Problem
Research questions and friction points this paper is trying to address.
Koopman operator
delay differential equations
infinite-dimensional dynamics
data-driven modeling
history discretization
On Data-Driven Koopman Representations of Nonlinear Delay Differential Equations
🤖 AI Summary
This work addresses the challenge of applying finite-dimensional Koopman operator methods to nonlinear delay differential equations, whose infinite-dimensional phase spaces hinder direct analysis. The paper presents the first finite-dimensional Koopman approximation framework with explicit error bounds, achieved through discretization of historical states and kernel-based reconstruction of the Koopman operator to enable learnable dynamical modeling. By integrating kernel extended dynamic mode decomposition (kEDMD), kernel interpolation, and data-driven regression, the approach introduces a theoretically grounded state reconstruction strategy. Numerical experiments demonstrate the convergence of the predictor with respect to both discretization resolution and training data volume, offering a reliable tool for high-accuracy prediction and control of delay systems.
📝 Abstract
This work establishes a rigorous bridge between infinite-dimensional delay dynamics and finite-dimensional Koopman learning, with explicit and interpretable error guarantees. While Koopman analysis is well-developed for ordinary differential equations (ODEs) and partially for partial differential equations (PDEs), its extension to delay differential equations (DDEs) remains limited due to the infinite-dimensional phase space of DDEs. We propose a finite-dimensional Koopman approximation framework based on history discretization and a suitable reconstruction operator, enabling a tractable representation of the Koopman operator via kernel-based extended dynamic mode decomposition (kEDMD). Deterministic error bounds are derived for the learned predictor, decomposing the total error into contributions from history discretization, kernel interpolation, and data-driven regression. Additionally, we develop a kernel-based reconstruction method to recover discretized states from lifted Koopman coordinates, with provable guarantees. Numerical results demonstrate convergence of the learned predictor with respect to both discretization resolution and training data, supporting reliable prediction and control of delay systems.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Koopman operator
delay differential equations
history discretization
kernel-based EDMD
error bounds
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