[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117248-en":3,"doc-seo-117248-105":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":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":27,"seo_description":14,"update_tm":28,"read_time":29},117248,2336464648322,"Aria","https://ap-avatar.wpscdn.com/avatar/2200025388227c56fec?_k=1778556882303663488",8,"Research & Report","Machine Learning approaches to classical density functional theory - Chapter summary","Recent developments and opportunities in machine learning for classical density functional theory are reviewed, focused on equilibrium properties of thermal nano- and microparticle systems with classical interactions. Machine learning enables constructing and improving the free energy functional—the central object of density functional theory—directly from simulation data, complementing physics- or intuition-driven functional design. The chapter also outlines future directions toward related functional formulations such as liquid state theory, electron density functional theory, and power functional theory for classical nonequilibrium systems.","Machine Learning approaches to classical density functional theory  \nAlessandro Simon and Martin Oettel  \narXiv :2406 .07345v1 [ cond-mat .stat-mech] 11 Jun 2024  \nThe following chapter is set to appear in:  \nte Vrugt, M. (Ed.), (2024), Artificial Intelligence and Intelligent Matter, published by Springer (Cham) .  \nAlessandro Simon  \nInstitute of Applied Physics, University [alessandro-rodolfo.simon@uni-tuebingen.de](alessandro-rodolfo.simon@uni-tuebingen.de)[ ](alessandro-rodolfo.simon@uni-tuebingen.de)Martin Oettel  \nInstitute of Applied Physics, University [martin.oettel@uni-tuebingen.de](martin.oettel@uni-tuebingen.de)  \nof  \nof  \nT¨ubingen, Germany, e-mail:  \nT¨ubingen, Germany, e-mail:  \nContents  \nMachine Learning approaches to classical density functional theory ................................... 1  \nAlessandro Simon and Martin Oettel 1 Introduction ................................. 4  \n2 Classical DFT: basic theory .................... 6  \n3 Model systems ................................ 9  \n3.1 Hard sphere system in 1D and 3D ......... 9  \n3.2 Lennard-Jones .......................... 10  \n3.3 Kern–Frenkel ........................... 11  \n4 Machine learning approaches ................... 12  \n4.1 Direct parameterization of the functional ... 12  \n4.2 Parameterization of c1 [ρ] ................. 22  \n4.3 Learning Fex by c2 matching ............. 25  \n4.4 Learning the map ρ ↔ Vext .............. 28  \n5 Outlook to related problems .................... 33  \n5.1 Liquid state theory ...................... 33  \n5.2 Electron DFT .......................... 34  \n5.3 Power functional theory ................. 37  \n6 Summary and conclusion ....................... 39  \nReferences ........................................ 41  \nAbstract In this chapter, we discuss recent advances and new opportunities through methods of machine learning for the field of classical density functional theory, dealing with the equilibrium properties of thermal nano– and microparticle systems having classical interactions. Machine learning methods offer the great potential to construct and/or improve the free energy functional  \n4  \n(the central object of density functional theory) from simulation data and thus they complement traditional physics– or intuition– based approaches to the free energy construction. We also give an outlook to machine learning efforts in related fields, such as liquid state theory, electron density functional theory and power functional theory as a functionally formulated approach to classical nonequilibrium systems.  \n1 Introduction  \nDensity functional theory (DFT) is a powerful reductionist scheme for classical and quantum many-body systems in equilibrium. The reduction comes about by the existence of a unique (free) energy functional, depending only on the one–body density of classical or quantum particles. From this functional, all other properties of interest, most notably higher order correlations can be derived. For quantum systems at zero temperature, T = 0, the unique mapping between an external potential Vext (r) and the particle density ρ(r) entails the existence of a unique energy functional E [ρ], not depending on Vext [1] . For finite T , the argument can be generalized to show the existence of a unique free energy functional F[ρ] both in the quantum case [2] and in the classical case [3] . However, in general the functional F[ρ] is different for differing internal Hamiltonians of the system (i.e. differing particle–particle interaction potentials) and it is not known except for a few exceptional cases (like the ideal gas and a system of one–dimensional (1D) hard rodsin the classical case) . Moreover, in the classical case there are a number of Hamiltonians of interest, ranging from those of atomic and molecular systems to those of polymeric and colloidal systems where the basic particles are macromolecular in nature and their particle–particle interactions are already coarse–grained.  \nConstructing classical fre","cbCaigRLLYMjU8wy","https://ap.wps.com/l/cbCaigRLLYMjU8wy","pdf",1936111,1,44,"English","en",105,"# Contents\n## Introduction\n## Classical DFT: basic theory\n## Model systems\n## Machine learning approaches\n## Outlook to related problems\n## Summary and conclusion\n## References","[{\"question\":\"What is the central quantity in classical density functional theory discussed in this chapter?\",\"answer\":\"The central object is the free energy functional, which depends on the one-body density. Once obtained, it allows deriving other properties and higher-order correlations.\"},{\"question\":\"How do machine learning methods help in classical DFT according to the chapter?\",\"answer\":\"They offer the potential to construct and/or improve the free energy functional from simulation data. This complements traditional approaches based on physical insight or intuition.\"},{\"question\":\"Which related fields does the chapter point to in its outlook?\",\"answer\":\"It discusses related efforts in liquid state theory, electron density functional theory, and power functional theory, framed as functionally formulated approaches for classical nonequilibrium systems.\"}]","Machine Learning approaches to classical density functional theory - Chapter summary | PDF",1785674655,111,{"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},"machine-learning-approaches-to-classical-density-functional-theory-chapter-summary","",{"@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/machine-learning-approaches-to-classical-density-functional-theory-chapter-summary/117248/",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","2026-08-02",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the central quantity in classical density functional theory discussed in this chapter?","Question",{"text":75,"@type":76},"The central object is the free energy functional, which depends on the one-body density. Once obtained, it allows deriving other properties and higher-order correlations.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do machine learning methods help in classical DFT according to the chapter?",{"text":80,"@type":76},"They offer the potential to construct and/or improve the free energy functional from simulation data. This complements traditional approaches based on physical insight or intuition.",{"name":82,"@type":73,"acceptedAnswer":83},"Which related fields does the chapter point to in its outlook?",{"text":84,"@type":76},"It discusses related efforts in liquid state theory, electron density functional theory, and power functional theory, framed as functionally formulated approaches for classical nonequilibrium systems.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]