[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81792-en":3,"doc-seo-81792-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},81792,137441390410,"Hazel","https://ap-avatar.wpscdn.com/avatar/2000252f4ab5702993?_k=1776741390130283984",8,"Research & Report","Compression of Polyconvex Envelopes of Isotropic Functions via Monotonic Input Convex Neural Networks","The document introduces a neural-network compression method for polyconvex envelopes of isotropic functions, built on a classical sufficient criterion for polyconvexity. It targets determinant-constrained energy densities in nonlinear elasticity and reduces computation versus approaches relying on necessary-and-sufficient characterizations. The method restricts evaluations to the positive octant in the signed singular value space. It uses input-convex neural networks with nonnegative weights to enforce convexity and monotonicity, while symmetry and inequality constraints are incorporated through the training loss. Numerical tests on Saint Venant–Kirchhoff energy show accurate practical approximations with improved efficiency.","arXiv :2607 .01055v1 [math .NA] 1 Jul 2026  \nCOMPRESSION OF POLYCONVEX ENVELOPES OF ISOTROPIC FUNCTIONS VIA MONOTONIC INPUT CONVEX NEURAL NETWORKS  \nT. NEUMEIER∗ , J. SALMON∗  \nAbstract. This work presents a novel neural-network compression approach for polyconvex envelopes of isotropic functions. The approach relies on a classical sufficient criterion for polyconvexity and is particularly suited for the representation of determinant-constrained energy densities arising in non-linear elasticity. Compared with existing compression methods based on the necessary and sufficient characterisation of polyconvex isotropic functions, the proposed framework reduces computational costs, due to the domain reduction through the restriction to the positive octant in the singed singular value space. The underlying neural-network architecture employs input-convex neural networks (ICNNs) with non-negative weight constraints to enforce the required convexity and monotonicity properties. The additional symmetry and inequality conditions characterising the polyconvex envelope are incorporated weakly through the loss function during training. Although the employed criterion is only sufficient and thus generally yields only a lower bound on the polyconvex envelope, numerical experiments based on the classical Saint Venant–Kirchhoff energy demonstrate that the proposed approach produces accurate approximations in practice while offering a computationally more efficient alternative  \nto existing methods.  \nKey words. Polyconvexity, input convex neural network, monotonicity, relaxation  \nAMS subject classifications. 49J45 , 49J10 , 74G65 , 74B20 , 68T07  \n1. Introduction  \nMany problems in non-linear elasticity can be formulated as variational minimisation problems of the form  \nI (u) = ZΩ W (∇u) dx,  \nwhere Ω ⊂ Rd in spatial dimension d ∈ {2, 3}, denotes the reference domain, u: Ω → Rd an admissible deformation, and W : Rd×d → R ∞ := R ∪ {∞} the energy density function. In practical applications, the function value W (F ) = ∞ is used to model physically inadmissible deformation states, for example through orientation-preserving determinant constraints.  \nMany constitutive models arising in non-linear elasticity, phase transformations, damage mechanics and fracture are inherently non-convex, see, for example,[Bal76; BJ87 ; Mül99 ; Kin+93 ; Bha03 ; Ped97 ; BO12 ; Rao86 ; BBŠ07] . Consequently, the associated variational problems may fail to admit minimisers, while numerical approximations often exhibit mesh dependence, reduced robustness and pronounced sensitivity with respect to discretisation and material parameters.  \nA classical remedy is provided by relaxation theory. That is, instead of the original nonconvex energy density W , one considers a suitable semiconvex envelope, thereby obtaining a relaxed problem that admits minimisers and captures the effective behaviour of oscillatory minimising sequences. Among the various notions of semiconvexity, polyconvexity, introduced by  \n∗ Institute of Mathematics, University of Augsburg, Universitätsstr. 12a, 86159 Augsburg, Germany  \nE-mail address: {timo.neumeier, [julian.salmon}@uni-a.de](julian.salmon}@uni-a.de).  \nDate: July 2, 2026 .  \nThe authors gratefully acknowledge funding from the German Research Foundation (DFG) within the Priority Programme 2256 Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials (project number 441154176, reference IDs PE1464/7-2 and PE2143/5-2) . Furthermore, the authors would also like to thank the Bavarian State Ministry of Science and the Arts for funding the Augsburg AI Production Network as part of the High-Tech Agenda Plus.  \n2 NEURAL NETWORK COMPRESSION OF POLYCONVEX ENVELOPES  \nBall in the works [Bal76; Bal77 ; Bal02], constitutes a physically meaningful sufficient condition for weak lower semicontinuity of the functional and therefore for the existence of minimisers [Dac08] . Consequently, the polyconvex envelope plays a central","cbCaiclOdqm51vm5","https://ap.wps.com/l/cbCaiclOdqm51vm5","pdf",1285544,1,16,"English","en",105,"# Introduction\n# Neural network compression of polyconvex envelopes","[{\"question\":\"What problem does the document address in nonlinear elasticity?\",\"answer\":\"Many energies in nonlinear elasticity are non-convex, leading to relaxation issues and computational challenges. The work focuses on polyconvex envelopes that enable a relaxed variational problem with minimizers.\"},{\"question\":\"How does the proposed neural network compression method reduce computational cost?\",\"answer\":\"It compresses computations by using a domain reduction to the positive octant in the signed singular value space, avoiding expensive repeated evaluations of polyconvex envelopes in lifted spaces.\"},{\"question\":\"What neural network architecture and constraints are used?\",\"answer\":\"The approach employs input-convex neural networks (ICNNs) with non-negative weight constraints to enforce convexity and monotonicity. Additional symmetry and inequality conditions are incorporated weakly via the training loss.\"}]",1784176181,40,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"compression-of-polyconvex-envelopes-of-isotropic-functions-via-monotonic-input-convex-neural-networks","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/compression-of-polyconvex-envelopes-of-isotropic-functions-via-monotonic-input-convex-neural-networks/81792/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What problem does the document address in nonlinear elasticity?","Question",{"text":74,"@type":75},"Many energies in nonlinear elasticity are non-convex, leading to relaxation issues and computational challenges. The work focuses on polyconvex envelopes that enable a relaxed variational problem with minimizers.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the proposed neural network compression method reduce computational cost?",{"text":79,"@type":75},"It compresses computations by using a domain reduction to the positive octant in the signed singular value space, avoiding expensive repeated evaluations of polyconvex envelopes in lifted spaces.",{"name":81,"@type":72,"acceptedAnswer":82},"What neural network architecture and constraints are used?",{"text":83,"@type":75},"The approach employs input-convex neural networks (ICNNs) with non-negative weight constraints to enforce convexity and monotonicity. 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