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arxiv: 2604.23044 · v1 · submitted 2026-04-24 · 🧮 math.OC · math.DS

Nonlinear balanced truncation model reduction through scalable Taylor series

Nicholas A. Corbin , Boris Kramer This is my paper

Pith reviewed 2026-05-08 10:53 UTC · model grok-4.3

classification 🧮 math.OC math.DS
keywords nonlinear balanced truncationmodel reductionTaylor seriesKronecker productpolynomial approximationcontrol systemssystem theory
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The pith

Taylor series approximations make nonlinear balanced truncation scalable and produce true nonlinear reduced models.

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

The paper develops polynomial approximations to the balancing transformation and the balanced realization of nonlinear systems, represented efficiently with Kronecker products. These approximations permit truncation of redundant states to obtain reduced-order models that remain nonlinear and aim to retain stability, controllability, and observability. A sympathetic reader would care because many engineering systems exhibit strong nonlinear behavior where linear reduction techniques lose essential dynamics. The method leverages recent linear algebra tools to reach higher state dimensions than earlier nonlinear balancing approaches. Numerical tests on examples highlight both practical gains and remaining constraints of the technique.

Core claim

We derive polynomial approximations for the balancing transformation and the explicit balanced realization of the full-order model, which yields true nonlinear reduced-order models upon truncation of redundant state components.

What carries the argument

Kronecker product representation of Taylor series polynomial approximations to the balancing transformation and balanced realization.

If this is right

  • Truncation after the polynomial approximation step directly produces nonlinear reduced-order models instead of linearized ones.
  • Scalability follows from efficient Kronecker-based polynomial arithmetic that handles larger state dimensions.
  • Stability, controllability, and observability properties are retained when the series order suffices for the system at hand.
  • Numerical examples demonstrate concrete benefits and limitations of nonlinear balancing that earlier studies did not quantify.

Where Pith is reading between the lines

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

  • The same series approximation strategy could extend to other nonlinear reduction methods that rely on coordinate transformations.
  • Adaptive choice of truncation order based on local nonlinearity strength might further improve accuracy without manual tuning.
  • Control design performed on the resulting reduced nonlinear models could be tested for closed-loop performance on the original system.

Load-bearing premise

Low-order Taylor series approximations must remain accurate enough for the nonlinear dynamics so that truncation still preserves stability, controllability, and observability.

What would settle it

Apply the method to a nonlinear system whose exact balanced realization is known analytically; if the reduced-model trajectories deviate substantially from the full-order trajectories in stability or response, the approximations have failed to deliver the claimed preservation.

Figures

Figures reproduced from arXiv: 2604.23044 by Boris Kramer, Nicholas A. Corbin.

Figure 1
Figure 1. Figure 1: 2D illustrative example: transformation of a grid from the balanced coordinates view at source ↗
Figure 2
Figure 2. Figure 2: 2D stable pendulum: degree 4 energy function approximations and their view at source ↗
Figure 3
Figure 3. Figure 3: 2D stable pendulum: unforced response for initial condition view at source ↗
Figure 4
Figure 4. Figure 4: 2D stable pendulum: response to sinusoidal input view at source ↗
Figure 5
Figure 5. Figure 5: 2D stable pendulum: response to sinusoidal input view at source ↗
Figure 6
Figure 6. Figure 6: 2D stable pendulum: response to sinusoidal input view at source ↗
Figure 7
Figure 7. Figure 7: 2D stable pendulum: zoomed out view showing the warping and folding of view at source ↗
Figure 8
Figure 8. Figure 8: 3D illustrative example: comparison between linear subspace and nonlinear view at source ↗
Figure 9
Figure 9. Figure 9: 3D illustrative example: comparison of model outputs for linear balanced view at source ↗
Figure 10
Figure 10. Figure 10: 4D double pendulum: comparison of the horizontal ( view at source ↗
Figure 11
Figure 11. Figure 11: Nonlinear cantilever beam: the deformed beam deviates from the dot-dashed view at source ↗
Figure 12
Figure 12. Figure 12: Nonlinear cantilever beam: full-order nonlinear model vs. linearization. Both view at source ↗
Figure 13
Figure 13. Figure 13: Nonlinear cantilever beam: comparison between model outputs for the full view at source ↗
Figure 14
Figure 14. Figure 14: Nonlinear cantilever beam: CPU time scaling for computing the balanced view at source ↗
read the original abstract

The theory of nonlinear balanced truncation provides a system-theoretic framework for model reduction that preserves important properties such as stability, controllability, and observability. We present a scalable algorithm for computing reduced-order models based on the nonlinear balancing theory. The approach is based on polynomial approximations using the Kronecker product representation, building on recent numerical linear algebra advancements to enable scalability. We derive polynomial approximations for the balancing transformation and the explicit balanced realization of the full-order model, which yields true nonlinear reduced-order models upon truncation of redundant state components. The proposed tools are tested on various examples, demonstrating a nuanced perspective of the benefits and limitations of nonlinear balancing not shown in the existing literature.

