[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117979-en":3,"doc-seo-117979-105":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},117979,4810365810221,"Aurora","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Machine learning of a density functional for anisotropic patchy particles","Anisotropic patchy particles provide a statistical model for associating fluids, and the work develops a density-functional-theory framework for the Kern–Frenkel model in flat wall geometries. The functional separates an orientationally averaged reference part from an orientational mean-field contribution. A kernel suitable for machine learning is built by expanding in orientational invariants while enforcing single-particle symmetries. Mean-field kernels are trained from hard-wall simulation data and assessed against the random-phase approximation, which fails near walls. Strengths and limitations are analyzed with perspectives toward fully learned density functionals.","arXiv :2311 .04358v1 [ cond-mat .stat-mech] 7 Nov 2023  \nMachine learning of a density functional for anisotropic patchy particles  \nAlessandro Simon,†,‡ Jens Weimar,† Georg Martius,‡ and Martin Oettel†  \n† Institute for Applied Physics, University of T¨ubingen, Auf der Morgenstelle 10, 72076  \nT¨ubingen, Germany  \n‡ Max Planck Institute for Intelligent Systems, Max-Planck-Ring 4, 72076 T¨ubingen,  \nGermany  \nAbstract  \nAnisotropic patchy particles have become an archetypical statistical model system for associating fluids. Here we formulate an approach to the Kern–Frenkel model via classical density functional theory to describe the positionally and orientationally resolved equilibrium density distributions in flat wall geometries. The density functional is split into a reference part for the orientationally averaged density and an orientational part in mean-field approximation. To bring the orientational part into a kernel form suitable for machine learning techniques, an expansion into orientational invariants and the proper incorporation of single-particle symmetries is formulated. The mean-field kernel is constructed via machine learning on the basis of hard wall simulation data. Results are compared to the well-known random-phase approximation which strongly understimates the orientational correlations close to the wall. Successes and shortcomings of the mean-field treatment of the orientational part are highlighted and perspectives are given for attaining a full density functional via machine learning.  \n1 Introduction  \nA useful model system for describing various phenomena in soft matter physics is given by patchy particles where particles with anisotropic repulsive core additionally interact via  \na certain number of attractive bonding sites distributed over their surface. These patchy particle systems show novel phenomena such as the existence of empty liquids 1,2 or of stable equilibrium gels. 3,4 The Kern-Frenkel (KF) model is an example for such a patchy particle model with a mathematical form for the pair potential that is easy to simulate. 5 In fact, many theoretical insights, especially on phase behavior, have been obtained by simulations. 6,7 An alternative approach to simulations for obtaining equilibrium properties of soft matter model systems is (classical) density functional theory (DFT) . It has been frequently and successfully employed for isotropic fluids, 8,9 in particular for hard particles in the form of the very precise fundamental measure theory (FMT), for a review see Ref. 10 For the specific case of patchy particles, FMT-based functionals have been derived 11,12 which are functionals of the orientationally averaged density profile. These functionals describe various properties of patchy particles reasonably well (e.g. phase diagrams and orientationally averaged pair correlations), but there are limitations which are clearly due to the neglect of orientational correlations (e.g. density profiles around hard tracer particles) . Thus, it is desirable to find free energy functionals of the full orientation-dependent density profile. However, there are no analytical methods known to construct such functionals for anisotropic particles that are similar in accuracy as FMT for hard particles. Hence, it seems to be promising  \nto turn to numerical and data-driven methods such as machine learning (ML) for this task.  \nFinding classical density functionals with ML methods is a rather recent development. Initial work has focused on model fluids in one dimension (1D), 13–16 whereas work on more realistic systems in 3D is scarce, for a study of the 3D Lennard-Jones (LJ) fluid at one supercritical temperature, see Ref. 17 and for a study of hard spheres in planar geometry see the very recent Ref. 18 Among these works, one can distinguish between approaches to learning integral kernels for the generally unknown, but sensibly parametrized part in the functional describing attractive interactions, 13,17 approache","cbCailMQL9c2TLW9","https://ap.wps.com/l/cbCailMQL9c2TLW9","pdf",2198655,1,26,"English","en",105,"# Abstract\n## Introduction\n## Density functional theory background\n## Machine learning approach and kernel construction\n## Comparison with random-phase approximation\n## Results, limitations, and outlook","[{\"question\":\"What model and physical setup does the paper address?\",\"answer\":\"The paper targets the Kern–Frenkel patchy-particle model and formulates a density functional for equilibrium density distributions resolved by position and orientation near flat walls.\"},{\"question\":\"How is the density functional structured in the proposed method?\",\"answer\":\"It splits into a reference part for the orientationally averaged density and an orientational part treated in a mean-field approximation.\"},{\"question\":\"Why is a kernel representation important for the machine learning part?\",\"answer\":\"The orientational contribution is reformulated into a kernel form by expanding in orientational invariants and incorporating single-particle symmetries, enabling machine-learning techniques to construct the mean-field kernel from simulation data.\"}]","Machine learning of a density functional for anisotropic patchy particles | 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model and physical setup does the paper address?","Question",{"text":75,"@type":76},"The paper targets the Kern–Frenkel patchy-particle model and formulates a density functional for equilibrium density distributions resolved by position and orientation near flat walls.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the density functional structured in the proposed method?",{"text":80,"@type":76},"It splits into a reference part for the orientationally averaged density and an orientational part treated in a mean-field approximation.",{"name":82,"@type":73,"acceptedAnswer":83},"Why is a kernel representation important for the machine learning part?",{"text":84,"@type":76},"The orientational contribution is reformulated into a kernel form by expanding in orientational invariants and incorporating single-particle symmetries, enabling machine-learning techniques to construct the mean-field kernel from simulation 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