[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84901-en":3,"doc-seo-84901-105":30,"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},84901,1099514068035,"Ezra","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Learning-Based Physics-Constrained Neural Kernel for Sound Field Estimation With Source-Position-Dependent Directional Weighting","A learning-based physics-constrained neural kernel is proposed for sound field estimation, reconstructing the spatial acoustic pressure distribution from discrete microphone measurements used across spatial audio and acoustic analysis. Kernel-regression methods provide a principled way to enforce governing-equation constraints via linear inference. To avoid overfitting from snapshot-only optimization, a source-position-dependent implicit neural representation is introduced for the directional weighting function, improving generalization to unseen source positions and matching target sound-field directivity.","# LEARNING-BASED PHYSICS-CONSTRAINED NEURAL KERNELFOR SOUND FIELD\n\nESTIMATION WITH SOURCE-POSITION-DEPENDENT DIRECTIONAL WEIGHTING  \nMattia Marella₁,2*  and Shoichi Koyama¹  \n¹National Institute of Informatics,Tokyo,Japan,²University of Ferrara,Ferrara,Italy  \narXiv:2607.06274v1[cs.SD]7 Jul 2026  \n## ABSTRACT\n\nA learning-based physics-constrained neural kernel for sound fieldestimation is proposed.Sound field estimation aims to estimate thespatial distribution of an acoustic field from a discrete set of mi-crophone measurements,which have a wide range of applications.Among existing sound field estimation methods,kernel-regression-based methods offer a flexible and principled framework for incor-porating physical constraints and allow inference through linear op-eration.It is also possible to adapt the kernel function to the tar-get acoustic environment by representing the directional weightingfunction as an implicit neural representation(INR)and optimizinghyperparameters using measurements.However,the kernel func-tion is generally optimized for single snapshot measurements of themicrophones,which can lead to strong overfitting and poor gener-alization.We propose a source-position-dependent INR for the di-rectional weighting function,enabling the kernel function to cap-ture common directional patterns and to generalize to unseen sourcepositions in the target acoustic environment.Experimental resultsindicate that our proposed method outperforms the snapshot-basedmethod by estimating a directional weighting function that matchesthe directivity of the target sound field.  \nIndex Terms—kernel regression,neural networks,physics-informed machine learning,sound field reconstruction,spatial audio  \n## 1.INTRODUCTION\n\nSound field estimation/reconstruction/interpolation is a fundamentaltask in audio signal processing and machine learning,aiming to esti-mate the spatial distribution of an acoustic field from a discrete set ofsensor(microphone)observations.It can be applied to a wide varietyof downstream tasks,for example,acoustic imaging,room acous-tic analysis 2,and spatial audio reproduction and control 34.  \nThere have been many studies on sound fied estimation 5].One of the most widely used techniques is the basis-expansion-basedmethod,which is based on the representation of the sound field asa linear combination of predefined basis functions,such as planewaves,spherical wave functions,and equivalent point sources 36].The expansion coefficients are then estimated from the microphonemeasurements using the least squares method.Sparse regulariza-tion techniques have also been applied to promote sparsity in thecoefficient space to improve reconstruction accuracy忆-9].Kernel-regression-based methods generalize the (finite-dimensional)basis-expansion-based method to an infinite-dimensional basis expansion,  \nthereby enabling the estimates to be constrained to the solution ofthe governng equation(wave and Helmholtz equations)11.  \nIn recent years,neural network (NN)-based methods have at-tracted attention because of their high representational power andinterpolation capabilities [12-16].Physics-informed (penalized andconstrained)approaches have also been investigated in the NN-basedmethods to avoid overfitting and increase the interpretability[17-  \n21.  \nAmong these current sound field estimation methods,thekernel-regression-based methods are particularly practical for ap-plications requiring real-time processing,as they satisfy physicalconstraints while enabling inference through linear operation.Todesign physics-constrained kernel functions,the Herglotz wavefunction [22,which is equivalent to a plane wave expansion,is used.To incorporate prior information regarding the directivity of the tar-get sound field,a directional weighting function can be introducedinto the Herglotz wave function [10.Although a fixed unimodalfunction or its linear combination has been used as the directionalweighting function,they are unable to fully capture the ch","cbCaio7D3n6qq7Ct","https://ap.wps.com/l/cbCaio7D3n6qq7Ct","pdf",2640774,2,1,5,"English","en",105,"# Abstract\n# Introduction\n# Problem Formulation","[{\"question\":\"What problem does the proposed method address in sound field estimation?\",\"answer\":\"It addresses overfitting and poor generalization caused by optimizing the kernel function using only single-snapshot microphone measurements.\"},{\"question\":\"How does the method improve generalization to unseen source positions?\",\"answer\":\"It models the directional weighting function with a source-position-dependent implicit neural representation, trained from data so common directional patterns are captured and adapted without fine-tuning.\"},{\"question\":\"What inputs and constraints does the approach rely on?\",\"answer\":\"The method reconstructs the acoustic field in a Helmholtz-governed, source-free region and uses acoustic transfer function data from known source positions, incorporating physical constraints through physics-constrained kernel regression.\"}]",1784199250,13,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"learning-based-physics-constrained-neural-kernel-for-sound-field-estimation-with-source-position-dependent-directional-weighting","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/learning-based-physics-constrained-neural-kernel-for-sound-field-estimation-with-source-position-dependent-directional-weighting/84901/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","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 the proposed method address in sound field estimation?","Question",{"text":75,"@type":76},"It addresses overfitting and poor generalization caused by optimizing the kernel function using only single-snapshot microphone measurements.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the method improve generalization to unseen source positions?",{"text":80,"@type":76},"It models the directional weighting function with a source-position-dependent implicit neural representation, trained from data so common directional patterns are captured and adapted without fine-tuning.",{"name":82,"@type":73,"acceptedAnswer":83},"What inputs and constraints does the approach rely on?",{"text":84,"@type":76},"The method reconstructs the acoustic field in a Helmholtz-governed, source-free region and uses acoustic transfer function data from known source positions, incorporating physical constraints through physics-constrained kernel 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