Efficient Parametric SVD of Koopman Operator for Stochastic Dynamical Systems

2025-07-09

AI Summary

Existing Koopman operator learning methods (e.g., VAMPnet, DPNet) rely on singular value decomposition (SVD) and inversion of empirical second-order moment matrices, leading to numerical instability, gradient bias, and poor scalability to large-scale systems. To address these limitations, we propose a scalable low-rank approximation framework that bypasses explicit matrix factorization, enabling stable and differentiable learning of Koopman singular functions. Our approach integrates principles from dynamic mode decomposition (DMD) with deep learning: it directly parameterizes the low-rank singular subspace via end-to-end optimization, eliminating numerically sensitive operations such as SVD and matrix inversion. Experiments demonstrate that the learned subspaces yield robust performance in spectral analysis and multi-step prediction, exhibit strong generalization across diverse dynamical systems, and significantly improve both training stability and scalability—particularly for high-dimensional and large-scale problems.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

The Koopman operator provides a principled framework for analyzing nonlinear dynamical systems through linear operator theory. Recent advances in dynamic mode decomposition (DMD) have shown that trajectory data can be used to identify dominant modes of a system in a data-driven manner. Building on this idea, deep learning methods such as VAMPnet and DPNet have been proposed to learn the leading singular subspaces of the Koopman operator. However, these methods require backpropagation through potentially numerically unstable operations on empirical second moment matrices, such as singular value decomposition and matrix inversion, during objective computation, which can introduce biased gradient estimates and hinder scalability to large systems. In this work, we propose a scalable and conceptually simple method for learning the top-k singular functions of the Koopman operator for stochastic dynamical systems based on the idea of low-rank approximation. Our approach eliminates the need for unstable linear algebraic operations and integrates easily into modern deep learning pipelines. Empirical results demonstrate that the learned singular subspaces are both reliable and effective for downstream tasks such as eigen-analysis and multi-step prediction.

Problem

• Learning Koopman operator singular functions stably
• Avoiding unstable matrix operations in deep learning
• Enhancing scalability for large stochastic dynamical systems

Innovation

• Uses low-rank approximation for Koopman operator
• Avoids unstable matrix operations in learning
• Integrates with modern deep learning pipelines