[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-124851-en":3,"doc-seo-124851-105":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},124851,549758252649,"Ivy","https://ap-avatar.wpscdn.com/avatar/8000253669c5317157?_k=1778319167496531819",8,"Research & Report","Inverting the Kohn-Sham equations with physics-informed machine learning","Electronic structure calculations explain matter at the quantum level and complement experiments in chemistry and materials science. Density functional theory (DFT) reduces the interacting-electron problem to a noninteracting Kohn-Sham system while preserving the same electron density. The exact exchange-correlation (XC) energy and its derivative determine the XC potential needed for this consistency. This work performs XC potential inversions from target densities using physics-informed machine learning, specifically physics informed neural networks (PINNs) and Fourier neural operators (FNOs), validated on one-dimensional atomic and molecular models and assessed for scalability potential.","arXiv :2312 . 15301v1 [physics .comp-ph] 23 Dec 2023  \nInverting the Kohn-Sham equations with physics-informed machine learning  \nVincent Martinetto 1 ,∗ , Karan Shah2 ,3 ,∗ , Attila Cangi2 ,3 ,†, Aurora Pribram-Jones 1 ,‡  \n1 Department of Chemistry and Biochemistry, University of California Merced, 5200 North Lake Rd., Merced, California 95343, USA  \n2 Center for Advanced Systems Understanding, 02826 G¨orlitz, Germany  \n3 Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany  \n∗ these authors have contributed equally.  \n† [a.cangi@hzdr.de](a.cangi@hzdr.de)  \n‡ [apj@ucmerced.edu](apj@ucmerced.edu)  \n27 December 2023  \nInverting the Kohn-Sham equations with physics-informed machine learning 2  \n1. Abstract  \nElectronic structure theory calculations offer an understanding of matter at the quantum level, complementing experimental studies in materials science and chemistry. One of the most widely used methods, density functional theory (DFT), maps a set of real interacting electrons to a set of fictitious noninteracting electrons that share the same probability density. Ensuring that the density remains the same depends on the exchange-correlation (XC) energy and, by a derivative, the XC potential. Inversions provide a method to obtain exact XC potentials from target electronic densities, in hopes of gaining insights into accuracy-boosting approximations. Neural networks provide a new avenue to perform inversions by learning the mapping from density to potential. In this work, we learn this mapping using physics-informed machine learning (PIML) methods, namely physics informed neural networks (PINNs) and Fourier neural operators (FNOs) . We demonstrate the capabilities of these two methods on a dataset of one-dimensional atomic and molecular models. The capabilities of each approach are discussed in conjunction with this proof-of-concept presentation. The primary finding of our investigation is that the combination of both approaches has the greatest potential for inverting the Kohn-Sham equations at scale.  \n2. Introduction  \nModern, high-performance computational resources have enabled large-scale electronic structure simulations of molecules, materials, and other systems of interest across biology, chemistry, physics, and beyond. Kohn-Sham (KS) Density Functional Theory (DFT) [1, 2] is the most widely used method due to its accuracy and computational efficiency. KS-DFT has helped solve major scientific and technological problems such as simulating chemical reactions, computing material properties, finding new catalysts, discovering drugs, and modeling microscopic environmental processes.  \nThe electronic structure problem is typically tackled by solving the non-relativistic Schr¨odinger equation for the molecular Hamiltonian within the Born-Oppenheimer approximation. KS-DFT simplifies this process by transforming the many-body problem into an effective single-particle problem, utilizing the  \none-to-one correspondence of the external potential and the electronic density [1] . The crux of KSDFT is producing an electronic density that matches the one obtained from solving the interacting manybody problem [2] . This is accomplished through an effective single-particle potential known as the KS potential, which is comprised of the external potential, the Hartree potential that accounts for electrostatic repulsion, and an exchange-correlation (XC) potential compensating for all interactions beyond the Hartree potential.  \nWhile an exact expression for the XC potential (vXC = δEXC /δn) is not known, useful approximations exist. These approximations are categorized based on increasing complexity and accuracy [3] . They include, for example, the local density approximation (LDA)[2, 4 , 5], which is derived from the interacting uniform electron gas [6]; generalized gradient approximations (GGAs)[7, 8 , 4 , 9], relying on density gradients; meta-GGAs [10, 11], which consider the kinetic energy density; and hybrid functionals, [1","cbCaiu4gLwiARRdk","https://ap.wps.com/l/cbCaiu4gLwiARRdk","pdf",933539,1,16,"English","en",105,"# Abstract\n# Introduction\n## Kohn-Sham DFT and the XC potential\n## Approximate XC functionals\n## Inverse problems and XC potential inversion\n## Machine learning approaches","[{\"question\":\"What problem does this document address in density functional theory?\",\"answer\":\"It addresses how to invert the Kohn-Sham framework to obtain the exchange-correlation (XC) potential from a target electronic density.\"},{\"question\":\"Which physics-informed machine learning methods are used for the inversion?\",\"answer\":\"The document uses physics informed neural networks (PINNs) and Fourier neural operators (FNOs) to learn the mapping from density to potential.\"},{\"question\":\"What do the authors conclude about combining PINNs and FNOs?\",\"answer\":\"The study finds that using both approaches together has the greatest potential for inverting the Kohn-Sham equations at scale.\"}]","Inverting the Kohn-Sham equations with physics-informed machine learning | 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problem does this document address in density functional theory?","Question",{"text":75,"@type":76},"It addresses how to invert the Kohn-Sham framework to obtain the exchange-correlation (XC) potential from a target electronic density.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which physics-informed machine learning methods are used for the inversion?",{"text":80,"@type":76},"The document uses physics informed neural networks (PINNs) and Fourier neural operators (FNOs) to learn the mapping from density to potential.",{"name":82,"@type":73,"acceptedAnswer":83},"What do the authors conclude about combining PINNs and FNOs?",{"text":84,"@type":76},"The study finds that using both approaches together has the greatest potential for inverting the Kohn-Sham equations at 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