On Data-Driven Koopman Representations of Nonlinear Delay Differential Equations
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2026
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| author | Rajkumar, Santosh Mohan Barman, Dibyasri Singh, Kumar Vikram Goswami, Debdipta |
| author_facet | Rajkumar, Santosh Mohan Barman, Dibyasri Singh, Kumar Vikram Goswami, Debdipta |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03086 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Data-Driven Koopman Representations of Nonlinear Delay Differential Equations Rajkumar, Santosh Mohan Barman, Dibyasri Singh, Kumar Vikram Goswami, Debdipta Systems and Control Machine Learning Dynamical Systems 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. |
| title | On Data-Driven Koopman Representations of Nonlinear Delay Differential Equations |
| topic | Systems and Control Machine Learning Dynamical Systems |
| url | https://arxiv.org/abs/2604.03086 |