[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-118181-en":3,"doc-seo-118181-105":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},118181,8796095461564,"Liam","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Designing Poisson Integrators Through Machine Learning","This paper develops a general framework for constructing Poisson integrators, numerical schemes that preserve the underlying Poisson geometry. The work assumes an integrable Poisson manifold and uses the relationship between Poisson diffeomorphisms and Lagrangian bisections to reformulate Poisson integrator design as a Hamilton–Jacobi PDE. The key contribution treats this PDE as an optimization problem, enabling practical approximations via machine-learning techniques, in line with physics-informed and data-driven modeling trends. The method is validated on a rigid body example.","arXiv :2403 .20139v1 [math-ph] 29 Mar 2024  \nDesigning Poisson Integrators Through Machine Learning  \nMiguel Vaquero ∗ David Mart´ın de Diego ∗∗ Jorge Cort´es ∗∗∗  \n∗ IE University, Segovia, 40001 Spain  \n([e-mail:mvaquero@faculty.ie.edu](e-mail:mvaquero@faculty.ie.edu)).  \n∗∗ Instituto de Ciencias Matem´aticas ICMAT, Madrid, 28049 (e-mail:  \n[david.martin@icmat.es](david.martin@icmat.es))  \n∗∗∗ University of California, San Diego, 9500 Gilman Dr, La Jolla,  \nCalifornia, 92093-0411, (e-mail: [cortes@ucsd.edu](cortes@ucsd.edu))  \nAbstract: This paper presents a general method to construct Poisson integrators, i.e. , integrators that preserve the underlying Poisson geometry. We assume the Poisson manifold isintegrable, meaning there is a known local symplectic groupoid for which the Poisson manifold serves as the set of units. Our constructions build upon the correspondence between Poisson diffeomorphisms and Lagrangian bisections, which allows us to reformulate the design of Poisson integrators as solutions to a certain PDE (Hamilton-Jacobi) . The main novelty of this work is to understand the Hamilton-Jacobi PDE as an optimization problem, whose solution can be easily approximated using machine learning related techniques. This research direction aligns with the current trend in the PDE and machine learning communities, as initiated by PhysicsInformed Neural Networks, advocating for designs that combine both physical modeling (the Hamilton-Jacobi PDE) and data.  \nKeywords: Poisson geometry, symplectic geometry, geometric integrators, optimization, machine learning.  \n1. INTRODUCTION  \nDue to their persistent presence in science and engineering, Hamiltonian systems have been intensively studied for centuries. Special attention has been given to all the structures able to describe Hamiltonian systems, namely symplectic and Poisson geometry. It is broadly recognized that geometry plays a pivotal role in the dynamical behavior of the aforementioned systems. Nonetheless, since Hamiltonian systems can describe a wide array of systems in nature, they usually show a high level of complexity that hinders their complete understanding. This fact has led several communities to the design of algorithms that seek to produce accurate simulations of Hamiltonian systems as an enabling tool for the analysis of their dynamical behavior and properties.  \nTo tackle this challenge, we follow here the geometric approach. The main observation is that numerical schemes sharing the same geometric properties as the original system usually enjoy better accuracy and more faithful qualitative description of the system compared to nongeometric algorithms. This philosophy has been broadly exploited when the underlying geometry is symplectic, see for instance Sanz-Serna and Calvo (1994); Hairer et al.(2010) .  \n⋆ The authors acknowledge financial support from the Spanish Ministry of Science and Innovation under grants PID2022-137909NBC21, RED2022-134301-TD, the Severo Ochoa Programme for Centres of Excellence in R&D (CEX2019-000904-S) and BBVA Foundation via the project “Mathematical optimization for a more efficient, safer and decarbonized maritime transport”.  \nNonetheless, when the Hamiltonian system is described using the more general Poisson setting, the situation is more subtle. See Cosserat (2023) and the references therein. This is mainly due to the fact that Poisson geometry is, in away, singular and irregular when compared to symplectic geometry. Moreover, to the best of the authors’ knowledge, there are no general methods for obtaining Poisson integrators 1 , although some recent attempts include those in Cosserat (2023) . In cases where the Poisson structure is linear and integrable, the references Ge (1991); Zhong and Marsden (1988); McLachlan et al. (2014); Ferraro et al. (2017) provide methods to generate them. These methods rely on the exploitation of the properties of the symplectic groupoid that integrates the Poisson manifold, or other symp","cbCaiihhLPY8FIEM","https://ap.wps.com/l/cbCaiihhLPY8FIEM","pdf",967384,1,5,"English","en",105,"# Introduction\n# A Quick Review of Geometry\n## Basic Notions on Symplectic and Poisson Geometry","[{\"question\":\"What does the paper mean by a Poisson integrator?\",\"answer\":\"A Poisson integrator is a numerical scheme approximating a Hamiltonian vector field on a Poisson manifold while preserving the underlying Poisson geometry.\"},{\"question\":\"How is Poisson integrator construction reduced to a PDE problem?\",\"answer\":\"Using the correspondence between Poisson diffeomorphisms and Lagrangian bisections, the paper reformulates the design task as solutions to a Hamilton–Jacobi PDE.\"},{\"question\":\"How does machine learning enter the method?\",\"answer\":\"The Hamilton–Jacobi PDE is cast as an optimization problem, and the resulting solution is approximated with machine learning–inspired techniques.\"}]","Designing Poisson Integrators Through Machine Learning | PDF",1785682053,13,{"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},"designing-poisson-integrators-through-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/designing-poisson-integrators-through-machine-learning/118181/",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 does the paper mean by a Poisson integrator?","Question",{"text":75,"@type":76},"A Poisson integrator is a numerical scheme approximating a Hamiltonian vector field on a Poisson manifold while preserving the underlying Poisson geometry.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is Poisson integrator construction reduced to a PDE problem?",{"text":80,"@type":76},"Using the correspondence between Poisson diffeomorphisms and Lagrangian bisections, the paper reformulates the design task as solutions to a Hamilton–Jacobi PDE.",{"name":82,"@type":73,"acceptedAnswer":83},"How does machine learning enter the method?",{"text":84,"@type":76},"The Hamilton–Jacobi PDE is cast as an optimization problem, and the resulting solution is approximated with machine learning–inspired techniques.","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},[93,97,101,105,109,114,119,122,127,130,134],{"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":21,"doc_module":4,"doc_module_name":46,"category_name":106,"show_sort_weight":107,"slug":108},"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":21,"slug":137},19,"General","general"]