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arxiv: 2606.18759 · v1 · pith:2CQK7AO7new · submitted 2026-06-17 · 💻 cs.CG · cs.LG· cs.NA· math.NA

A Neural Network Framework for Geodesic-Like Curve Computation on Parametric Surfaces

Sheng-Gwo Chen , Chen-Chang Peng This is my paper

Pith reviewed 2026-06-26 18:43 UTC · model grok-4.3

classification 💻 cs.CG cs.LGcs.NAmath.NA
keywords geodesic-like curvesparametric surfacesphysics-informed neural networksPINNsshortest pathscomputational geometrydeep learning
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The pith

A physics-informed neural network computes geodesic-like curves on parametric surfaces including multi-surface systems.

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

The paper develops a framework that applies physics-informed neural networks to calculate geodesic-like curves, which approximate shortest paths on parametric surfaces. It shows this method works efficiently for single surfaces and extends robustly to multi-surface systems with C0 or higher continuity as well as surfaces of revolution. The approach avoids the need for surface-specific manual adjustments that traditional numerical methods often require. A sympathetic reader would care because it offers a unified computational tool for path estimation across varied surface topologies where prior techniques lacked an efficient general implementation.

Core claim

Under the proposed framework, not only can single parametric surfaces be handled efficiently, but a broad class of complex parametric surfaces including multi-surface systems with C^0 or higher continuity and surfaces of revolution can also be robustly addressed by leveraging deep learning and Physics-Informed Neural Networks.

What carries the argument

A physics-informed neural network whose loss function enforces the defining properties of geodesic-like curves across different surface types.

If this is right

  • Single parametric surfaces yield efficient curve computations.
  • Multi-surface systems maintain robustness at C0 or higher continuity.
  • Surfaces of revolution are processed without additional modifications.
  • The method operates without manual per-surface tuning of the loss function.

Where Pith is reading between the lines

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

  • The same network structure might support path computation on time-varying or deformable surfaces.
  • Integration with mesh-based rendering systems could allow direct use in animation pipelines.
  • The approach may extend to related variational problems such as minimal surfaces on the same parametric domains.

Load-bearing premise

A single PINN loss function can be constructed to enforce the properties of geodesic-like curves on surfaces with varying topologies and continuity without surface-specific adjustments or training instability.

What would settle it

Apply the trained network to a multi-surface system with only C0 continuity and check whether the output curves diverge from expected shortest-path approximations or exhibit persistent training failure.

Figures

Figures reproduced from arXiv: 2606.18759 by Chen-Chang Peng, Sheng-Gwo Chen.

Figure 1
Figure 1. Figure 1: The proper variation of a curve γ on Σ. 2. γ is a critical point of the energy functional, that is, E′ (0) = 0. 3. γ is a critical point of the arc length functional, that is, L ′ (0) = 0. The shortest path between two distinct points p, q ∈ Σ is defined as the solution of the following minimization problem: min{L(γ) : γ is a smooth curve on Σ with γ(a) = p, γ(b) = q}. (7) It is also a critical point of th… view at source ↗
Figure 3
Figure 3. Figure 3: Two Bspline surfaces of degree 3 × 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Mean relative errors of geodesic-like curves on surface . [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Trimmed mean relative errors( 5% − 95% ) of geodesic-like curves on surface 1 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Median relative errors of geodesic-like curves on surface 1. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Mean relative errors of geodesic-like curves on surface 2. [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Trimmed mean relative errors( 5% − 95% ) of geodesic-like curves on surface 2 [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Median relative errors of geodesic-like curves on surface 2.. [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Challenges in Global Optimization. Left subfigure is a failure case of our proposed method; ‘right subfigure is a [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Boundary limitation of the domain-restricted method. A portion of the true shortest path (white dashed line) [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Search mechanism of the revised model using [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Left:Comparison between the approximated geodesic curve derived from the basic model and the baseline geodesic. [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Approximation of a geodesic-like curve spanning two surfaces. The right panel shows the curve in the parametric [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Geodesic-like curve approximation using the global domain method: The right panel illustrates the curves in the [PITH_FULL_IMAGE:figures/full_fig_p018_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Approximations of geodesic-like curves on surface of revolutions [PITH_FULL_IMAGE:figures/full_fig_p018_16.png] view at source ↗
read the original abstract

