Pith. sign in

REVIEW 2 major objections 5 minor 22 references

This paper argues that physics-informed neural networks can solve the asymptotic Plateau problem when the geometry is hard-coded into the network architecture and the residual is evaluated with compiled second-order jets, cutting training c

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-03 00:32 UTC pith:HKO7YYMC

load-bearing objection Useful how-to for geometry-aware PINNs; the speedup is credible, but the 'asymptotically minimal by construction' claim rests on a criterion borrowed from the companion paper that this note does not prove. the 2 major comments →

arxiv 2607.28733 v1 pith:HKO7YYMC submitted 2026-07-30 math.DG cs.AIhep-thmath.AP

A user's guide to PINNs in geometric analysis: lessons from the asymptotic Plateau problem

Tancredi Schettini Gherardini This is my paper
classification math.DG cs.AIhep-thmath.AP MSC 53A1053C4268T07
keywords physics-informed neural networksasymptotic Plateau problemminimal surfaceshyperbolic spacehard constraintsautomatic differentiationgraph compilationHOMFLY polynomial
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This note argues that physics-informed neural networks can reliably solve the asymptotic Plateau problem—finding near-minimal discs in hyperbolic space spanning a prescribed knot—provided two principles are followed. First, the geometry must be hard-coded into the architecture: a stereobiharmonic extension of the boundary curve, a boundary-defining function, and a decay factor make the model satisfy the boundary condition and the boundary asymptotics exactly for every parameter value, leaving a single-component loss that measures minimality only. Second, the residual must be evaluated by propagating second-order jets forward through the network and compiling the computational graph once, instead of using nested reverse-mode automatic differentiation; on identical hardware this speeds up each training step by roughly 40–50 times. If correct, the method turns a computation that would need a cluster into a laptop-scale run with residuals typically in the 10⁻⁷–10⁻⁴ range, and the lessons transfer to other geometric variational problems.

Core claim

On the paper's own terms: the geometry of the problem must be encoded in the architecture of the model, not learned. The model wraps a plain multi-layer perceptron in a composite map u_θ = (ρ e^{NN_X}, ext(γ) + ρ^k NN^Y), where ρ is a boundary defining function, ext(γ) is a stereobiharmonic extension of the knot, and k = 2. This guarantees exact boundary values and orthogonal, hence asymptotically minimal, approach to the boundary for every value of the learnable parameters, leaving a single-component loss that measures minimality only. The second thesis: evaluating the residual via explicit forward propagation of second-order jets—exact chain, Leibniz, and coordinate rules—and compiling the

What carries the argument

The hard-constrained ansatz (3.1): u_θ = (ρ e^{NN_X}, ext(γ) + ρ^k NN^Y), with the stereographic boundary defining function ρ_st = (1−r²)/(1+r²) and a stereobiharmonic extension ext(γ) that satisfies both Dirichlet and Neumann data. This parametrises away the constraint manifold so that boundary values, positivity, and decay hold identically for all parameters. The computational engine is second-order jet propagation through the MLP—affine and tanh jet rules applied layer by layer—followed by Leibniz assembly and one-time graph compilation with static shapes, which reduces the residual to a branch-free tensor program.

Load-bearing premise

The load-bearing premise is that the k=2 hard-constrained model is asymptotically minimal by construction, which rests on an equivalence—orthogonality at infinity iff the radial derivative of the boundary trace is tangent to the knot—quoted from the companion paper and not proved here; a secondary premise is that the 40–50× speedup measured on one benchmark extrapolates to other low-dimensional PINN problems.

What would settle it

Construct a knot with a nontrivial second Fourier mode and evaluate, at θ=0, the radial derivative ∂_r(ext(γ)+ρ²NN^Y) along the boundary circle; if it is not tangent to γ beyond truncation error, the architectural claim that asymptotic minimality holds identically is false. Independent check: run Table 1's timing protocol on a second low-dimensional PINN problem with different batch sizes; the 40–50× factor is an empirical claim that must reproduce.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Boundary-weight tuning disappears: with constraints parametrised into the architecture, the loss has no boundary term, eliminating the fragile balancing of soft-constrained PINNs.
  • Typical training residuals fall in the 10⁻⁷–10⁻⁴ range for a family of knots with no problem-specific tuning, with the ten-crossing knot reaching at most 5×10⁻⁴.
  • A complete high-precision training run takes roughly 40 minutes on a laptop CPU; the same schedule through the naive pipeline extrapolates to more than a day.
  • The jet-propagation and compilation techniques of §4 apply verbatim to any PINN whose domain has low dimension—the typical situation for parametrised curves, surfaces, and maps in geometric analysis.
  • The signed self-intersection numbers computed from the trained discs match HOMFLY coefficients in every tested case, corroborating the conjecture the companion paper probes.

