[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-124585-en":3,"doc-seo-124585-105":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},124585,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1784791008015729253",8,"Research & Report","M-ENIAC - A machine learning recreation of the first successful numerical weather forecasts","M-ENIAC recreates the first successful numerical weather forecasts by replacing traditional numerical discretizations with machine learning solvers. The study asks how ENIAC-era forecasts would have changed under learning-based methods and implements physics-informed neural networks to solve the barotropic vorticity equation on the sphere. Results show physics-informed neural networks yield an easier and more accurate framework for solving meteorological equations compared with the original ENIAC solver.","arXiv:2304.09070v1 [[physics. ao-ph](physics. ao-ph)] 18 Apr 2023  \nM-ENIAC: A machine learning recreation of the 􀀌rst successful numerical weather forecasts  \nR􀁿udiger Brechty and Alex Bihloz  \ny Department of Mathematics, University of Hamburg, Hamburg, Germany  \nz Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's (NL) A1C 5S7, Canada  \nE-mail: [rbrecht@uni-hamburg.de](rbrecht@uni-hamburg.de), [abihlo@mun.ca](abihlo@mun.ca)  \nIn 1950 the 􀀌rst successful numerical weather forecast was obtained by solving the barotropic vorticity equation using the Electronic Numerical Integrator and Computer (ENIAC), which marked the beginning of the age of numerical weather prediction. Here, we ask the question of how these numerical forecasts would have turned out, if machine learning based solvers had been used instead of standard numerical discretizations. Speci􀀌cally, we recreate these numerical forecasts using physics-informed neural networks. We show that physics-informed neural networks provide an easier and more accurate methodology for solving meteorological equations on the sphere, as compared to the ENIAC solver.  \n1 Introduction  \nNumerical weather prediction is the backbone of modern meteorology [2] . Without numerical weather forecasts it would be impossible to predict the weather in advance with as much accuracy as is possible today. While numerical weather prediction is being taken for granted nowadays, it was 􀀌rst shown feasible by the seminal work of Charney, Fj􀀜rtoft and von Neumann [10], although the dream of numerical weather prediction is much older, dating back to Richardson [28] more than 100 years ago. The 􀀌rst successful numerical weather forecast was carried out on the Electronic Numerical Integrator and Computer (ENIAC), using a highly simpli􀀌ed version of the governing equations of the atmosphere. This simpli􀀌ed equation, the so-called barotropic vorticity equation, was solved using a straightforward 􀀌nite-di􀀋erence method, and the resulting accuracy of the forecasts was indeed quite underwhelming [21] . Still, in comparison to the numerical forecast attempted by Richardson thirty years earlier, the forecasts by Charney, Fj􀀜rtoft and von Neumann were a resounding success, that eventually lead to the quiet revolution of numerical weather prediction [2] .  \nThe ENIAC forecasts were recreated in the paper [21], and later carried out on a cellphone [22] to showcase the dramatic improvements of processing powers, and the potential that this holds for new avenues for numerical weather forecasting. With the recent breakthrough in training deep neural networks [19], and the enormous interest in both using neural networks for solving di􀀋erential equations [27] and meteorology in general, we argue it is natural to revisit the ENIAC integrations again, this time using machine learning.  \nMore speci􀀌cally, in this paper we ask the following speculative question: How would the 􀀌rst successful numerical weather forecasts have turned out, had machine-learning based numerical solvers be used instead of classical numerical integration? We refer to this task as M-ENIAC, a machine learning recreation of the ENIAC forecasts.  \nWhile the recreation of the ENIAC forecasts using neural networks may seem contrived, we do believe this is a timely problem to consider for several reasons. As most other 􀀌elds of the mathematical sciences, meteorology has also seen a substantial increase of interest in machine learning and deep learning. While neural networks have been considered in this 􀀌eld as early as the 1990s, see [12] for a historical review, the most recent surge in breakthroughs using deep neural networks has sparked a renewed interest in these networks in meteorology in the past few years, with several groups working on integrating deep neural networks into operational weather forecasting. Indeed, recently it was shown that a Transformer based weather forecasting model can outperform the state","cbCairwBYO0OdlfX","https://ap.wps.com/l/cbCairwBYO0OdlfX","pdf",679541,1,10,"English","en",105,"# Introduction\n## Numerical weather prediction and the ENIAC benchmark\n## Motivation for machine learning based solvers\n## Organization of the paper","[{\"question\":\"What question does M-ENIAC investigate?\",\"answer\":\"It asks how the first successful numerical weather forecasts would have turned out if machine learning based numerical solvers were used instead of classical numerical integration.\"},{\"question\":\"Which model approach does the paper use for the meteorological equations?\",\"answer\":\"The paper recreates the forecasts using physics-informed neural networks to solve the barotropic vorticity equation on the sphere.\"},{\"question\":\"What conclusion does the paper draw compared with the ENIAC solver?\",\"answer\":\"Physics-informed neural networks provide an easier and more accurate methodology for solving meteorological equations on the sphere than the ENIAC solver.\"}]","M-ENIAC - A machine learning recreation of the first successful numerical weather forecasts | PDF",1785893171,25,{"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},"m-eniac-a-machine-learning-recreation-of-the-first-successful-numerical-weather-forecasts","",{"@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/m-eniac-a-machine-learning-recreation-of-the-first-successful-numerical-weather-forecasts/124585/",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","2026-08-05",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 question does M-ENIAC investigate?","Question",{"text":75,"@type":76},"It asks how the first successful numerical weather forecasts would have turned out if machine learning based numerical solvers were used instead of classical numerical integration.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which model approach does the paper use for the meteorological equations?",{"text":80,"@type":76},"The paper recreates the forecasts using physics-informed neural networks to solve the barotropic vorticity equation on the sphere.",{"name":82,"@type":73,"acceptedAnswer":83},"What conclusion does the paper draw compared with the ENIAC solver?",{"text":84,"@type":76},"Physics-informed neural networks provide an easier and more accurate methodology for solving meteorological equations on the sphere than the ENIAC solver.","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},[93,97,101,105,110,115,120,123,128,131,134],{"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":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":21,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":21,"slug":133},"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]