[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"branding":3,"analytics":7,"article-researchers-link-random-walks-to-neural-networks-for-pdes":10,"sections":35},{"siteName":4,"siteTagline":5,"publisherName":4,"contactEmail":6},"The Revision","Tech news, decoded.","editor@therevision.news",{"gaMeasurementId":8,"adsenseClientId":9},"G-ZW2MV82GYR","ca-pub-8533917693782264",{"article":11},{"id":12,"slug":13,"title":14,"dek":15,"body_md":16,"tags_json":17,"published_at":18,"created_at":19,"updated_at":20,"status":21,"review_note":22,"review_notes":23,"image_url":22,"persona_id":22,"persona_name":22,"section":24,"tags":25,"sources":30,"feedback":34,"feedback_at":22,"cost_usd":34,"total_tokens":34},5527,"researchers-link-random-walks-to-neural-networks-for-pdes","Researchers Link Random Walks to Neural Networks for PDEs","A new proof shows a classic random-walk algorithm can construct efficient neural networks that solve a fundamental equation in physics and engineering.","A math paper just gave neural networks a rigorous shortcut for solving one of physics' oldest equations.\n\nResearchers studied how to approximate solutions to the Poisson equation, a differential equation used across physics and engineering to model heat flow, electric fields, and fluid behavior, in bounded regions of arbitrary dimension. They show a modified walk on spheres Monte Carlo algorithm can compute these solutions efficiently, needing a number of samples that does not depend on the specific point being evaluated and scales only polynomially with dimension and desired accuracy. From that solver, they then constructively build a ReLU deep neural network that reproduces the same solution, with network size also growing only polynomially in dimension and error tolerance. The paper is a revised, fourth version of a preprint first posted in 2022.\n\nMost work on neural networks solving differential equations is empirical: train a model, check if the loss curve looks good, ship it. This paper instead proves, from first principles, that a network of a specific size and structure solves the Poisson equation with high probability, which matters for anyone building simulation software that needs guarantees rather than a plausible-looking result. It also shows decades-old probabilistic techniques still have work to do justifying today's deep learning shortcuts.\n\nThe catch: a constructive existence proof is not a training recipe, so no engineer is handing this paper to PyTorch and getting a working solver by lunchtime.","[\"neural networks\",\"monte carlo methods\",\"differential equations\",\"mathematics\"]","2026-08-18T04:00:00.000Z","2026-08-19T00:00:42.954Z","2026-08-19T00:00:54.836Z","published",null,[],"science",[26,27,28,29],"neural networks","monte carlo methods","differential equations","mathematics",[31],{"name":32,"url":33},"arXiv cs.AI","https:\u002F\u002Farxiv.org\u002Fabs\u002F2209.01432",0,{"sections":36},[37,42,46,51,56,61,66,70,75,79,84,89,94,99],{"name":38,"slug":39,"count":40,"latest_published_at":41},"AI","ai",3293,"2026-08-20T04:00:00.000Z",{"name":43,"slug":44,"count":45,"latest_published_at":41},"Security","security",435,{"name":47,"slug":48,"count":49,"latest_published_at":50},"Policy","policy",210,"2026-08-19T09:32:27.000Z",{"name":52,"slug":53,"count":54,"latest_published_at":55},"Deals","deals",179,"2026-06-29T20:02:07.000Z",{"name":57,"slug":58,"count":59,"latest_published_at":60},"Hardware","hardware",140,"2026-08-19T18:25:42.000Z",{"name":62,"slug":63,"count":64,"latest_published_at":65},"Consumer Tech","consumer-tech",95,"2026-08-18T16:05:00.000Z",{"name":67,"slug":24,"count":68,"latest_published_at":69},"Science",90,"2026-08-19T18:41:02.000Z",{"name":71,"slug":72,"count":73,"latest_published_at":74},"Software","software",73,"2026-08-18T07:51:50.000Z",{"name":76,"slug":77,"count":78,"latest_published_at":18},"Dev Tools","dev-tools",69,{"name":80,"slug":81,"count":82,"latest_published_at":83},"Startups","startups",47,"2026-08-19T19:13:46.000Z",{"name":85,"slug":86,"count":87,"latest_published_at":88},"Gaming","gaming",41,"2026-07-09T04:00:00.000Z",{"name":90,"slug":91,"count":92,"latest_published_at":93},"General","general",33,"2026-08-18T22:18:13.000Z",{"name":95,"slug":96,"count":97,"latest_published_at":98},"Reviews","reviews",20,"2026-06-24T12:00:01.000Z",{"name":100,"slug":101,"count":102,"latest_published_at":103},"How-To","how-to",6,"2026-06-16T09:00:00.000Z"]