Where Pith is reading between the lines

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

  • The 40–50× factor is measured on one architecture and one CPU; the margin is likely to shrink on GPUs running double precision or when batch sizes grow, so the qualitative advice (forward jets plus compilation) matters more than the exact factor.
  • The recipe's scope is bounded by its own precondition: hard constraints require an explicit parametrisation of the constraint manifold; for domains and boundary data without such a parametrisation, penalty methods remain the fallback and the weighting problem returns.
  • If residuals can be pushed below the current range, the same pipeline could seed computer-assisted existence proofs, where the trained map serves as an approximate solution for rigorous a posteriori contraction arguments—a direction the paper names as a natural next step.
  • A natural test of the 'asymptotically minimal by construction' claim: measure ∂_r(ext(γ)+ρ²NN^Y) tangency on a trained model for a knot with nontrivial Fourier content; the paper's Remark 3.1 warns that the exact boundary condition is only modulo the degree-15 Fourier truncation of γ.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This proceedings note is a methodological companion to the author's joint paper with M. Usula (arXiv:2605.26234) on using physics-informed neural networks to construct near-minimal discs in hyperbolic 4-space with a prescribed knot at infinity. The note distills two lessons. First, the geometry should be hard-coded into the network architecture: the ansatz u_θ = (ρ e^{NN^X}, ext(γ)+ρ^k NN^Y) encodes positivity, the boundary curve γ, and the expected asymptotic orthogonality, leaving a single-component interior loss. Second, the residual evaluation should be engineered: the paper advocates forward propagation of second-order jets through the MLP and compiling the residual computation graph once, reporting a 40–50× speedup per training step in Table 1. The paper also lists six transferable guidelines for geometric PINN users. No new theorems are claimed here; the geometric validity of the construction is inherited from [18], and the numerical evidence for Fine's conjecture is summarized from the companion paper.

Significance. If the claims hold, the note is a genuinely useful resource for geometric analysts who want to use PINNs: it gives a concrete, reproducible architecture for a PDE problem whose boundary condition lives at infinity, and it identifies two computational bottlenecks with practical fixes. The public code, the closed-form jet-propagation rules (4.1)–(4.2), and the internal consistency check that four residual pipelines agree to 5×10^{-14} are concrete strengths that support the methodological advice. However, the central 'asymptotically minimal by construction' guarantee rests on a geometric criterion that is cited from [18] but not stated or proved here, and the headline speedup factor is measured in a single configuration with no variance. The note is therefore valuable but needs to make the load-bearing assumptions explicit before the central claims can be accepted as stated.

major comments (2)
  1. [§3.2 (Eq. (3.1))] The central claim that the model is 'asymptotically minimal by construction' rests entirely on the assertion that orthogonality to the boundary is equivalent to ∂_r(ext(γ)+ρ^k NN^Y)|_{∂D^2} being tangent to γ, cited to [18, §3.2] with no statement of hypotheses or proof. This is load-bearing: it is the only justification for dropping the boundary term from the loss and for claiming that the residual vanishes at the boundary before training. Please state this criterion as a lemma, including the regularity required of ρ, ext, γ and the behavior of NN^X near the boundary, and either prove it or give an explicit theorem reference. In particular, the model's first component is ρ e^{NN^X}, so the equivalence must be stated for the full map, not merely for the Y-component; the current sentence does not allow the reader to check this. Remark 3.1 already concedes that the boundary condition is ex
  2. [§4.4, Table 1] The quantitative headline 'factor of roughly forty to fifty' is supported by a single benchmark: one model, one laptop CPU, one initialisation, median of 20 steps after warm-up, with no variance or repeated runs. Because the abstract and §6 state this as a general result of the two techniques, please either report repeated timings (several seeds and independent runs) with a range, or qualify the claim as 'in our benchmark'. The residual agreement to 5×10^{-14} is a good correctness check, but it does not quantify timing variability.
minor comments (5)
  1. [§3.1 / Remark 3.1] The degree-15 Fourier truncation caveat should appear where the 'exact boundary condition' claim is first made, not only in a remark at the end of the section. Alternatively, quantify the truncation error for the knots used.
  2. [§3.2] The biharmonic-extension formula for Γ is stated without derivation. A one-line verification that Δ²Γ=0 and that the boundary conditions are satisfied would make the note more self-contained and would help readers implement the method.
  3. [§4.2, Eq. (4.2)] In the jet-propagation rule, the notation (J_a ⊗ J_a)_{ij}=∂_i a ∂_j a should be written out explicitly, since the standard matrix outer product convention could be confused with the symmetric product used here.
  4. [§6] The claim that the jet/compilation techniques apply 'verbatim to any PINN whose domain has low dimension' is stronger than the evidence. The compilation strategy is shape-static and tied to a fixed collocation pool; adaptive-sampling or variable-batch-size workflows would need recompilation and may not enjoy the same gains. Suggest softening to a heuristic.
  5. [§4.4 / Table 1 caption] The text says the four pipelines 'agree pairwise to 5×10^{-14}' while the caption gives 'maximum pairwise deviation 5.0×10^{-14} on a residual field of maximum modulus ≈53'; state the same pair of numbers in both places.

