{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,23]],"date-time":"2026-10-23T03:01:18Z","timestamp":1792724478605,"version":"4.5.0"},"reference-count":59,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"name":"DOE PhILMs","award":["de-sc0019453"],"award-info":[{"award-number":["de-sc0019453"]}]},{"name":"DARPA-AIRA","award":["HR00111990025"],"award-info":[{"award-number":["HR00111990025"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM Rev."],"published-print":{"date-parts":[[2021,1,1]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently. Here, we present an overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiation. The PINN algorithm is simple, and it can be applied to different types of PDEs, including integro-differential equations, fractional PDEs, and stochastic PDEs. Moreover, from an implementation point of view, PINNs solve inverse problems as easily as forward problems. We propose a new residual-based adaptive refinement (RAR) method to improve the training efficiency of PINNs. For pedagogical reasons, we compare the PINN algorithm to a standard finite element method. We also present a Python library for PINNs, DeepXDE, which is designed to serve both as an educational tool to be used in the classroom as well as a research tool for solving problems in computational science and engineering. Specifically, DeepXDE can solve forward problems given initial and boundary conditions, as well as inverse problems given some extra measurements. DeepXDE supports complex-geometry domains based on the technique of constructive solid geometry and enables the user code to be compact, resembling closely the mathematical formulation. We introduce the usage of DeepXDE and its customizability, and we also demonstrate the capability of PINNs and the user-friendliness of DeepXDE for five different examples. More broadly, DeepXDE contributes to the more rapid development of the emerging scientific machine learning field.<\/jats:p>","DOI":"10.1137\/19m1274067","type":"journal-article","created":{"date-parts":[[2021,2,4]],"date-time":"2021-02-04T09:27:06Z","timestamp":1612430826000},"page":"208-228","source":"Crossref","is-referenced-by-count":2411,"title":["DeepXDE: A Deep Learning Library for Solving Differential Equations"],"prefix":"10.1137","volume":"63","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5476-5768","authenticated-orcid":true,"given":"Lu","family":"Lu","sequence":"first","affiliation":[{"name":"Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xuhui","family":"Meng","sequence":"additional","affiliation":[{"name":"Division of Applied Mathematics, Brown University, Providence, RI 02912 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhiping","family":"Mao","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9713-7120","authenticated-orcid":true,"given":"George Em","family":"Karniadakis","sequence":"additional","affiliation":[{"name":"Division of Applied Mathematics, Brown University, Providence, RI 02912 USA, and Pacific Northwest National Laboratory, Richland, WA 99354 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,2,4]]},"reference":[{"key":"ref1","unstructured":"M. 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