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pith:KKSFSMDI

pith:2026:KKSFSMDI5XOMNAU3ISRSX3UT5L

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Numerical exploration of the range of shape functionals using neural networks

Eloi Martinet, Ilias Ftouhi

Invertible neural networks based on gauge functions parametrize convex bodies to numerically chart the attainable ranges of their shape functionals.

arxiv:2602.14881 v2 · 2026-02-16 · math.OC · cs.AI

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Claims

C1strongest claim

We introduce a parametrization of convex bodies in arbitrary dimensions using a specific invertible neural network architecture based on gauge functions, allowing an intrinsic conservation of the convexity of the sets during the shape optimization process. ... The effectiveness of the method is demonstrated on several diagrams involving both geometric and PDE-type functionals for convex bodies of R^2 and R^3.

C2weakest assumption

The chosen invertible neural-network architecture based on gauge functions is sufficiently expressive to densely cover the space of all convex bodies so that the sampled diagrams accurately reflect the true attainable ranges.

C3one line summary

A gauge-function neural network parametrization of convex bodies combined with Riesz-energy particle optimization enables numerical exploration of Blaschke-Santaló diagrams for volume, perimeter, torsional rigidity, Willmore energy, and Neumann eigenvalues.

References

57 extracted · 57 resolved · 0 Pith anchors

[1] V. Agostiniani and L. Mazzieri. Monotonicity formulas in potential theory.Calc. Var. Partial Differential Equations, 59(1):6, Feb. 2020 2020

[2] P. R. S. Antunes and B. Bogosel. Parametric shape optimization using the support function.Comput. Optim. Appl., 82(1):107–138, May 2022 2022

[3] P. R. S. Antunes and P. Freitas. New bounds for the principal Dirichlet eigenvalue of planar regions.Experimental Mathematics, 15(3):333–342, March 2006 2006

[4] P. R. S. Antunes and A. Henrot. On the range of the first two dirichlet and neumann eigenvalues of the laplacian. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 4 2010

[5] M. S. Ashbaugh and R. D. Benguria. Universal bounds for the low eigenvalues of Neumann Laplacians in n dimen- sions.SIAM J. Math. Anal., 467(3):557–570, 1993 1993

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Cited by

1 paper in Pith

Parametrizing Convex Sets Using Sublinear Neural Networks

Receipt and verification

First computed	2026-05-17T23:39:16.118280Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

52a4593068eddcc6829b44a32bee93eae7b3d9c70ea4a6fb4ff634ab94fadb21

Aliases

arxiv: 2602.14881 · arxiv_version: 2602.14881v2 · doi: 10.48550/arxiv.2602.14881 · pith_short_12: KKSFSMDI5XOM · pith_short_16: KKSFSMDI5XOMNAU3 · pith_short_8: KKSFSMDI

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curl -sH 'Accept: application/ld+json' https://pith.science/pith/KKSFSMDI5XOMNAU3ISRSX3UT5L \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 52a4593068eddcc6829b44a32bee93eae7b3d9c70ea4a6fb4ff634ab94fadb21

Canonical record JSON

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    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.OC",
    "submitted_at": "2026-02-16T16:10:58Z",
    "title_canon_sha256": "7659ba1e172ee80e341807dc84497c3ec7d3bf2cf7db33b0ef37c48c636e7ec7"
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