Circularity Check

0 steps flagged

No significant circularity: the geometric ansatz is constructed, the speedup is measured, and the one self-cited lemma is analytic support rather than an input-output loop.

full rationale

The paper's derivation chain is methodological rather than predictive-evaluative. No parameter is fitted to the Fine/HOMFLY data in this note; that comparison is reported in the companion paper [18] and is not used to define the architecture or the loss. The central efficiency claim (40–50×) is a measured benchmark, not a fitted quantity: Table 1 compares four pipelines on identical collocation points, identical weights, and identical hardware, with pairwise agreement to 5×10^-14, so the speedup is externally verifiable rather than derived from its own conclusion. The 'asymptotically minimal by construction' claim is an explicit ansatz: equation (3.1) forces the boundary restriction to equal γ and, through the stereobiharmonic extension plus the decay exponent k=2, makes the radial derivative of the surface map tangent to γ at the boundary, so the residual vanishes there before training. This is a construction, not a prediction, and it does not reduce an output to an input in the sense of circularity. The paper does rely on a self-cited lemma from [18, §3.2] (the orthogonality-tangency equivalence), and Remark 3.1 honestly qualifies the 'exact' boundary condition as holding up to Fourier truncation. These are rigor/verification concerns about external support, not circular steps: the cited lemma is a parameter-free geometric computation, and the note's own measured speedup and architectural design remain independent content. No fitted target is renamed as a prediction, and no load-bearing conclusion is identical by construction to its premises. Score 0.

Axiom & Free-Parameter Ledger

5 free parameters · 2 axioms · 0 invented entities

The central claim rests on a small set of hand-chosen design parameters (ansatz decay, Fourier truncation, ρ_st, static shapes) plus two theorems imported from the literature ([15] and [18, §3.2]). No free constants are fitted to the HOMFLY data in this note; the network weights are optimization variables minimizing the residual (2.4).

free parameters (5)
  • decay exponent k = 2
    Chosen by hand in ansatz (3.1); k=2 makes the network correction decay too fast to alter boundary orthogonality, so the extension alone must carry the right asymptotics. Reported in §3.2; not learned and not derived in this note beyond that heuristic.
  • Fourier truncation degree = 15
    The 'exact' boundary condition in (3.1) is exact only with respect to the degree-15 truncation of γ; Remark 3.1 acknowledges this. Truncation error is claimed negligible for the explicit trigonometric knots used.
  • stereographic boundary defining function ρ_st = (1-r^2)/(1+r^2)
    Hand-selected so that θ=0 reproduces the totally geodesic unknot solution (3.2); a modeling choice that anchors initialization and was used throughout.
  • Network architecture = 4 hidden layers of width 64, tanh, 12,932 parameters
    Hand-selected default architecture for all experiments; hyperparameters are not optimized or justified in this note.
  • Collocation pool and batch sizes = 2^14 pool, 2^10 batch
    Fixed shapes chosen for static graph compilation; a design decision described in §4.3.
axioms (2)
  • standard math Marx-Kuo regularity theorem: minimal p-submanifolds of H^{n+1} with smooth boundary at infinity meet the boundary orthogonally and admit a polyhomogeneous expansion.
    Invoked in §2.2 and §3.2 to justify that exact boundary orthogonality is the right target and that the model can be required to have it; not proved in this note.
  • domain assumption Orthogonality criterion from [18, §3.2]: for maps of the form (3.1), orthogonality at infinity holds iff ∂_r(ext(γ)+ρ^k NN^Y) is tangent to the curve along ∂D².
    Load-bearing for the claim that the k=2 ansatz is 'asymptotically minimal by construction' for every value of the learnable parameters; stated as 'one computes' and referenced to the companion paper but not derived here.

