a priori. Recent machine-learning approaches address this limitation by learning the dictionary from data, predominantly using artificial neural network (ANN) autoencoder architectures. Although kernel methods offer an alternative with greater interpretability and tractability for theoretical analysis, they have received little attention in this setting.

The authors introduce extended dynamic mode decomposition with kernel-based dictionary learning (EDMD-kDL), a kernel method for directly learning finite-dimensional Koopman embeddings from data. The approach combines a collocation method with ideas from bilevel optimization, simultaneously learning the kernel dictionary and the corresponding Koopman approximation.

Key Results

In a series of numerical experiments—including global sea surface temperature prediction and learning directly from video data—EDMD-kDL matches or outperforms state-of-the-art ANN-based methods in all tested scenarios.
Unlike standard kernel methods, the size of the kernel matrix in EDMD-kDL depends on the number of collocation points rather than the size of the training set, making the method scalable to large datasets by design.