DGM swMATH ID: 39282 Software Authors: Sirignano, Justin; Spiliopoulos, Konstantinos Description: DGM: a deep learning algorithm for solving partial differential equations. High-dimensional PDEs have been a longstanding computational challenge. We propose to solve high-dimensional PDEs by approximating the solution with a deep neural network which is trained to satisfy the differential operator, initial condition, and boundary conditions. Our algorithm is meshfree, which is key since meshes become infeasible in higher dimensions. Instead of forming a mesh, the neural network is trained on batches of randomly sampled time and space points. The algorithm is tested on a class of high-dimensional free boundary PDEs, which we are able to accurately solve in up to 200 dimensions. The algorithm is also tested on a high-dimensional Hamilton-Jacobi-Bellman PDE and Burgers’ equation. The deep learning algorithm approximates the general solution to the Burgers’ equation for a continuum of different boundary conditions and physical conditions (which can be viewed as a high-dimensional space). We call the algorithm a “Deep Galerkin method (DGM)” since it is similar in spirit to Galerkin methods, with the solution approximated by a neural network instead of a linear combination of basis functions. In addition, we prove a theorem regarding the approximation power of neural networks for a class of quasilinear parabolic PDEs. Homepage: https://paperswithcode.com/paper/dgm-a-deep-learning-algorithm-for-solving Source Code: https://github.com/alialaradi/DeepGalerkinMethod Dependencies: Python Keywords: partial differential equations; machine learning; deep learning; high-dimensional partial differential equations Related Software: Adam; TensorFlow; PDE-Net; DeepXDE; PyTorch; FPINNs; DiffSharp; ImageNet; AlexNet; L-BFGS; torchdiffeq; DeepONet; GitHub; U-Net; PPINN; Matlab; PhyGeoNet; NSFnets; Keras; PINNsNTK Cited in: 205 Publications Standard Articles 1 Publication describing the Software, including 1 Publication in zbMATH Year DGM: a deep learning algorithm for solving partial differential equations. Zbl 1416.65394Sirignano, Justin; Spiliopoulos, Konstantinos 2018 all top 5 Cited by 445 Authors 13 Jentzen, Arnulf 8 Karniadakis, George Em 6 Cai, Zhiqiang 6 Liu, Min 6 Sirignano, Justin A. 5 Chen, Jingshuang 5 Hutzenthaler, Martin 5 Petersen, Philipp Christian 5 Spiliopoulos, Konstantinos V. 4 Becker, Sebastian 4 Darbon, Jerome 4 Grohs, Philipp 4 Perdikaris, Paris G. 4 Warin, Xavier 4 Ying, Lexing 3 Arridge, Simon R. 3 Beck, Christian 3 Cheridito, Patrick 3 E, Weinan 3 Fan, Yuwei 3 Han, Jiequn 3 Huang, Jianguo 3 Hwang, Hyung Ju 3 Kruse, Thomas 3 Kutyniok, Gitta 3 Meng, Tingwei 3 Pham, Huyên 3 Rabczuk, Timon 3 Raslan, Mones 3 Tartakovsky, Daniel M. 3 Wang, Haoqin 3 Wang, Sifan 3 Yan, Zhenya 3 Zhuang, Xiaoying 2 Alajlan, Naif 2 Ayed, Ibrahim 2 Cheng, Lin 2 de Bézenac, Emmanuel 2 Dong, Bin 2 Fedkiw, Ronald P. 2 Feliu-Fabà, Jordi 2 Gallinari, Patrick 2 Gao, Han 2 Germain, Maximilien 2 Gu, Yiqi 2 Gühring, Ingo 2 Hall, Eric Joseph 2 Hao, Wenrui 2 He, Cuiyu 2 Hu, Wei-Fan 2 Jang, Jin Woo 2 Jiao, Yuling 2 Kalise, Dante 2 Katsoulakis, Markos A. 2 Koutsourelakis, Phaedon-Stelios 2 Kunisch, Karl 2 Lai, Mingchih 2 Lai, Yanming 2 Lee, Jaeyong 2 Li, Dingwei 2 Li, Jian 2 Lin, Guang 2 Lin, Lin 2 Lin, Te-Sheng 2 Liu, Wing Kam 2 Lu, Jianfeng 2 Lu, Lu 2 Lu, Xiliang 2 Maass, Peter 2 Meng, Xuhui 2 Nguyen-Thanh, Vien Minh 2 Omella, Ángel Javier 2 Parand, Kourosh 2 Pardo, David 2 Saporito, Yuri F. 2 Schneider, Reinhold 2 Schönlieb, Carola-Bibiane 2 Schwab, Christoph 2 Soize, Christian 2 Stinchcombe, Adam R. 2 Stuart, Andrew M. 2 Sun, Luning 2 Tang, Shaoqiang 2 Taverniers, Søren 2 von Wurstemberger, Philippe 2 Wang, Jianxun 2 Wang, Li 2 Wang, Zhongjian 2 Xiao, Heng 2 Xu, Jinchao 2 Yang, Chao 2 Yang, Haizhao 2 Yang, Jerry Zhijian 2 Yue, Jing 2 Zabaras, Nicholas J. 2 Zepeda-Núñez, Leonardo 2 Zhang, Lei 2 Zhang, Wen 2 Zhang, Zhiwen 2 Zhou, Chao ...and 345 more Authors all top 5 Cited in 72 Serials 54 Journal of Computational Physics 19 Computer Methods in Applied Mechanics and Engineering 15 Communications in Computational Physics 10 Journal of Scientific Computing 9 SIAM Journal on Scientific Computing 5 Computers & Mathematics with Applications 3 Applied Mathematics and Computation 3 Mathematics and Computers in Simulation 3 Physica D 3 Chaos 3 Quantitative Finance 3 Analysis and Applications (Singapore) 3 Research in the Mathematical Sciences 3 SN Partial Differential Equations and Applications 3 SIAM Journal on Mathematics of Data Science 2 Physics Letters. 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