[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-122752-en":3,"doc-seo-122752-105":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":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},122752,8796095360427,"Lucas Martin","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","Physics-inspired machine learning of localized intensive properties","Machine learning models are widely used to predict molecular and materials properties, yet the prevalent local energy, size-extensive paradigm can fail for intensive and spatially localized electronic quantities. This article examines strategies for learning localized intensive properties, using HOMO energies in organic molecules as a representative case. By analyzing pooling functions in atomistic neural networks, the study introduces an orbital weighted average (OWA) approach to improve accurate prediction of orbital energies and their spatial locations.","Open Access Article . Pu on 10 Apr 2023. Down on 5/23/2023blished il loaded 7:40:50 AM .  \nChemical Science  \nEDGE ARTICLE  \nView Article Online View Journal | View Issue  \nCite this: Chem. Sci., 2023, 14, 4913  \nAll publication charges for this article have been paid for by the Royal Society of Chemistry  \nReceived 14th February 2023  \nAccepted 10th April 2023 DOI: 10.1039/d3sc00841j[rsc.li/chemical-science](rsc.li/chemical-science)  \nPhysics-inspired machine learning of localized intensive properties†  \nKe Chen, abc Christian Kunkel,  a Bingqing Cheng,  c Karsten Reuter aband Johannes T. Margraf  *a  \nMachine learning (ML) has been widely applied to chemical property prediction, most prominently for the energies and forces in molecules and materials. The strong interest in predicting energies in particular has led to a ‘ local energy ’ -based paradigm for modern atomistic ML models, which ensures sizeextensivity and a linear scaling of computational cost with system size. However, many electronic properties (such as excitation energies or ionization energies) do not necessarily scale linearly with system size and may even be spatially localized. Using size-extensive models in these cases can lead to large errors. In this work, we explore different strategies for learning intensive and localized properties, using HOMO energies in organic molecules as a representative test case. In particular, we analyze the pooling functions that atomistic neural networks use to predict molecular properties, and suggest an orbital weighted average (OWA) approach that enables the accurate prediction of orbital energies and locations.  \n1. Introduction  \nDue to their great potential for accelerating materials discovery and design, there has been signi􀀁cant interest in machine learning (ML) models that enable the fast and accurate prediction of molecular and materials properties.1–5 Consequently, a wide range of neural network (NN) and Kernel ML methods have been developed and applied to systems ranging from isolated molecules to complex amorphous solids.6–14  \nIn this context, many state-of-the-art approaches exploit the approximately local nature of chemical interactions. This is achieved by representing chemical structures in terms of the element of each atom and the types and positions of the atoms in its immediate surrounding (the chemical environment) .15–17 This is, e.g., commonly used when developing ML interatomic potentials, where the total energy is then obtained as a sum of local atomic contributions (see Fig. 1) .  \nThere are two distinct but related advantages to this approach. On one hand, locality ensures that the computational cost of the model asymptotically displays linear scaling with the size of the system, allowing for instance the routine application of ML potentials to systems with a thousand atoms or more. On  \naFritz-Haber-Institut der Max-Planck-Gesellscha􀀁, Faradayweg 4-6, D-14195 Berlin, Germany. E-mail: margraf@􀀃i-berlin.mpg.de  \nbChair for Theoretical Chemistry and Catalysis Research Center, Technische Universitt München, Lichtenbergstraße 4, D-85747 Garching, Germany  \ncInstitute of Science and Technology, Am Campus 1, 3400 Klosterneuburg, Austria † Electronic supplementary information (ESI) available: Details on structure generation, model hyperparameters, additional learning curves, and further details on the LocalOrb dataset. See DOI: [https://doi.org/10.1039/d3sc00841j](https://doi.org/10.1039/d3sc00841j)  \nthe other hand, the summation of atomic contributions ensures size-extensivity, which is o􀀁en desirable, if not a key requirement as in the case of interatomic potentials.  \nSimply put, size-extensivity means that predicted properties (e.g. energies) scale linearly upon trivial extensions of the system size, e.g. when describing ideal crystals in larger periodic supercells or replicating non-interacting molecules. This allows size-extensive ML models to be trained on small molecules or simulation cells and l","cbCaicdtnXu7UDxD","https://ap.wps.com/l/cbCaicdtnXu7UDxD","pdf",1520006,1,10,"English","en",105,"# Introduction\n## Locality and size-extensive modeling in atomistic ML\n## Intensive and localized electronic properties\n## Pooling functions and average pooling limitations","[{\"question\":\"Why can size-extensive atomistic ML models produce large errors for electronic properties?\",\"answer\":\"Some electronic properties, such as excitation energies and orbital energies, do not scale linearly with system size and can be spatially localized. Size-extensive pooling assumptions can therefore lead to unphysical extrapolations for larger systems.\"},{\"question\":\"What pooling concept is commonly used for intensive properties, and what is its drawback?\",\"answer\":\"Average pooling is often used because it keeps predictions constant under trivial scaling. However, it can still yield unphysical results when the target property is localized and the system has low symmetry.\"},{\"question\":\"What is the proposed orbital weighted average (OWA) approach?\",\"answer\":\"The orbital weighted average approach enhances atomistic neural networks by using an additional model to predict the weights used in the pooling function. This enables accurate prediction of both orbital energies and their spatial locations for localized properties.\"}]","Physics-inspired machine learning of localized intensive properties | PDF",1785812709,25,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":28},"physics-inspired-machine-learning-of-localized-intensive-properties","",{"@graph":36,"@context":85},[37,54,68],{"@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/physics-inspired-machine-learning-of-localized-intensive-properties/122752/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-04",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},"Why can size-extensive atomistic ML models produce large errors for electronic properties?","Question",{"text":75,"@type":76},"Some electronic properties, such as excitation energies and orbital energies, do not scale linearly with system size and can be spatially localized. Size-extensive pooling assumptions can therefore lead to unphysical extrapolations for larger systems.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What pooling concept is commonly used for intensive properties, and what is its drawback?",{"text":80,"@type":76},"Average pooling is often used because it keeps predictions constant under trivial scaling. However, it can still yield unphysical results when the target property is localized and the system has low symmetry.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the proposed orbital weighted average (OWA) approach?",{"text":84,"@type":76},"The orbital weighted average approach enhances atomistic neural networks by using an additional model to predict the weights used in the pooling function. 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