On Data-Driven Koopman Representations of Nonlinear Delay Differential Equations

TL;DR

This paper develops a rigorous, data-driven Koopman framework for delay differential equations, providing explicit error bounds and demonstrating convergence for reliable prediction and control.

Contribution

It introduces a finite-dimensional Koopman approximation for DDEs using history discretization and kernel methods, with provable error guarantees.

Findings

Convergence of the learned predictor with respect to discretization and data size.

Explicit error bounds decomposing discretization, interpolation, and regression errors.

Numerical results confirming the effectiveness of the approach.

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.