{"paper":{"title":"Numerical exploration of the range of shape functionals using neural networks","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Invertible neural networks based on gauge functions parametrize convex bodies to numerically chart the attainable ranges of their shape functionals.","cross_cats":["cs.AI"],"primary_cat":"math.OC","authors_text":"Eloi Martinet, Ilias Ftouhi","submitted_at":"2026-02-16T16:10:58Z","abstract_excerpt":"We introduce a novel numerical framework for the exploration of Blaschke--Santal\\'o diagrams, which are efficient tools characterizing the possible inequalities relating some given shape functionals. 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. To achieve a uniform sampling inside the diagram, and thus a satisfying description of it, we introduce an interacting particle system that min"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Invertible neural networks based on gauge functions parametrize convex bodies to numerically chart the attainable ranges of their shape functionals.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"6db8d55bc436c2ccd3f9da4d1b967e14dcbea10d0cc0f3449b3bf06a8b591597"},"source":{"id":"2602.14881","kind":"arxiv","version":2},"verdict":{"id":"002684eb-ab08-4831-8a5f-87c4191feffb","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T21:50:31.590254Z","strongest_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.","one_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.","pipeline_version":"pith-pipeline@v0.9.0","weakest_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.","pith_extraction_headline":"Invertible neural networks based on gauge functions parametrize convex bodies to numerically chart the attainable ranges of their shape functionals."},"references":{"count":57,"sample":[{"doi":"","year":2020,"title":"V. Agostiniani and L. Mazzieri. 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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 4","work_id":"129d5396-ef49-458f-b1b1-281a161baab0","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1993,"title":"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","work_id":"bf0c7aef-e092-4026-8978-31e7cd4dbc75","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":57,"snapshot_sha256":"5a7ee1c19cc80f5ead0902e495fc01108ba01067d0b6889ce39e26f93fc86342","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"327707c9e37102ff0ce9fb22f2483b9b20875b6c46a5792d9d8ad93b01307e3e"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}