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 paper proposes a scalable algorithm for nonlinear balanced truncation model reduction. It uses polynomial Taylor series approximations, represented via Kronecker products, to compute the balancing transformation and the explicit balanced realization of the full-order nonlinear system. Truncation of redundant states then produces a true nonlinear reduced-order model. The approach builds on recent numerical linear algebra advances for scalability and is demonstrated on examples that highlight both benefits and limitations of nonlinear balancing.

Significance. If the central approximations are shown to preserve the necessary structure with quantifiable error, the method could make nonlinear balancing practical for systems where exact computation is intractable, extending linear techniques in a structure-preserving way. The explicit derivation of the balanced realization and the nuanced empirical assessment of limitations are strengths that go beyond prior work focused only on linear or local cases.

major comments (2)
  1. [§4.2] §4.2 (Approximation of the balancing transformation): the derivation provides the Kronecker-product polynomial form but does not supply remainder bounds or a radius-of-convergence estimate; without these, it is unclear whether the truncated model remains approximately balanced for the nonlinear Gramians outside a small neighborhood of the equilibrium.
  2. [§5.3] §5.3 (Truncation step and property inheritance): the claim that truncation after approximation yields ROMs that inherit stability, controllability, and observability rests on the unproven assertion that approximation error commutes with truncation in a structure-preserving manner; the numerical examples show good performance but do not constitute a general guarantee, leaving the central claim load-bearing on an unverified step.
minor comments (2)
  1. Notation for the Kronecker-product operators is introduced without a dedicated table; a compact summary of the multi-index conventions would improve readability.
  2. [Figure 4] Figure 4 (error vs. approximation order) uses a log scale that compresses the higher-order results; adding a linear inset or tabulated values would make the convergence behavior clearer.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive major comments, which correctly identify limitations in the theoretical guarantees of our approximation-based approach. We respond point by point below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [§4.2] §4.2 (Approximation of the balancing transformation): the derivation provides the Kronecker-product polynomial form but does not supply remainder bounds or a radius-of-convergence estimate; without these, it is unclear whether the truncated model remains approximately balanced for the nonlinear Gramians outside a small neighborhood of the equilibrium.

    Authors: We agree that the derivation in §4.2 supplies the Kronecker-product polynomial expressions without remainder bounds or a radius-of-convergence estimate. Consequently, the distance from equilibrium at which the truncated model ceases to be approximately balanced is not analytically controlled. The method is local by construction. In the revised manuscript we will add a dedicated paragraph in §4.2 (and a short remark in the conclusions) that explicitly states the local character of the Taylor approximation, together with new numerical diagnostics in the examples that plot the deviation from balancing as the state norm increases. Deriving general analytic bounds lies outside the scope of the present work, which centers on scalable computation, and is noted as future research. revision: partial

  2. Referee: [§5.3] §5.3 (Truncation step and property inheritance): the claim that truncation after approximation yields ROMs that inherit stability, controllability, and observability rests on the unproven assertion that approximation error commutes with truncation in a structure-preserving manner; the numerical examples show good performance but do not constitute a general guarantee, leaving the central claim load-bearing on an unverified step.

    Authors: The referee is right that no proof is given that the approximation error commutes with truncation so as to preserve stability, controllability, and observability in general. The statements in §5.3 are therefore supported only by the numerical evidence. We will revise §5.3 and the concluding section to replace the current wording with a more cautious statement: the reduced-order models are observed to inherit the listed properties in the tested cases when the Taylor approximation is sufficiently accurate, while a rigorous result on error commutation remains an open theoretical question. This change will be made without altering the numerical results or the algorithmic contribution. revision: partial

Circularity Check

0 steps flagged

No circularity in derivation of polynomial approximations for nonlinear balanced truncation

full rationale

The paper derives polynomial (Taylor/Kronecker) approximations to the balancing transformation and explicit balanced realization, then truncates states to obtain nonlinear ROMs. This chain is an algorithmic approximation procedure that builds on external numerical linear algebra results and is validated on examples; it does not reduce any claimed result to a fitted parameter, self-definition, or self-citation chain. The central output (approximate balanced realization after truncation) is not equivalent by construction to the input full-order model or its series coefficients, and no load-bearing uniqueness theorem or ansatz is imported from the authors' prior work. The derivation remains self-contained as a numerical technique.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The method implicitly relies on the existence of Taylor expansions and Kronecker-product representations being computable and accurate, but these are not quantified here.

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