Closing Trajectories: Equation-Free Cyclic Animation via Koopman Surrogates

Peter Yichen Chen; Shixun Huang; Siyuan Chen; Yue Chang; Zhecheng Wang

arxiv: 2605.23462 · v1 · pith:Q3V5OSH5new · submitted 2026-05-22 · 💻 cs.GR

Closing Trajectories: Equation-Free Cyclic Animation via Koopman Surrogates

Shixun Huang , Siyuan Chen , Yue Chang , Zhecheng Wang , Peter Yichen Chen This is my paper

Pith reviewed 2026-05-25 02:41 UTC · model grok-4.3

classification 💻 cs.GR

keywords cyclic animationKoopman surrogateequation-freequadratic programmingperiodic trajectoriescomputer graphicsdata-driven simulation

0 comments

The pith

Cyclic animation is synthesized equation-free by learning a Koopman surrogate from data and solving a linearly constrained quadratic program for a periodic control force.

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

The paper establishes a method to produce seamless repeating trajectories from a short observed sequence when the governing equations are unavailable and the sequence endpoints do not close. A Koopman surrogate is identified directly from the data; a Fourier-parameterized time-varying force is then optimized subject to an exact periodicity constraint on the state, turning the task into a linearly constrained quadratic program. The program is solved efficiently via a structured KKT system. A sympathetic reader would care because this removes the need for either exact endpoint matching or an analytic model, enabling cyclic playback for cloth, fluids, and deformable bodies in games and VR from limited observations.

Core claim

We identify a Koopman surrogate from the observed trajectory and compute a cyclic trajectory by applying a Fourier-parameterized, time-varying control force under a hard temporal periodicity constraint. The resulting formulation reduces cyclic synthesis to a linearly constrained quadratic program that can be solved efficiently through a structured KKT system.

What carries the argument

Koopman surrogate learned from the finite observed trajectory, used to formulate and solve the linearly constrained quadratic program for the control force that enforces periodicity.

If this is right

The approach applies directly to N-body systems, cloth, deformable objects, and shallow water without requiring their governing equations.
Cyclic synthesis reduces to a single structured quadratic program whose KKT system can be factored once and reused.
Endpoint mismatch in the input sequence is automatically corrected by the optimized control force.
The same surrogate and constraint formulation works across rigid, deformable, and fluid examples.

Where Pith is reading between the lines

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

The same quadratic-program structure could be reused across multiple short clips of the same object to produce families of cycles with different periods.
If the Koopman surrogate is replaced by a learned neural operator, the method would become a general template for data-driven periodic control.
The periodicity constraint can be relaxed to a soft penalty to handle quasi-periodic or slowly drifting motions.

Load-bearing premise

The Koopman surrogate learned from the finite observed trajectory remains sufficiently accurate when the added control force drives the system to a new periodic orbit that was not present in the training data.

What would settle it

Apply the method to a known integrable system such as a simple pendulum with a recorded non-closed trajectory and verify whether the synthesized closed orbit satisfies the true equations of motion to within integrator tolerance.

Figures

Figures reproduced from arXiv: 2605.23462 by Peter Yichen Chen, Shixun Huang, Siyuan Chen, Yue Chang, Zhecheng Wang.

**Figure 1.** Figure 1: Wave propagation. Top: the input free-surface sequence is non-cyclic, and a visible mismatch appears across the temporal boundary between Frame [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: Cute dragon. Top: the input deformable sequence is non-cyclic, resulting in an open trajectory whose final frame does not return to the initial [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: N-body Dynamics. A 5-body system of different mass interacting [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: Pinned-cloth. Top: the input sequence is non-cyclic, since the final [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: Parrot. Top: the input sequence is non-cyclic, as the wing-flapping [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: Interactive local control on the parrot sequence. Top: The synthesized cyclic sequence without user control. Bottom: applying a downward local edit at [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

read the original abstract

Cyclic animation is widely used in computer graphics and interactive content.It supports seamless playback in games, VR, and interactive simulation,where short clips must repeat smoothly over long durations. Achievingphysically plausible cyclic synthesis from an input sequence is challengingbecause the endpoint states of the observed sequence rarely match exactly,and the governing equations of the underlying system are often unavailable.We therefore propose an equation-free framework that identiffes a Koopmansurrogate from the observed trajectory and computes a cyclic trajectory byapplying a Fourier-parameterized, time-varying control force under a hardtemporal periodicity constraint. The resulting formulation reduces cyclicsynthesis to a linearly constrained quadratic program that can be solvedefffciently through a structured KKT system. Our method is applicable toa diverse range of examples, including N-body systems, cloth, deformableobjects, shallow water, etc.

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.