pith-pipeline@v1.3.0-alltime-deepseek · 10684 in / 15846 out tokens · 161073 ms · 2026-08-03T00:32:01.832614+00:00 · methodology

0 comments
read the original abstract

This proceedings contribution elaborates on the findings of arXiv:2605.26234v2: a joint work with Marco Usula, where we introduced a machine learning framework based on physics-informed neural networks (PINNs), aimed at constructing near-minimal discs in hyperbolic space asymptotic to a prescribed knot at infinity. We used this method to provide numerical evidence for a conjecture of Joel Fine relating minimal surfaces in $H^{4}$ to the coefficients of the HOMFLY polynomial. This is a methodological companion to that paper, based on a presentation given at the 2026 edition of the workshop "DANGER: Data, Numbers, and Geometry". Rather than reviewing the results, which are presented extensively in the preprint above, we discuss the two aspects of the framework which, in our experience, determined whether the method worked at all. First, the geometry of the problem must be encoded in the architecture of the model, so that the boundary condition and asymptotics at infinity hold exactly for every value of the learnable parameters - leaving us with a single-component loss function; second, the evaluation of the PDE residual must be engineered with care to ensure that complete trainings can be performed in a reasonable time. On the latter point, we describe two implementation techniques which are not spelled out in detail in the original paper: replacing nested reverse-mode automatic differentiation with the forward propagation of second-order jets, and compiling the computational graph of the residual once instead of rebuilding it at every optimisation step. Together, on identical hardware, these two changes reduce the cost of a training step by a factor of roughly forty to fifty. We hope these methodological discussions can be useful for researchers in differential geometry and geometric analysis who wish to deploy PINNs on problems of their own.

Figures

Figures reproduced from arXiv: 2607.28733 by Tancredi Schettini Gherardini.

Figure 3
Figure 3. Figure 3: shows the practical meaning of these choices: the pointwise residual [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Pointwise squared residual |τ (uθ)| 2 for a (perturbed) unknot boundary, be￾fore training (left) and after training (right). Note the colour scales. The residual vanishes along ∂D2 even before training: the model is asymptotically minimal by construction, and the optimiser only works in the interior. Figure reproduced from [18]. Remark 3.1. Two comments are needed, for the sake of completeness. First, th… view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

22 extracted references · 4 linked inside Pith

  1. [1]

    Anderson

    Michael T. Anderson. Complete minimal varieties in hyperbolic space.Invent. Math., 69:477–494, 1982

    1982

  2. [2]

    PyTorch 2: Faster machine learning through dynamic Python bytecode trans- formation and graph compilation

    Jason Ansel, Edward Yang, Horace He, Natalia Gimelshein, Animesh Jain, Michael Voznesensky, Bin Bao, Peter Bell, David Berard, Evgeni Burovski, et al. PyTorch 2: Faster machine learning through dynamic Python bytecode trans- formation and graph compilation. InProceedings of the 29th ACM International Conference on Architectural Support for Programming Lan...

    2024

  3. [3]

    Pearlmutter, Alexey Andreyevich Radul, and Jeffrey Mark Siskind

    Atılım Günes Baydin, Barak A. Pearlmutter, Alexey Andreyevich Radul, and Jeffrey Mark Siskind. Automatic differen- tiation in machine learning: a survey.Journal of Machine Learning Research, 18(153):1–43, 2018

    2018

  4. [4]

    Stapleton

    Gianfranco Cortes, Maria Esteban-Casadevall, Yueqing Feng, Jonas Henkel, Edward Hirst, Tancredi Schettini Gherardini, and Alexander G. Stapleton. A Machine Learning Approach to the Nirenberg Problem.arXiv preprint arXiv:2602.12368, 2026

    arXiv 2026

  5. [5]

    Knots, minimal surfaces and J-holomorphic curves.arXiv preprint arXiv:2112.07713, 2021

    Joel Fine. Knots, minimal surfaces and J-holomorphic curves.arXiv preprint arXiv:2112.07713, 2021

    Pith/arXiv arXiv 2021

  6. [6]

    Freyd, D

    P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, and A. Ocneanu. A new polynomial invariant of knots and links.Bull. Am. Math. Soc., New Ser., 12:239–246, 1985

    1985

  7. [7]

    Computer-assisted proofs in PDE: a survey.SeMA Journal, 76(3):459–484, 2019

    Javier Gomez-Serrano. Computer-assisted proofs in PDE: a survey.SeMA Journal, 76(3):459–484, 2019

    2019

  8. [8]

