[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-118724-en":3,"doc-seo-118724-105":30,"detail-sidebar-cat-0-en-105":92},{"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":27,"seo_description":14,"update_tm":28,"read_time":29},118724,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","A smooth basis for atomistic machine learning","Investigates how to construct atomic-density representations for atomistic machine learning by choosing an optimal radial basis. The study expands the neighbor atom density using spherical harmonics for angular symmetry, but addresses the open question of radial basis selection by solving the Laplacian eigenvalue problem inside a sphere around the central atom. Results show Laplacian eigenstate bases yield the smoothest possible basis of a given size and extend via tensor products to smooth higher-order correlations. Unsupervised metrics outperform several common basis sets and match competitive data-driven alternatives; supervised tests show comparable or better performance when optimized to dataset-specific atom-density correlations, highlighting function smoothness as a key overlooked factor.","arXiv :2209 .01948v1 [physics .chem-ph] 5 Sep 2022  \nA smooth basis for atomistic machine learning  \nFilippo Bigi,1 Kevin Huguenin-Dumittan,2 Michele Ceriotti,2 and David E. Manolopoulos1  \n1) Physical and Theoretical Chemistry Laboratory, South Parks Road, Oxford OX1 3QZ, UK  \n2) Laboratory of Computational Science and Modeling, Institute of Materials,􀀓  \nEcole Polytechnique F􀀓ed􀀓erale de Lausanne, 1015 Lausanne, Switzerland  \nMachine learning frameworks based on correlations of interatomic positions begin with a discretized description of the density of other atoms in the neighbourhood of each atom in the system. Symmetry considerations support the use of spherical harmonics to expand the angular dependence of this density, but there is as yet no clear rationale to choose one radial basis over another. Here we investigate the basis that results from the solution of the Laplacian eigenvalue problem within a sphere around the atom of interest. We show that this generates the smoothest possible basis of a given size within the sphere, and that a tensor product of Laplacian eigenstates also provides the smoothest possible basis for expanding any higher-order correlation of the atomic density within the appropriate hypersphere. We consider several unsupervised metrics of the quality of a basis for a given dataset, and show that the Laplacian eigenstate basis has a performance that is much better than some widely used basis sets and is competitive with data-driven bases that numerically optimize each metric. In supervised machine learning tests, we 􀀌nd that the optimal function smoothness of the Laplacian eigenstates leads to comparable or better performance than can be obtained from a data-driven basis of a similar size that has been optimized to describe the atom-density correlation for the speci􀀌c dataset. We conclude that the smoothness of the basis functions is a key and hitherto largely overlooked aspect of successful atomic density representations.  \nI. INTRODUCTION  \nMachine learning (ML) has gained increasing importance in the 􀀌eld of atomistic modeling during the last decade. Successful applications have involved both supervised learning (often in the form of regression of atomic-scale properties) and unsupervised learning (e.g., clustering of large compound/structure databases) . Most ML methods, whether supervised or unsupervised, rely on representations of the atomic structures of interest. These representations are constructed as a set of numerical descriptors (or features) which act as the inputs to the ML model. Although several classes of descriptors are now available, their performance is often limited by the choice of basis that is used to project the atomic density correlations 1 . Asa result, the identi􀀌cation of a suitable basis is a key step in building more e􀀋ective descriptors and, ultimately, ML models.  \nDesirable properties of atomic-scale descriptors include equivariance, uniqueness, interpretability, and low computational cost2 . However, most of these are properties of broad classes of descriptors, which will not be our focus here. Instead we shall con􀀌ne our attention to properties that are directly a􀀋ected by the choice of basis, such as low dimensionality and high information content of the descriptor space, sensitivity to changes in the atomic con􀀌guration, and good performance on regression tasks.  \nThe spherical harmonics are (almost3 ) always used as the angular basis for atomic density expansions 1,4{8 . This is because of their symmetry properties with respect to rotation and inversion, which make it possible to generate features with the desired equivariant behavior for a speci􀀌c ML task. However, the best choice of the radial basis is still an open question, with a variety of di􀀋erent radial bases  \ncurrently in use in di􀀋erent packages for atomic-scale representations and machine learning4,9{12 .  \nRadial bases have traditionally been regarded as a component of atomic density-based approa","cbCaidmyyEwCxF7Z","https://ap.wps.com/l/cbCaidmyyEwCxF7Z","pdf",1328691,1,29,"English","en",105,"# Abstract\n# Introduction\n## Atomistic modeling and descriptor representations\n## Role of spherical harmonics and open radial-basis question\n# Theory\n## Density expansion and density correlations","[{\"question\":\"Why does the paper focus on choosing a radial basis for atomic density representations?\",\"answer\":\"ML performance depends on how atomic density correlations are projected into a descriptor space, and while angular symmetry is often handled by spherical harmonics, the radial basis choice remains an open question affecting descriptor quality.\"},{\"question\":\"What basis does the paper propose for improving smoothness of descriptors?\",\"answer\":\"It proposes using Laplacian eigenstates within a sphere around each atom, derived from the Laplacian eigenvalue problem, and shows that their tensor products provide smooth bases for higher-order density correlations.\"},{\"question\":\"How do the Laplacian eigenstate bases perform compared with common and data-driven bases?\",\"answer\":\"Unsupervised quality metrics show much better performance than some widely used basis sets and competitiveness with data-driven bases optimized numerically; supervised tests also achieve comparable or better results than similarly sized data-driven bases for describing atom-density correlations.\"}]","A smooth basis for atomistic machine learning | PDF",1785719916,73,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"a-smooth-basis-for-atomistic-machine-learning","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/a-smooth-basis-for-atomistic-machine-learning/118724/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-04","2026-08-03",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"Why does the paper focus on choosing a radial basis for atomic density representations?","Question",{"text":76,"@type":77},"ML performance depends on how atomic density correlations are projected into a descriptor space, and while angular symmetry is often handled by spherical harmonics, the radial basis choice remains an open question affecting descriptor quality.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What basis does the paper propose for improving smoothness of descriptors?",{"text":81,"@type":77},"It proposes using Laplacian eigenstates within a sphere around each atom, derived from the Laplacian eigenvalue problem, and shows that their tensor products provide smooth bases for higher-order density correlations.",{"name":83,"@type":74,"acceptedAnswer":84},"How do the Laplacian eigenstate bases perform compared with common and data-driven bases?",{"text":85,"@type":77},"Unsupervised quality metrics show much better performance than some widely used basis sets and competitiveness with data-driven bases optimized numerically; supervised tests also achieve comparable or better results than similarly sized data-driven bases for describing atom-density correlations.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":93},[94,98,102,106,111,116,121,124,129,132,136],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":46,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":46,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":46,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":46,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":46,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":46,"category_name":138,"show_sort_weight":107,"slug":139},19,"General","general"]