Desk Editor's Note private letter to a colleague

The paper turns cyclic animation into a QP by fitting a Koopman surrogate then optimizing a Fourier control under hard periodicity, but the whole thing rests on the surrogate staying accurate outside the training data.

read the letter

The main takeaway is that the paper reduces cyclic animation synthesis to a quadratic program by learning a Koopman surrogate and optimizing a Fourier control under periodicity constraints. This is done without access to the system's equations. It does a good job framing the challenge of mismatched endpoints in observed sequences and offering an equation-free alternative. The applicability to N-body, cloth, deformable objects, and shallow water suggests they tested it across different simulation types, which is a plus for a graphics method. The soft spot is the reliance on the surrogate's accuracy for the controlled system. The Koopman model is fit to the input trajectory, but the new periodic orbit involves states and controls not in the training set. For nonlinear systems like cloth or fluids, the linear lifted dynamics may not extrapolate reliably, and any mismatch would mean the solved trajectory isn't cyclic in reality. The abstract mentions no error bounds or closed-loop validation, so this assumption carries the method. This work is for computer graphics practitioners and researchers building animation tools or simulation pipelines. Someone needing to generate seamless loops from short clips might find the QP approach practical. It deserves a serious referee because the formulation is concrete and targets a common issue in the field, even if the generalization question needs addressing in review.

Referee Report

2 major / 1 minor

Summary. The paper introduces an equation-free method for creating cyclic animations from input sequences using Koopman surrogates. It learns a linear model from the observed trajectory and optimizes a Fourier-parameterized control force to enforce periodicity, reducing the task to a linearly constrained quadratic program solved via KKT conditions. The method is positioned as applicable to diverse physical systems such as N-body simulations, cloth, deformable objects, and shallow water equations.

Significance. If the Koopman surrogate generalizes reliably to the controlled periodic orbits, the approach could enable seamless cyclic synthesis without requiring the underlying governing equations, offering a practical tool for animation in games, VR, and interactive simulations across multiple domains in computer graphics.

major comments (2)

[Abstract] Abstract: the reduction of cyclic synthesis to a linearly constrained QP solved via a structured KKT system is asserted without derivation, error analysis, or demonstration that the surrogate remains valid under the added control.
[Abstract] Abstract (central claim): the Koopman surrogate learned from one finite observed trajectory is assumed to remain predictive under the optimized time-varying Fourier control that produces a new periodic orbit absent from training data; no validation or accumulation analysis is provided for systems such as cloth or shallow water where small errors can violate the hard periodicity constraint over cycles.

minor comments (1)

[Abstract] Abstract contains typos: 'identiffes' and 'efffciently'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and recommendation for major revision. We address each major comment point by point below, outlining the revisions we will implement to improve clarity and rigor.

read point-by-point responses

Referee: [Abstract] Abstract: the reduction of cyclic synthesis to a linearly constrained QP solved via a structured KKT system is asserted without derivation, error analysis, or demonstration that the surrogate remains valid under the added control.

Authors: We agree that the abstract presents the QP reduction concisely without derivation details. The full derivation, showing how the Fourier-parameterized control is incorporated into the Koopman model to yield a linearly constrained quadratic program solved via the structured KKT system, appears in Section 3. In the revision we will expand the abstract with a brief outline of this reduction and add an explicit error analysis subsection (with supporting figures) demonstrating surrogate validity under the added control input. revision: yes
Referee: [Abstract] Abstract (central claim): the Koopman surrogate learned from one finite observed trajectory is assumed to remain predictive under the optimized time-varying Fourier control that produces a new periodic orbit absent from training data; no validation or accumulation analysis is provided for systems such as cloth or shallow water where small errors can violate the hard periodicity constraint over cycles.

Authors: The manuscript already contains numerical demonstrations on cloth and shallow-water examples in which the synthesized periodic orbits remain stable over repeated cycles. Nevertheless, we acknowledge that dedicated long-horizon accumulation analysis would strengthen the central claim. In the revised manuscript we will add quantitative error-accumulation plots and discussion for these systems, confirming that the hard periodicity constraint is satisfied within acceptable tolerances. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on external Koopman identification without self-referential reduction

full rationale

The provided abstract and context describe an equation-free method that identifies a Koopman surrogate from observed trajectory data and then solves a linearly constrained QP for periodicity. No equations are exhibited that define the surrogate in terms of the closed trajectory or that rename a fitted parameter as a prediction. The load-bearing step is the assumption that the learned surrogate extrapolates to controlled periodic orbits, but this is an empirical generalization claim rather than a definitional or self-citation reduction. No self-citation load-bearing, uniqueness theorem, or ansatz smuggling is visible. The derivation is therefore self-contained against external benchmarks for the purpose of this analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities. The method implicitly assumes that a finite observed trajectory yields a Koopman operator accurate enough for controlled extrapolation to a periodic orbit, but this is not formalized.

