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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
Dependencies: Python
Keywords: partial differential equations; machine learning; deep learning; high-dimensional partial differential equations
Related Software: Adam; TensorFlow; PDE-Net; DeepXDE; PyTorch; FPINNs; ImageNet; AlexNet; DiffSharp; L-BFGS; DeepONet; torchdiffeq; GitHub; NSFnets; U-Net; Matlab; Keras; FEniCS; Wasserstein GAN; PINNsNTK
Referenced in: 195 Publications
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Referenced by 422 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 Grohs, Philipp
4 Perdikaris, Paris G.
4 Warin, Xavier
4 Ying, Lexing
3 Arridge, Simon R.
3 Beck, Christian
3 Cheridito, Patrick
3 Darbon, Jerome
3 E, Weinan
3 Fan, Yuwei
3 Han, Jiequn
3 Huang, Jianguo
3 Hwang, Hyung Ju
3 Kruse, Thomas
3 Kutyniok, Gitta
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 Gühring, Ingo
2 Hall, Eric Joseph
2 He, Cuiyu
2 Jang, Jin Woo
2 Jiao, Yuling
2 Kalise, Dante
2 Katsoulakis, Markos A.
2 Koutsourelakis, Phaedon-Stelios
2 Kunisch, Karl
2 Lai, Yanming
2 Lee, Jaeyong
2 Li, Dingwei
2 Lin, Lin
2 Liu, Wing Kam
2 Lu, Jianfeng
2 Lu, Lu
2 Lu, Xiliang
2 Maass, Peter
2 Meng, Tingwei
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, Jian-Xun
2 Wang, Li
2 Wang, Zhongjian
2 Xiao, Heng
2 Xu, Jinchao
2 Yang, Haizhao
2 Yang, Jerry Zhijian
2 Zabaras, Nicholas J.
2 Zepeda-Núñez, Leonardo
2 Zhang, Lei
2 Zhang, Zhiwen
2 Zhou, Chao
1 Abdeljawad, Ahmed
1 Aghaei, A. A.
1 Ainsworth, Mark
1 Al-Aradi, Ali
1 Amendt, Maurice
1 Amini Niaki, Sina
1 Anitescu, Cosmin
1 Aragon-Calvo, Miguel A.
1 Bacchi, Raphael D. A.
1 Baharifard, Fatemeh
...and 322 more Authors
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Referenced in 71 Serials

51 Journal of Computational Physics
18 Computer Methods in Applied Mechanics and Engineering
14 Communications in Computational Physics
9 Journal of Scientific Computing
8 SIAM Journal on Scientific Computing
4 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. A
2 Journal of Computational and Applied Mathematics
2 Constructive Approximation
2 European Journal of Applied Mathematics
2 European Journal of Mechanics. A. Solids
2 Foundations of Computational Mathematics
2 SIAM Journal on Financial Mathematics
2 Advances in Mathematical Physics
1 Computers and Fluids
1 Nonlinearity
1 Wave Motion
1 Mathematics of Computation
1 Chaos, Solitons and Fractals
1 Applied Mathematics and Optimization
1 Journal of the Korean Mathematical Society
1 Journal of Optimization Theory and Applications
1 Mathematics of Operations Research
1 Numerische Mathematik
1 SIAM Journal on Numerical Analysis
1 Chinese Annals of Mathematics. Series B
1 Applied Numerical Mathematics
1 Computational Mechanics
1 Journal of Economic Dynamics & Control
1 Neural Networks
1 Machine Learning
1 International Journal of Computer Mathematics
1 SIAM Journal on Applied Mathematics
1 SIAM Review
1 Journal of Mathematical Imaging and Vision
1 Journal of Nonlinear Science
1 Computational and Applied Mathematics
1 Advances in Applied Clifford Algebras
1 Advances in Computational Mathematics
1 Electronic Journal of Probability
1 Finance and Stochastics
1 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations
1 Communications in Nonlinear Science and Numerical Simulation
1 Journal of Machine Learning Research (JMLR)
1 Journal of Numerical Mathematics
1 SIAM Journal on Applied Dynamical Systems
1 Multiscale Modeling & Simulation
1 Acta Numerica
1 International Journal of Numerical Analysis and Modeling
1 Journal of Statistical Mechanics: Theory and Experiment
1 Oberwolfach Reports
1 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis
1 SIAM Journal on Imaging Sciences
1 Discrete and Continuous Dynamical Systems. Series S
1 Numerical Mathematics: Theory, Methods and Applications
1 Science China. Mathematics
1 Statistics and Computing
1 Journal of Computational Dynamics
1 East Asian Journal on Applied Mathematics
1 Results in Applied Mathematics
1 Foundations of Data Science
1 Frontiers of Mathematical Finance

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