    SIAM, Philadelphia, PA, 2nd edition, 2008

    Andreas Griewank and Andrea Walther.Evaluating Derivatives: Principles and Techniques of Algorithmic Differenti- ation. SIAM, Philadelphia, PA, 2nd edition, 2008

    2008

  9. [9]

    Metric flows with neural networks.Machine Learning: Science and Technology, 5(4):045020, oct 2024

    James Halverson and Fabian Ruehle. Metric flows with neural networks.Machine Learning: Science and Technology, 5(4):045020, oct 2024

    2024

  10. [10]

    Physics-informed neural network solves minimal surfaces in curved spacetime.Machine Learning: Science and Technology, 7(1):015013, jan 2026

    Koji Hashimoto, Koichi Kyo, Masaki Murata, Gakuto Ogiwara, and Norihiro Tanahashi. Physics-informed neural network solves minimal surfaces in curved spacetime.Machine Learning: Science and Technology, 7(1):015013, jan 2026

    2026

  11. [11]

    Sá Earp, and Tomás S

    Edward Hirst, Henrique N. Sá Earp, and Tomás S. R. Silva. Minimising Willmore Energy via Neural Flow.arXiv preprint arXiv:2604.04321, 2026

    Pith/arXiv arXiv 2026

  12. [12]

    Stapleton

    Edward Hirst, Tancredi Schettini Gherardini, and Alexander G. Stapleton. AInstein: Numerical Einstein Metrics via Machine Learning.AI Sci., 1(2):025001, 2025

    2025

  13. [13]

    Kingma and Jimmy Ba

    Diederik P. Kingma and Jimmy Ba. Adam: A Method for Stochastic Optimization.Proceedings of ICLR, 2015

    2015

  14. [14]

    Liu and Jorge Nocedal

    Dong C. Liu and Jorge Nocedal. On the limited memory BFGS method for large scale optimization.Mathematical Programming, 45(3):503–528, 1989

    1989

  15. [15]

    Variations of renormalized volume for minimal submanifolds of Poincaré-Einstein manifolds.Commun

    Jared Marx-Kuo. Variations of renormalized volume for minimal submanifolds of Poincaré-Einstein manifolds.Commun. Anal. Geom., 33(1):17–29, 2025

    2025

  16. [16]

    Non-uniqueness and symmetries for the Nirenberg problem using computer assistance.arXiv preprint arXiv:2603.29544, 2026

    Daniel Platt. Non-uniqueness and symmetries for the Nirenberg problem using computer assistance.arXiv preprint arXiv:2603.29544, 2026

    arXiv 2026

  17. [17]

    Raissi, P

    M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.J. Comput. Phys., 378:686–707, 2019

    2019

  18. [18]

    Minimal surfaces, knots, and neural networks, 2026

    Tancredi Schettini Gherardini and Marco Usula. Minimal surfaces, knots, and neural networks, 2026. arXiv preprint arXiv:2605.26234

    Pith/arXiv arXiv 2026

  19. [19]

    Biharmonic maps between conformally compact manifolds.Calc

    Marco Usula. Biharmonic maps between conformally compact manifolds.Calc. Var. Partial Differ. Equ., 65(2):24, 2026. Id/No 59

    2026

  20. [20]

    Isometric embeddings of conformally compact manifolds into hyperbolic spaces.International Mathematics Research Notices, 2026(10):rnag098, 05 2026

    Marco Usula. Isometric embeddings of conformally compact manifolds into hyperbolic spaces.International Mathematics Research Notices, 2026(10):rnag098, 05 2026

    2026

  21. [21]

    Yongji Wang, Mehdi Bennani, James Martens, Sébastien Racanière, Sam Blackwell, Alex Matthews, Stanislav Nikolov, Gonzalo Cao-Labora, Daniel S. Park, Martin Arjovsky, Daniel Worrall, Chongli Qin, Ferran Alet, Borislav Kozlovskii, A USER’S GUIDE TO PINNS IN GEOMETRIC ANALYSIS: LESSONS FROM THE ASYMPTOTIC PLATEAU PROBLEM 10 Nenad Tomašev, Alex Davies, Pushme...

    arXiv 2025

  22. [22]

    Approximating High-Dimensional Minimal Surfaces with Physics-Informed Neural Net- works.arXiv preprint arXiv:2309.02589, 2023

    Steven Zhou and Xiaojing Ye. Approximating High-Dimensional Minimal Surfaces with Physics-Informed Neural Net- works.arXiv preprint arXiv:2309.02589, 2023

    Pith/arXiv arXiv 2023