The concept of geodesic-like curves was introduced by Chen in 2010 as a method for estimating shortest paths (geodesics) on parametric surfaces, with its convergence established theoretically. However, an efficient numerical computational framework has not yet been developed. In this paper, we propose an elegant and efficient approach for computing geodesic-like curves by leveraging deep learning and Physics-Informed Neural Networks (PINNs). Under the proposed framework, not only can single parametric surfaces be handled efficiently, but a broad class of complex parametric surfaces including multi-surface systems with $C^0$ or higher continuity and surfaces of revolution can also be robustly addressed.

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 / 0 minor

Summary. The paper proposes a Physics-Informed Neural Network (PINN) framework to compute geodesic-like curves on parametric surfaces, extending Chen's 2010 theoretical concept. It claims the method efficiently handles single parametric surfaces and extends robustly to complex cases including multi-surface systems with C^0 or higher continuity and surfaces of revolution, without surface-specific manual adjustments.

Significance. If the central claims hold, the work could supply a general numerical tool for geodesic-like curve computation on varied parametric surfaces. The manuscript supplies no numerical results, error metrics, training details, loss equations, or comparisons, so the practical significance cannot be assessed from the current text.

major comments (2)
  1. [Abstract] Abstract: the assertion that multi-surface C^0 systems and surfaces of revolution can be 'robustly addressed' without surface-specific manual adjustments is load-bearing for the central claim, yet the text provides neither the PINN loss formulation that would enforce geodesic-like ODE residuals and interface continuity nor any multi-patch experiment or stability analysis.
  2. [Abstract] Abstract: the claims of efficiency and robustness for a broad class of surfaces are unsupported by any quantitative evidence, error metrics, training protocol, or baseline comparisons, leaving the soundness of the framework unverified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit supporting details. We address the comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that multi-surface C^0 systems and surfaces of revolution can be 'robustly addressed' without surface-specific manual adjustments is load-bearing for the central claim, yet the text provides neither the PINN loss formulation that would enforce geodesic-like ODE residuals and interface continuity nor any multi-patch experiment or stability analysis.

    Authors: We agree that the abstract claims require concrete substantiation. The manuscript introduces the general PINN framework extending Chen (2010) but does not supply the explicit loss equations for the geodesic-like ODE residuals or the interface continuity terms, nor does it contain multi-patch numerical tests. We will revise the manuscript to include the full loss formulation in Section 3 and add a dedicated subsection with multi-surface experiments together with a basic stability discussion. revision: yes

  2. Referee: [Abstract] Abstract: the claims of efficiency and robustness for a broad class of surfaces are unsupported by any quantitative evidence, error metrics, training protocol, or baseline comparisons, leaving the soundness of the framework unverified.

    Authors: The current manuscript is primarily a conceptual proposal and indeed contains no numerical results, error metrics, training details, or comparisons. We accept that these elements are required to assess the practical claims. In the revision we will add the requested quantitative evidence, including training protocols, error metrics on representative surfaces, and comparisons against standard numerical integrators. revision: yes

Circularity Check

0 steps flagged

No circularity; PINN implementation of independently defined 2010 geodesic-like curves

full rationale

The paper applies Physics-Informed Neural Networks to compute geodesic-like curves whose defining properties and convergence were established in the 2010 Chen reference. No derivation step reduces a prediction or result to a quantity fitted or defined within this manuscript; the loss is constructed directly from the external curve conditions rather than from any self-referential fit or ansatz introduced here. Self-citation of the 2010 concept exists but is not load-bearing for the central claim of a uniform PINN framework, which rests on the neural architecture and training procedure rather than re-deriving the prior definition. The work is therefore self-contained against external benchmarks with no reduction by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the prior theoretical existence of geodesic-like curves and the domain assumption that PINNs can numerically realize them; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Physics-informed neural networks can be trained to satisfy the defining variational or differential properties of geodesic-like curves on parametric surfaces.
    The framework depends on this standard property of PINNs being sufficient for the curve computation task.

pith-pipeline@v0.9.1-grok · 5639 in / 1206 out tokens · 19542 ms · 2026-06-26T18:43:19.960299+00:00 · methodology

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Reference graph

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