pith-pipeline@v0.9.0 · 5681 in / 1171 out tokens · 21612 ms · 2026-05-25T02:41:01.998496+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We therefore propose an equation-free framework that identifies a Koopman surrogate from the observed trajectory and computes a cyclic trajectory by applying a Fourier-parameterized, time-varying control force under a hard temporal periodicity constraint. The resulting formulation reduces cyclic synthesis to a linearly constrained quadratic program
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the learned reduced surrogate ˆK∈R r×r ... ˜z_{t+1} = ˆK˜z_t +u_t ... ˜z_{T+1}=˜z_1

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: 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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and Brunton, Steven L

Proctor, Joshua L. and Brunton, Steven L. and Kutz, J. Nathan , title =. SIAM Journal on Applied Dynamical Systems , volume =. 2018 , doi =. https://doi.org/10.1137/16M1062296 , abstract =

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and Brunton, Steven L

Proctor, Joshua L. and Brunton, Steven L. and Kutz, J. Nathan , title =. SIAM Journal on Applied Dynamical Systems , volume =. 2016 , doi =. https://doi.org/10.1137/15M1013857 , abstract =

work page doi:10.1137/15m1013857 2016

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Jia, Shiyang and Wang, Stephanie and Li, Tzu-Mao and Chern, Albert , title =. Proc. ACM Comput. Graph. Interact. Tech. , month = aug, articleno =. 2023 , issue_date =. doi:10.1145/3606938 , abstract =

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work page doi:10.1145/1576246.1531359 2009

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Witkin, Andrew and Kass, Michael , title =. SIGGRAPH Comput. Graph. , month = jun, pages =. 1988 , issue_date =. doi:10.1145/378456.378507 , abstract =

work page doi:10.1145/378456.378507 1988

[7] [7]

Proceedings of the 2007 Symposium on Interactive 3D Graphics and Games , pages =

Heck, Rachel and Gleicher, Michael , title =. Proceedings of the 2007 Symposium on Interactive 3D Graphics and Games , pages =. 2007 , isbn =. doi:10.1145/1230100.1230123 , abstract =

work page doi:10.1145/1230100.1230123 2007

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Motion graphs , year =

Kovar, Lucas and Gleicher, Michael and Pighin, Fr\'. Motion graphs , year =. ACM Trans. Graph. , month = jul, pages =. doi:10.1145/566654.566605 , abstract =

work page doi:10.1145/566654.566605

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Journal of fluid mechanics , volume=

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2025 , journal =

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2025

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Hodgins , title =

Edilson de Aguiar and Leonid Sigal and Adrien Treuille and Jessica K. Hodgins , title =. ACM Trans. Graph. (SIGGRAPH) , volume =. 2010 , doi =

2010

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Proceedings of the Special Interest Group on Computer Graphics and Interactive Techniques Conference (SIGGRAPH '25) , publisher =

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work page doi:10.1145/3721238.3730829

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Proceedings of the Special Interest Group on Computer Graphics and Interactive Techniques Conference (SIGGRAPH '25) , publisher =

Reenact Anything: Semantic Video Motion Transfer Using Motion-Textual Inversion , author =. Proceedings of the Special Interest Group on Computer Graphics and Interactive Techniques Conference (SIGGRAPH '25) , publisher =. doi:10.1145/3721238.3730826 , isbn =

work page doi:10.1145/3721238.3730826

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Interactive editing of deformable simulations , year =

Barbi. Interactive editing of deformable simulations , year =. ACM Trans. Graph. , month = jul, articleno =. doi:10.1145/2185520.2185566 , abstract =

work page doi:10.1145/2185520.2185566

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B. O. Koopman , title =. Proceedings of the National Academy of Sciences , volume =. 1931 , doi =

1931

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Proceedings of the 2002 ACM SIGGRAPH/Eurographics Symposium on Computer Animation , pages =

Park, Sang Il and Shin, Hyun Joon and Shin, Sung Yong , title =. Proceedings of the 2002 ACM SIGGRAPH/Eurographics Symposium on Computer Animation , pages =. 2002 , isbn =. doi:10.1145/545261.545279 , abstract =

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and Buchanan, J

Mizuguchi, M. and Buchanan, J. and Calvert, T. , year =

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, title =

de Aguiar, Edilson and Sigal, Leonid and Treuille, Adrien and Hodgins, Jessica K. , title =. ACM SIGGRAPH 2010 Papers , articleno =. 2010 , isbn =. doi:10.1145/1833349.1778843 , abstract =

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2023 , url=

Peter Yichen Chen and Jinxu Xiang and Dong Heon Cho and Yue Chang and G A Pershing and Henrique Teles Maia and Maurizio M Chiaramonte and Kevin Thomas Carlberg and Eitan Grinspun , booktitle=. 2023 , url=

2023

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