[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-116977-en":3,"doc-seo-116977-105":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},116977,8796095360427,"Lucas Martin","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","Solution of Schrödinger Equation for Quantum Systems via Physics-Informed Neural Networks","Machine learning achievements across technical domains have driven scientific interest in scientific machine learning. By combining expertise from machine learning and computational science, physics-aware models can incorporate physical constraints during training. Physics-Informed Neural Networks embed prior knowledge such as physical laws, symmetries, and conservation principles, learning to simulate complex phenomena while respecting underlying physics. This thesis uses Physics-Informed Neural Networks to efficiently simulate one-electron quantum systems through direct eigenvalue-solution of the Schrödinger equation, avoiding discretization and enabling evaluation of ground-state energy, wavefunctions, and electron density, benchmarked against quantum chemistry literature.","UNIVERSITÀ DEGLI STUDI DI PADOVA  \nDipartimento di Fisica e Astronomia “Galileo Galilei”  \nMaster Degree in Physics of Data  \nThesis  \nSolution of Schrödinger Equation for Quantum Systems via Physics-Informed Neural Networks  \nInternal supervisor, Università di Padova  \nProf. Marco Baiesi  \nExternal supervisors, Fraunhofer IISB  \nPhilipp Brendel  \nDr. Simon Mundinar  \nCandidate  \nPaolo Zinesi  \nAcademic Year 2022/2023  \nAbstract  \nThe numerous successes achieved by machine learning techniques in many technical areas have sparked interest in the scientific community for their application in science. By merging the knowledge of machine learning experts and computational scientists, the field of scientific machine learning has shown its ability to greatly improve the performance of existing computational methods. One possible approach to developing physics-aware machine learning is the inclusion of physical constraints in the training of a machine learning model. Physics-Informed Neural Networks are an example of such an approach, as they can incorporate prior physical knowledge into their architecture, enabling them to learn and simulate complex phenomena while respecting the underlying physics principles. Possible constraints are physical laws, symmetries, and conservation laws. Compared to other machine learning models, PhysicsInformed Neural Networks do not require substantial input data, with the exception of initial and boundary conditions to correctly formalize the problem.  \nIn this Thesis, we exploit the advantages of Physics-Informed Neural Networks to efficiently simulate one-electron quantum systems. The simulations rely on the direct solution of the eigenvalue equation represented by the Schrödinger equation. Traditional methods for solving the Schrödinger equation often rely on approximations and can become computationally expensive for nontrivial systems. The mesh-free Physics-Informed Neural Networks approach avoids the need for discretization, as the residuals computed with respect to the physical constraints are minimized during training for a given set of points within the domain.  \nThe solution of the Schrödinger equation allows one to calculate important physical quantities of the physical system under study, such as the ground state energy, the electronic wavefunction, and the associated electron density. These quantities are compared with the estimations present in the quantum chemistry literature to assess the performance of the Physics-Informed Machine Learning approach.  \nContents  \nIntroduction vii  \n1 Ab initio simulations of quantum systems 1  \n1.1 Schrödinger equation ........................ 2  \n1.2 Hartee-Fock and post-Hartree-Fock methods ........... 3  \n1.3 Variational Monte Carlo ...................... 4  \n1.3.1 Variational Monte Carlo and machine learning ...... 6  \n2 Physics-Informed Machine Learning 9  \n2.1 Biases ................................ 10  \n2.2 Physics-Informed Neural Networks ................ 13  \n2.2.1 Mathematical formulation ................. 13  \n2.2.2 Losses ............................ 14  \n2.2.3 Network ........................... 15  \n2.2.4 Training ........................... 16  \n2.3 Motivations and advantages .................... 17  \n2.4 Limitations ............................. 19  \n2.5 Software ............................... 20  \n3 Quantum systems simulations with physics-informed neural networks 21  \n3.1 Literature review of existing physics-informed approaches .... 21  \n3.2 Proposed approach and methodology ............... 24  \n3.2.1 Architecture ......................... 25  \n3.2.2 Training ........................... 33  \n4 Results 35  \n4.1 Metrics and datasets ........................ 35  \n4.2 One-dimensional H atom ...................... 37  \n4.2.1 No eigenvalue estimation .................. 37  \n4.2.2 With eigenvalue estimation ................. 38  \n4.3 Three-dimensional H atom ..................... 43  \n4.3.1 No eigenvalue estimation .................. 44  ","cbCailnMiyXioJkr","https://ap.wps.com/l/cbCailnMiyXioJkr","pdf",2757313,1,84,"English","en",105,"# Introduction\n# Ab initio simulations of quantum systems\n## Schrödinger equation\n## Hartee-Fock and post-Hartree-Fock methods\n## Variational Monte Carlo\n### Variational Monte Carlo and machine learning\n# Physics-Informed Machine Learning\n## Biases\n## Physics-Informed Neural Networks\n### Mathematical formulation\n### Losses\n### Network\n### Training\n## Motivations and advantages\n## Limitations\n## Software\n# Quantum systems simulations with physics-informed neural networks\n## Literature review of existing physics-informed approaches\n## Proposed approach and methodology\n### Architecture\n### Training\n# Results\n## Metrics and datasets\n## One-dimensional H atom\n### No eigenvalue estimation\n### With eigenvalue estimation\n## Three-dimensional H atom\n### No eigenvalue estimation\n### With eigenvalue estimation\n## Importance of training distributions\n## Three-dimensional H molecule\n### Eigenvalue estimation\n# Conclusions and Outlooks\n# Bibliography","[{\"question\":\"What is the key idea behind Physics-Informed Neural Networks in this thesis?\",\"answer\":\"Physics-Informed Neural Networks incorporate physical constraints directly into training so the model learns solutions that respect underlying physics principles such as laws, symmetries, and conservation.\"},{\"question\":\"How does the proposed method solve the Schrödinger equation?\",\"answer\":\"The approach targets a direct solution of the eigenvalue equation represented by the Schrödinger equation, minimizing residuals with respect to physical constraints during training on points within the domain.\"},{\"question\":\"Which physical quantities are used to evaluate the method’s performance?\",\"answer\":\"The thesis computes ground state energy, the electronic wavefunction, and electron density, and compares these results with estimations reported in quantum chemistry literature.\"}]","Solution of Schrödinger Equation for Quantum Systems via Physics-Informed Neural Networks | PDF",1785672933,212,{"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},"solution-of-schrodinger-equation-for-quantum-systems-via-physics-informed-neural-networks","",{"@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/solution-of-schrodinger-equation-for-quantum-systems-via-physics-informed-neural-networks/116977/",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 key idea behind Physics-Informed Neural Networks in this thesis?","Question",{"text":75,"@type":76},"Physics-Informed Neural Networks incorporate physical constraints directly into training so the model learns solutions that respect underlying physics principles such as laws, symmetries, and conservation.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed method solve the Schrödinger equation?",{"text":80,"@type":76},"The approach targets a direct solution of the eigenvalue equation represented by the Schrödinger equation, minimizing residuals with respect to physical constraints during training on points within the domain.",{"name":82,"@type":73,"acceptedAnswer":83},"Which physical quantities are used to evaluate the method’s performance?",{"text":84,"@type":76},"The thesis computes ground state energy, the electronic wavefunction, and electron density, and compares these results with estimations reported in quantum chemistry literature.","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"]