[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85013-en":3,"doc-seo-85013-105":29,"detail-sidebar-cat-0-en-105":91},{"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":20,"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},85013,13056703019404,"Miles","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Neural Operator-enabled Topology-informed Evolutionary Strategy for PDE-Constrained Optimization","Inverse design of physical systems governed by partial differential equations faces computational cost driven by high-dimensionality and non-convex design spaces. Existing generative inverse-design models can be fragile in robustness and transferability, while evolutionary strategies remain robust but struggle at scale. This work proposes NOTES, combining a DeepONet-based neural operator with CMA-ES to optimize in a compact latent space that encodes topology-aware priors. On nanophotonic beam-deflector problems, dimensionality drops from 256 to 25 while maintaining efficiency above 95%, and achieves strong gains on structural compliance.","Neural Operator-enabled Topology-informed Evolutionary Strategy  \nfor PDE-Constrained Optimization Xiangming Huang, Guannan Zhang, Lu Lu, Raphal Pestourie  \narXiv :2607 .07682v 1 [ cs .LG] 8 Jul 2026  \nThis is the accepted manuscript version for publication by IEEE.  \n© 2026 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.  \nI. ABSTRACT  \nThe inverse design of physical systems governed by partial differential equations is computationally demanding due to the high dimensionality and non-convexity of design spaces. Generative models for inverse design often lack robustness and transferability, whereas evolutionary strategies are robust but struggle in high-dimensional spaces. This paper introducesa Neural Operator–enabled Topology-informed Evolutionary Strategy (NOTES) that integrates dimensionality reduction, representation learning, and evolutionary optimization for efficient and transferable inverse design. NOTES couples a DeepONet-based neural operator with the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) to perform global optimization in a compact latent space that encodes topologyaware priors while discovering high-performance designs for unseen operating conditions. Applied to nanophotonic beamdeflector inverse design governed by Maxwell’s equations, NOTES reduces the design dimensionality from 256 to 25 and consistently achieves over 95% efficiency, outperforming CMA-ES, topology optimization, and other baselines. Applied to structural optimization, NOTES discovers designs that achieve compliance down to 246 . By decoupling topology learning of a DeepONet from the governing physics in a PDEsolver, NOTES provides a flexible and transferable framework for the inverse design of physical systems.  \nII. INTRODUCTION  \nWe introduce a Neural Operator-enabled Topologyinformed Evolutionary Strategy (NOTES) that integrates implicit neural reparameterization and evolutionary optimization to provide an efficient, flexible, and transferable framework for inverse design of physical systems. The method unifies a DeepONet-based neural operator [1] with the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [2] to perform global optimization in a compact latent space, which encodes topology-aware priors and reduces the number of optimization variables. Although NOTES does not embed the governing partial-differential equation (PDE) explicitly into the neural operator loss, the framework remains physicsinformed through the construction of its training dataset. The DeepONet is trained exclusively on designs obtained from  \ndirect PDE-constrained optimization of the target physical systems, so the learned latent representation is biased toward geometries that encode physically meaningful high-performance features. In this sense, physical information enters NOTES indirectly through the data distribution rather than through a residual-based loss function [3] . In addition, DeepONet allows architectural inductive biases to be incorporated into the representation, illustrated here through binarization constraints and, in prior work, through hard enforcement of physical priors such as boundary conditions and invariance [4] .  \nNOTES is particularly well suited for PDE-constrained inverse design problems where relevant high-performance designs are already available, allowing transferable geometric features to be learned from existing data. By combining representation learning with evolutionary optimization, the framework aims to make evolutionary strategies practical for high-dimensional topology optimization problems that are otherwise prohibitively expensive for direct search. In addition to improving opti","cbCaitWWNvmKhUE2","https://ap.wps.com/l/cbCaitWWNvmKhUE2","pdf",2850497,1,22,"English","en",105,"# Abstract\n# Introduction\n# Prior Work\n## Evolutionary Strategy for Inverse Design\n## Representation Learning for Inverse Design\n## Neural Operator","[{\"question\":\"What problem does NOTES address in inverse design under PDE constraints?\",\"answer\":\"Inverse design under PDE constraints is difficult due to high-dimensional and non-convex design spaces, making search expensive and optimization unstable across conditions. NOTES targets efficiency and transferability in this setting.\"},{\"question\":\"How does NOTES combine a neural operator with an evolutionary strategy?\",\"answer\":\"NOTES couples a DeepONet-based neural operator with CMA-ES. The neural operator provides a compact latent representation with topology-aware priors, while CMA-ES performs global optimization within that latent space.\"},{\"question\":\"What dimensionality reduction and performance results does the paper report?\",\"answer\":\"For nanophotonic beam-deflector inverse design, the approach reduces design dimensionality from 256 to 25 and consistently achieves over 95% efficiency. For structural optimization, it reports compliance reduction down to 246.\"}]",1784200274,55,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":27},"neural-operator-enabled-topology-informed-evolutionary-strategy-for-pde-constrained-optimization","",{"@graph":35,"@context":85},[36,53,68],{"@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/neural-operator-enabled-topology-informed-evolutionary-strategy-for-pde-constrained-optimization/85013/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does NOTES address in inverse design under PDE constraints?","Question",{"text":75,"@type":76},"Inverse design under PDE constraints is difficult due to high-dimensional and non-convex design spaces, making search expensive and optimization unstable across conditions. NOTES targets efficiency and transferability in this setting.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does NOTES combine a neural operator with an evolutionary strategy?",{"text":80,"@type":76},"NOTES couples a DeepONet-based neural operator with CMA-ES. The neural operator provides a compact latent representation with topology-aware priors, while CMA-ES performs global optimization within that latent space.",{"name":82,"@type":73,"acceptedAnswer":83},"What dimensionality reduction and performance results does the paper report?",{"text":84,"@type":76},"For nanophotonic beam-deflector inverse design, the approach reduces design dimensionality from 256 to 25 and consistently achieves over 95% efficiency. 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