[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-120051-en":3,"doc-seo-120051-105":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},120051,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Accelerating wavepacket propagation with machine learning","Discusses the use of Fourier neural operators (FNO), a machine learning technique, as an efficient alternative for solving the time-dependent Schrödinger equation (TDSE) via approximate solutions of partial differential equations. Demonstrates accurate and faithful modeling of wavepacket propagation for an anharmonic potential and a tunneling system using the density representation. Shows FNOs can replace traditional TDSE solvers when quantum dynamics results are required repeatedly, such as for parameter optimization and control tasks, enabling faster integration with Markov-chain Monte Carlo for inverse problems in laser control.","DOI: 10.1002/jcc.27443  \nRES EARCH A RTICLE  \nAccelerating wavepacket propagation with machine learning  \nKanishka Singh 1,2  | Ka Hei Lee 1,3 | Daniel Peláez 4  | Annika Bande 1,5,6   \n1Theory of Electron Dynamics and Spectroscopy, Helmholtz-Zentrum Berlin für Materialien und Energie GmbH,  \nBerlin, Germany  \n2Institute of Chemistry and Biochemistry, Freie Universität Berlin, Berlin, Germany 3Fachbereich Physik, Freie Universität Berlin, Berlin, Germany  \n4CNRS, Institut des Sciences Moléculaires  \nd'Orsay, Université Paris-Saclay, Orsay, France 5Institute of Inorganic Chemistry, Leibniz University Hannover, Hannover, Germany 6Cluster of Excellence PhoenixD, Leibniz University Hannover, Hannover, Germany  \nCorrespondence  \nAnnika Bande, Theory of Electron Dynamicsand Spectroscopy, Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, HahnMeitner-Platz 1, 10409 Berlin, Germany. Email: annika. bande@helmholtz-berlin.de  \nFunding information  \nHelmholtz Einstein International Berlin Research School in Data Science (HEIBRiDS)  \nAbstract  \nIn this work, we discuss the use of a recently introduced machine learning (ML) technique known as Fourier neural operators (FNO) as an efficient alternative to the traditional solution of the time-dependent Schrödinger equation (TDSE) . FNOs are ML models which are employed in the approximated solution of partial differential equations. For a wavepacket propagating in an anharmonic potential and for a tunneling system, we show that the FNO approach can accurately and faithfully model wavepacket propagation via the density. Additionally, we demonstrate that FNOs can be a suitable replacement for traditional TDSE solvers in cases where the results of the quantum dynamical simulation are required repeatedly such as in the case of parameter optimization problems (e.g., control) . The speed-up from the FNO method allows for its combination with the Markov-chain Monte Carlo approach in applications that involve solving inverse problems such as optimal and coherent laser control of the outcome of dynamical processes.  \nKEYWOR DS  \nFourier neural operators, machine learning, quantum dynamics  \n1 | INTRODUCTION  \nAn accurate theoretical description of molecular phenomena requires the solution of the Schrödinger equation (SE) in any of its variants, time-independent (TISE)1 or time-dependent (TDSE) .2 The latter is more general in nature owing to the fact that not only time-dependent processes can be visualized but also, at the same time, timeindependent quantities such as eigenstates or spectra can be computed.3,4 The efficient solution of the SE (in terms of CPU time and storage) has constituted and still constitutes one of the greatest bottlenecks to its widespread use. There are two main limiting aspects in this regard: (i) the need for storing large amounts of information, as epitomized by the exponential growth in the number of data points with dimensionality, and (ii) the actual integration of the SE (i.e., application of a propagator) or, equivalently, the numerical solution of some kind of equations of motion (EOM) . The former issue has been successfully overcome through the use of efficient (large) tensor decomposition  \nschemes (see for instance References 5–12) . Concerning the latter aspect, clever and sophisticated integration schemes,3,13,14 many of them relying on different data structures (e.g., tensor networks15) have been developed over the past fifty years.  \nML approaches are modern numerical computer science techniques that are in terms of efficiency a true alternative to conventional algorithms in many domains. They have recently shown success in replacing the quantum-mechanical evaluation of the TISE and obtaining molecular properties at a faster speed without losing accuracy.16,17 Deep-learning approaches involving complex neural networks can be used for the accurate generation of potential energy surfaces18–23 and to predict chemical reaction outcomes,24,25 or for property","cbCaiek6kRnN3hc7","https://ap.wps.com/l/cbCaiek6kRnN3hc7","pdf",2644616,1,14,"English","en",105,"# Abstract\n## Introduction\n## Background: Schrödinger equation and numerical bottlenecks\n## Machine learning for quantum dynamics and PDEs\n## Data-driven PDE solving approaches","[{\"question\":\"What machine learning method is proposed to accelerate wavepacket propagation?\",\"answer\":\"The work uses Fourier neural operators (FNO) to provide an efficient approximate solution for the time-dependent Schrödinger equation.\"},{\"question\":\"For which physical scenarios is the FNO approach demonstrated?\",\"answer\":\"It is demonstrated for a wavepacket propagating in an anharmonic potential and for a tunneling system.\"},{\"question\":\"How can FNOs help in repeated quantum-dynamics evaluations?\",\"answer\":\"The speed-up allows FNOs to replace traditional TDSE solvers when results are needed repeatedly, such as in parameter optimization and control problems.\"}]","Accelerating wavepacket propagation with machine learning | PDF",1785727896,35,{"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},"accelerating-wavepacket-propagation-with-machine-learning","",{"@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/accelerating-wavepacket-propagation-with-machine-learning/120051/",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","2026-08-03",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 machine learning method is proposed to accelerate wavepacket propagation?","Question",{"text":75,"@type":76},"The work uses Fourier neural operators (FNO) to provide an efficient approximate solution for the time-dependent Schrödinger equation.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"For which physical scenarios is the FNO approach demonstrated?",{"text":80,"@type":76},"It is demonstrated for a wavepacket propagating in an anharmonic potential and for a tunneling system.",{"name":82,"@type":73,"acceptedAnswer":83},"How can FNOs help in repeated quantum-dynamics evaluations?",{"text":84,"@type":76},"The speed-up allows FNOs to replace traditional TDSE solvers when results are needed repeatedly, such as in parameter optimization and control problems.","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"]