{"title":"椭圆随机 PDE 随机 Galerkin 近似的深度学习方法","authors":"Fabio Musco, Andrea Barth","doi":"arxiv-2409.08063","DOIUrl":null,"url":null,"abstract":"This work considers stochastic Galerkin approximations of linear elliptic\npartial differential equations with stochastic forcing terms and stochastic\ndiffusion coefficients, that cannot be bounded uniformly away from zero and\ninfinity. A traditional numerical method for solving the resulting\nhigh-dimensional coupled system of partial differential equations (PDEs) is\nreplaced by deep learning techniques. In order to achieve this,\nphysics-informed neural networks (PINNs), which typically operate on the strong\nresidual of the PDE and can therefore be applied in a wide range of settings,\nare considered. As a second approach, the Deep Ritz method, which is a neural\nnetwork that minimizes the Ritz energy functional to find the weak solution, is\nemployed. While the second approach only works in special cases, it overcomes\nthe necessity of testing in variational problems while maintaining mathematical\nrigor and ensuring the existence of a unique solution. Furthermore, the\nresidual is of a lower differentiation order, reducing the training cost\nconsiderably. The efficiency of the method is demonstrated on several model\nproblems.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"30 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Deep learning methods for stochastic Galerkin approximations of elliptic random PDEs\",\"authors\":\"Fabio Musco, Andrea Barth\",\"doi\":\"arxiv-2409.08063\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This work considers stochastic Galerkin approximations of linear elliptic\\npartial differential equations with stochastic forcing terms and stochastic\\ndiffusion coefficients, that cannot be bounded uniformly away from zero and\\ninfinity. A traditional numerical method for solving the resulting\\nhigh-dimensional coupled system of partial differential equations (PDEs) is\\nreplaced by deep learning techniques. In order to achieve this,\\nphysics-informed neural networks (PINNs), which typically operate on the strong\\nresidual of the PDE and can therefore be applied in a wide range of settings,\\nare considered. As a second approach, the Deep Ritz method, which is a neural\\nnetwork that minimizes the Ritz energy functional to find the weak solution, is\\nemployed. While the second approach only works in special cases, it overcomes\\nthe necessity of testing in variational problems while maintaining mathematical\\nrigor and ensuring the existence of a unique solution. Furthermore, the\\nresidual is of a lower differentiation order, reducing the training cost\\nconsiderably. The efficiency of the method is demonstrated on several model\\nproblems.\",\"PeriodicalId\":501162,\"journal\":{\"name\":\"arXiv - MATH - Numerical Analysis\",\"volume\":\"30 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Numerical Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.08063\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.08063","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Deep learning methods for stochastic Galerkin approximations of elliptic random PDEs
This work considers stochastic Galerkin approximations of linear elliptic
partial differential equations with stochastic forcing terms and stochastic
diffusion coefficients, that cannot be bounded uniformly away from zero and
infinity. A traditional numerical method for solving the resulting
high-dimensional coupled system of partial differential equations (PDEs) is
replaced by deep learning techniques. In order to achieve this,
physics-informed neural networks (PINNs), which typically operate on the strong
residual of the PDE and can therefore be applied in a wide range of settings,
are considered. As a second approach, the Deep Ritz method, which is a neural
network that minimizes the Ritz energy functional to find the weak solution, is
employed. While the second approach only works in special cases, it overcomes
the necessity of testing in variational problems while maintaining mathematical
rigor and ensuring the existence of a unique solution. Furthermore, the
residual is of a lower differentiation order, reducing the training cost
considerably. The efficiency of the method is demonstrated on several model
problems.