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Sci. Comput."],"published-print":{"date-parts":[[2023,6,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>This paper proposes Friedrichs learning as a novel deep learning methodology that can learn the weak solutions of PDEs via a minimax formulation, which transforms the PDE problem into a minimax optimization problem to identify weak solutions. The name \u201cFriedrichs learning\u201d is to highlight the close relation between our learning strategy and Friedrichs theory on symmetric systems of PDEs. The weak solution and the test function in the weak formulation are parameterized as deep neural networks in a mesh-free manner, which are alternately updated to approach the optimal solution networks approximating the weak solution and the optimal test function, respectively. Extensive numerical results indicate that our mesh-free Friedrichs learning method can provide reasonably good solutions for a wide range of PDEs defined on regular and irregular domains, where conventional numerical methods such as finite difference methods and finite element methods may be tedious or difficult to be applied, especially for those with discontinuous solutions in high-dimensional problems.<\/jats:p>","DOI":"10.1137\/22m1488405","type":"journal-article","created":{"date-parts":[[2023,6,14]],"date-time":"2023-06-14T03:46:46Z","timestamp":1686714406000},"page":"A1271-A1299","source":"Crossref","is-referenced-by-count":25,"title":["Friedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learning"],"prefix":"10.1137","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0082-7908","authenticated-orcid":true,"given":"Fan","family":"Chen","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jianguo","family":"Huang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chunmei","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Florida, Gainesville, FL 32611 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8408-1754","authenticated-orcid":true,"given":"Haizhao","family":"Yang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Maryland, College Park, MD 20742 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,6,14]]},"reference":[{"key":"ref1","volume-title":"Pure Appl. 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