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arxiv: 2604.03086 · v1 · submitted 2026-04-03 · 📡 eess.SY · cs.LG· cs.SY· math.DS

On Data-Driven Koopman Representations of Nonlinear Delay Differential Equations

Santosh Mohan Rajkumar , Dibyasri Barman , Kumar Vikram Singh , Debdipta Goswami This is my paper

Pith reviewed 2026-05-13 19:13 UTC · model grok-4.3

classification 📡 eess.SY cs.LGcs.SYmath.DS
keywords Koopman operatordelay differential equationskernel methodsextended dynamic mode decompositionerror boundsdata-driven modelingsystem identification
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The pith

History discretization and kernel methods turn infinite-dimensional delay differential equations into finite Koopman models with explicit error bounds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a finite-dimensional Koopman approximation for nonlinear delay differential equations by discretizing the system's history and applying kernel-based extended dynamic mode decomposition. This approach handles the infinite-dimensional phase space of DDEs while providing deterministic error bounds that separate contributions from discretization, interpolation, and regression. A kernel reconstruction operator recovers the discretized states from the lifted Koopman coordinates with provable accuracy. Numerical experiments show the predictor's error decreases as the discretization resolution increases and as more training data is added. The framework aims to support reliable prediction and control tasks for delay systems.

Core claim

A finite-dimensional Koopman approximation framework for nonlinear DDEs is obtained via history discretization and a suitable reconstruction operator, represented tractably through kernel-based extended dynamic mode decomposition, with deterministic error bounds that decompose the total error into history discretization, kernel interpolation, and data-driven regression contributions, together with a kernel-based reconstruction method that recovers discretized states from lifted coordinates under provable guarantees.

What carries the argument

History discretization combined with kernel-based extended dynamic mode decomposition, which reduces infinite-dimensional delay dynamics to a finite lifted space while enabling explicit error decomposition and state reconstruction.

If this is right

  • The learned predictor converges with increasing discretization resolution and training data volume.
  • Explicit decomposition of error sources allows interpretable assessment of approximation quality.
  • Kernel reconstruction recovers discretized states from lifted Koopman coordinates with guarantees.
  • The finite representation supports prediction and control applications for nonlinear delay systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The discretization approach could adapt to other infinite-dimensional systems such as certain PDEs by choosing suitable history or spatial grids.
  • The error bounds may guide selection of discretization parameters to balance accuracy and computational cost in applications.
  • Integration with control design methods could yield data-driven controllers that explicitly account for the delay-induced error terms.

Load-bearing premise

History discretization combined with kernel interpolation produces controllable error for the nonlinear DDEs of interest, allowing the derived bounds to remain useful in practice.

What would settle it

If the prediction error does not decrease with finer history discretization resolution or larger training data sets, in a manner inconsistent with the derived convergence and bounds, the practical utility of the approximation would be falsified.

Figures

Figures reproduced from arXiv: 2604.03086 by Debdipta Goswami, Dibyasri Barman, Kumar Vikram Singh, Santosh Mohan Rajkumar.

Figure 1
Figure 1. Figure 1: Step-wise mean prediction error µzk for (26) (τd = 1 sec): (a) increasing M (hX = 0.057), (b) increasing p, (c) decreasing ρ (p = 169); shaded area depicts the range from worst to best error. 0 1 2 3 4 5 Time (s) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 x(t) True Predicted (a) (τd = 1 sec) 0 1 2 3 4 5 Time (s) 0.6 0.8 1.0 1.2 1.4 1.6 1.8 x 1(t) True Predicted (b) (τd = 1.64 sec) 0 1 2 3 4 5 Time (s) 0.0 0.5 1.0… view at source ↗
Figure 2
Figure 2. Figure 2: Evolution of current value of the state function ( [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a framework for approximating the Koopman operator for nonlinear delay differential equations (DDEs) using history discretization combined with kernel-based extended dynamic mode decomposition (kEDMD). It derives deterministic error bounds decomposing the total error into history discretization, kernel interpolation, and regression terms, develops a kernel-based reconstruction from lifted coordinates, and provides numerical evidence of convergence with respect to discretization resolution and data size.

