# On Data-Driven Koopman Representations of Nonlinear Delay Differential Equations

arXiv: https://arxiv.org/abs/2604.03086 · HTML: https://mlsanity.com/paper/2604.03086/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 31 references that are in the map.
- First posted on arXiv 3 Apr 2026.

## Builds on (most cited first, 9 of 9)

- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [Temporally Consistent Koopman Autoencoders for Forecasting Dynamical Systems](https://mlsanity.com/paper/2403.12335/) (2024)
- [An overview of Koopman-based control: From error bounds to closed-loop guarantees](https://mlsanity.com/paper/2509.02839/) (2025)
- [Equivariance and partial observations in Koopman operator theory for partial differential equations](https://mlsanity.com/paper/2307.15325/) (2023)
- [Kernel-based Koopman approximants for control: Flexible sampling, error analysis, and stability](https://mlsanity.com/paper/2412.02811/) (2024)
- [On minimizing the training set fill distance in machine learning regression](https://mlsanity.com/paper/2307.10988/) (2023)
- [Kernel-based error bounds of bilinear Koopman surrogate models for nonlinear data-driven control](https://mlsanity.com/paper/2503.13407/) (2025)
- [Online Delay Estimation and Adaptive Compensation in Wireless Networked System: An Embedded Control Design](https://mlsanity.com/paper/1912.00343/) (2019)
- [Real-Time Linear MPC for Quadrotors on SE(3): An Analytical Koopman-based Realization](https://mlsanity.com/paper/2409.12374/) (2024)

## Abstract (arXiv)

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 reliable prediction of nonlinear delay systems, with potential relevance to future control applications.