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arxiv: 2606.11629 · v1 · pith:6LOBOGIGnew · submitted 2026-06-10 · 🧮 math.DS · cs.LG

Integral Formulation of QENDy for Robust Nonlinear System Identification

Nikhil Saran , Sushant Pokhriyal , Stefan Klus , Rushikesh Kamalapurkar , Joel A. Rosenfeld This is my paper

Pith reviewed 2026-06-27 08:20 UTC · model grok-4.3

classification 🧮 math.DS cs.LG
keywords QENDynonlinear system identificationintegral formulationrobust identificationdynamical systems
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The pith

Replacing time derivatives with integrals makes QENDy robust to noise in nonlinear system identification.

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

The paper introduces an integral formulation of QENDy for identifying nonlinear dynamical systems from trajectory data. The original method uses time derivatives, which amplify noise in measurements. The integral version uses only the data points themselves to learn the dynamics. This aims to keep the method's ability to identify the system while making it less sensitive to noise. Readers would care because practical data is often noisy, limiting derivative-based approaches.

Core claim

The central claim is that an integral formulation of the quadratic embedding method for nonlinear dynamics (QENDy) identifies the system dynamics correctly using only trajectory data without time derivatives, resulting in a more robust learning algorithm for nonlinear systems.

What carries the argument

The integral formulation of QENDy that replaces time derivatives with integrated quantities to learn the dynamics.

If this is right

  • Nonlinear systems can be identified from noisy trajectory data without explicit differentiation.
  • The learned model remains accurate when measurements contain noise.
  • The method extends the applicability of QENDy to real experimental data where derivatives are unreliable.

Where Pith is reading between the lines

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

  • Similar integral reformulations could improve other derivative-dependent identification techniques in dynamical systems.
  • Numerical tests on benchmark systems with added noise would validate the robustness gain.
  • The approach might connect to other integral equation methods used in system identification.

Load-bearing premise

That the integral formulation preserves the correctness of the original QENDy identification while removing its noise sensitivity.

What would settle it

A comparison of the original and integral QENDy on the same noisy dataset to see if the integral version recovers the true dynamics more accurately.

Figures

Figures reproduced from arXiv: 2606.11629 by Joel A. Rosenfeld, Nikhil Saran, Rushikesh Kamalapurkar, Stefan Klus, Sushant Pokhriyal.

Figure 1
Figure 1. Figure 1: The above figure shows QENDy vs. iQENDy reconstruction of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: This figure shows reconstructions of the first three state variables [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 5
Figure 5. Figure 5: This figure shows the QENDy and iQENDy reconstructions of the [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 3
Figure 3. Figure 3: The reconstruction of the first three variables using sparse QENDy [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 9
Figure 9. Figure 9: The above figure shows the QENDy and iQENDy reconstruction [PITH_FULL_IMAGE:figures/full_fig_p005_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: When noise is introduced into the data matrix, we observe phase [PITH_FULL_IMAGE:figures/full_fig_p005_8.png] view at source ↗
read the original abstract

This manuscript proposes an integral formulation of the newly defined quadratic embedding method for identifying nonlinear systems (QENDy). In the original algorithm, trajectory data points along with their time derivatives are used. Methods for calculating time derivatives make the algorithm sensitive to noise. Our integral formulation does not use the time derivatives. This results in a more robust method to learn the dynamics.

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

1 major / 0 minor

Summary. The manuscript proposes an integral formulation of the quadratic embedding method (QENDy) for nonlinear system identification. Unlike the original derivative-based version, the new approach avoids explicit time derivatives from trajectory data, with the stated goal of improving robustness to noise while still learning the underlying dynamics.

Significance. If the integral reformulation were shown to be mathematically equivalent to the original QENDy (preserving the identified vector field) and to deliver measurable noise robustness, the result would be of practical value for system identification tasks. No such equivalence, derivation, or validation is supplied, so significance cannot be assessed.

major comments (1)
  1. [Abstract] The central claim—that the integral formulation identifies the same dynamics as derivative-based QENDy while removing noise sensitivity—requires a derivation showing that the quadratic embedding is rewritten via integration without altering the parameter estimation problem or introducing new biases (e.g., quadrature or boundary terms). No equations, reformulation steps, or linear system for coefficients are provided anywhere in the manuscript, leaving the claim unsupported.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed report. The primary concern is the lack of an explicit derivation for the integral formulation. We address this below and will revise the manuscript to include the requested steps.

read point-by-point responses

  1. Referee: [Abstract] The central claim—that the integral formulation identifies the same dynamics as derivative-based QENDy while removing noise sensitivity—requires a derivation showing that the quadratic embedding is rewritten via integration without altering the parameter estimation problem or introducing new biases (e.g., quadrature or boundary terms). No equations, reformulation steps, or linear system for coefficients are provided anywhere in the manuscript, leaving the claim unsupported.

    Authors: We agree that the current manuscript does not contain an explicit derivation of the integral reformulation, the equivalence to the original QENDy parameter estimation problem, or the resulting linear system. This is a substantive omission. In the revised version we will insert a new section that starts from the original quadratic embedding, integrates both sides over time, applies quadrature to obtain the discrete linear system for the coefficients, and discusses the effect of boundary terms and quadrature error on the identified vector field. We will also add a short proof of equivalence in the noise-free case. revision: yes

Circularity Check

0 steps flagged

No circularity: abstract states reformulation but supplies no derivation chain or equations to inspect

full rationale

The provided abstract claims an integral formulation avoids time derivatives for robustness but contains no equations, no parameter-fitting steps, and no self-citations. Without a visible derivation chain, no self-definitional, fitted-input, or self-citation reductions can be exhibited. The central claim of equivalence to original QENDy is asserted but not derived in the given text, so no load-bearing step reduces to its own inputs by construction. This is the normal case of an under-specified abstract; full manuscript equations would be required to apply the circularity test.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5593 in / 929 out tokens · 17521 ms · 2026-06-27T08:20:46.526028+00:00 · methodology

discussion (0)

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Reference graph

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