Significance. If the derived bounds are valid and practical, this work offers a rigorous and interpretable approach to data-driven modeling of DDEs, which are prevalent in control and dynamical systems. The decomposition of errors into interpretable components is a notable strength, potentially allowing users to balance discretization and data requirements. The extension from ODEs/PDEs to DDEs addresses an important gap, and the provision of reconstruction guarantees enhances applicability to prediction and control tasks.

major comments (2)
  1. [Error bound derivation] The discretization error is claimed to be controllable and proportional to mesh size, but the analysis must address potential amplification by the nonlinear delay operator. For DDEs with only locally Lipschitz nonlinearities, the bound may include an interaction term that depends on the trajectory and could prevent uniform convergence as the history mesh is refined, contrary to the abstract's assertion of provable guarantees.
  2. [Numerical experiments] The numerical results demonstrate convergence but lack quantitative tables comparing the observed errors to the theoretical bounds or reporting the individual error components (discretization vs. regression), making it difficult to verify the tightness and practical utility of the derived guarantees.
minor comments (2)
  1. [Notation] The definition of the reconstruction operator and its kernel could be presented with more explicit formulas to improve readability.
  2. [References] Ensure all relevant prior work on Koopman operators for infinite-dimensional systems is cited for context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and insightful comments on our manuscript. The feedback highlights important aspects of the error analysis and experimental validation. We address each major comment point by point below and will revise the manuscript accordingly to improve clarity and completeness.

read point-by-point responses

  1. Referee: The discretization error is claimed to be controllable and proportional to mesh size, but the analysis must address potential amplification by the nonlinear delay operator. For DDEs with only locally Lipschitz nonlinearities, the bound may include an interaction term that depends on the trajectory and could prevent uniform convergence as the history mesh is refined, contrary to the abstract's assertion of provable guarantees.

    Authors: We appreciate this careful reading. Our derivation of the discretization error bound explicitly uses the local Lipschitz continuity of the nonlinearity and the delay operator. The analysis is performed on compact invariant sets containing the trajectories of interest, where the local Lipschitz constant is finite; this yields an O(h) bound (h = mesh size) whose prefactor depends on this constant and the trajectory length but remains independent of h. Consequently, the bound guarantees convergence as h → 0 for any fixed bounded trajectory. We agree, however, that the dependence on the local Lipschitz factor and the restriction to compact sets should be stated more explicitly to preclude misinterpretation. In the revised manuscript we will insert a clarifying remark immediately after the discretization-error theorem, reiterating the local-Lipschitz assumption and the compact-set hypothesis, and we will adjust the abstract wording to read “provable guarantees under standard local-Lipschitz and bounded-trajectory assumptions.” These changes preserve the original results while addressing the referee’s concern. revision: yes

  2. Referee: The numerical results demonstrate convergence but lack quantitative tables comparing the observed errors to the theoretical bounds or reporting the individual error components (discretization vs. regression), making it difficult to verify the tightness and practical utility of the derived guarantees.

    Authors: We fully agree that quantitative verification of the error decomposition would strengthen the paper. In the revised version we will add a new table (Table 2) in Section 4. For each combination of history-mesh resolution and training-set size we will report: (i) the observed total prediction error on a held-out test trajectory, (ii) the separately estimated discretization error (obtained by comparing the discretized DDE solution to the true DDE solution), (iii) the kernel-interpolation error, and (iv) the regression error. We will also list the corresponding theoretical upper bounds derived in the paper and compute the ratio of observed to bound values, thereby allowing readers to assess tightness and the relative contribution of each error source. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained with independently derived error bounds

full rationale

The paper constructs a finite-dimensional Koopman approximation for DDEs via history discretization and a reconstruction operator, then applies kEDMD to obtain the lifted representation. Deterministic error bounds are explicitly decomposed into discretization, interpolation, and regression contributions using standard approximation theory arguments applied to the proposed operators. No step reduces a claimed prediction to a fitted parameter by construction, invokes a self-citation as the sole justification for a uniqueness or convergence claim, or renames an empirical pattern as a new derivation. The bounds are stated to follow from the framework assumptions rather than being calibrated to the target prediction error, rendering the central claims independent of the inputs they bound.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on standard domain assumptions about approximability of DDEs by finite history segments and properties of kernel methods; no new entities are postulated and no free parameters are explicitly fitted in the abstract.

free parameters (1)
  • history discretization resolution
    Number of past points retained; controls the discretization error term but value is chosen by user.
axioms (1)
  • domain assumption Delay differential equations admit finite-history approximations whose error can be bounded independently of the Koopman lift.
    This assumption enables reduction from infinite-dimensional dynamics to a tractable finite-dimensional problem.

pith-pipeline@v0.9.0 · 5487 in / 1136 out tokens · 41880 ms · 2026-05-13T19:13:31.537190+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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    Relation between the paper passage and the cited Recognition theorem.

    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... decomposing the total error into contributions from history discretization, kernel interpolation, and data-driven regression

What do these tags mean?
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The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
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The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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