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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
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; DeepXDE; TensorFlow; DeepONet; FPINNs; PyTorch; DiffSharp; PDE-Net; hp-VPINNs; ImageNet; AlexNet; L-BFGS; XPINNs; PINNsNTK; GitHub; torchdiffeq; MscaleDNN; NSFnets; PhyGeoNet; VarNet
Cited in: 358 Documents
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Cited by 752 Authors

17 Jentzen, Arnulf
11 Karniadakis, George Em
8 Cai, Zhiqiang
8 Liu, Min
7 Petersen, Philipp Christian
7 Yan, Zhenya
6 Chen, Jingshuang
6 Hutzenthaler, Martin
6 Rabczuk, Timon
6 Sirignano, Justin A.
6 Spiliopoulos, Konstantinos V.
5 Becker, Sebastian
5 Darbon, Jerome
5 Grohs, Philipp
5 Huang, Jianguo
5 Hwang, Hyung Ju
5 Jiao, Yuling
5 Li, Jian
5 Pardo, David
5 Wang, Li
5 Warin, Xavier
5 Zhou, Zijian
4 Beck, Christian
4 E, Weinan
4 Hao, Wenrui
4 Liu, Wing Kam
4 Meng, Tingwei
4 Nguwi, Jiang Yu
4 Perdikaris, Paris G.
4 Privault, Nicolas
4 Wang, Haoqin
4 Yang, Chao
4 Ying, Lexing
4 Zhang, Wen
4 Zhuang, Xiaoying
3 Alajlan, Naif
3 Arridge, Simon R.
3 Chen, Jingrun
3 Cheridito, Patrick
3 Cui, Tao
3 Dong, Bin
3 Fan, Yuwei
3 Han, Jiequn
3 Kruse, Thomas
3 Kutyniok, Gitta
3 Lai, Yanming
3 Lee, Jaeyong
3 Lu, Lu
3 Lu, Xiliang
3 Maass, Peter
3 Meng, Xuhui
3 Muga, Ignacio
3 Omella, Ángel Javier
3 Parand, Kourosh
3 Pham, Huyên
3 Raslan, Mones
3 Schwab, Christoph
3 Tartakovsky, Daniel M.
3 von Wurstemberger, Philippe
3 Wan, Xiaoliang
3 Wang, Chunmei
3 Wang, Sifan
3 Wang, Zhongjian
3 Xu, Jinchao
3 Yamada, Toshihiro
3 Yang, Haizhao
3 Yang, Jerry Zhijian
3 Yang, Jiang
3 Yue, Jing
3 Zhang, Zhiwen
2 Ainsworth, Mark
2 Ayed, Ibrahim
2 Basir, Shamsulhaq
2 Chen, Minxin
2 Cheng, Lin
2 Cuomo, Salvatore
2 de Bézenac, Emmanuel
2 Dinh Dũng
2 Dong, Justin
2 Fedkiw, Ronald P.
2 Feliu-Fabà, Jordi
2 Gallinari, Patrick
2 Gao, Han
2 Germain, Maximilien
2 Ghodsi, Ali
2 Giampaolo, Fabio
2 Gibou, Frédéric
2 Grüne, Lars
2 Gu, Yiqi
2 Gühring, Ingo
2 Guo, Hongwei
2 Guo, Jiachen
2 Hall, Eric Joseph
2 Han, Jihun
2 He, Cuiyu
2 Hornung, Fabian
2 Hu, Ruimeng
2 Hu, Wei-Fan
2 Hu, Xiaozhe
2 Huang, Haoxiang
...and 652 more Authors
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Cited in 103 Serials

77 Journal of Computational Physics
33 Computer Methods in Applied Mechanics and Engineering
23 Communications in Computational Physics
19 Journal of Scientific Computing
18 SIAM Journal on Scientific Computing
15 Computers & Mathematics with Applications
6 Applied Mathematics and Computation
6 Journal of Computational and Applied Mathematics
6 SN Partial Differential Equations and Applications
5 Physica D
5 Computational Mechanics
5 Quantitative Finance
5 Analysis and Applications (Singapore)
4 Mathematics and Computers in Simulation
4 Communications in Nonlinear Science and Numerical Simulation
4 SIAM Journal on Financial Mathematics
4 East Asian Journal on Applied Mathematics
3 Physics Letters. A
3 Chaos, Solitons and Fractals
3 Neural Networks
3 Chaos
3 Foundations of Computational Mathematics
3 International Journal of Computational Methods
3 International Journal of Numerical Analysis and Modeling
3 Oberwolfach Reports
3 Research in the Mathematical Sciences
3 SIAM Journal on Mathematics of Data Science
2 Inverse Problems
2 Constructive Approximation
2 European Journal of Applied Mathematics
2 International Journal of Computer Mathematics
2 Computational and Applied Mathematics
2 Advances in Computational Mathematics
2 Mathematical Finance
2 European Journal of Mechanics. A. Solids
2 Multiscale Modeling & Simulation
2 Acta Numerica
2 Advances in Mathematical Physics
2 Results in Applied Mathematics
2 Foundations of Data Science
1 Computers and Fluids
1 Journal of Mathematical Analysis and Applications
1 Nonlinearity
1 Wave Motion
1 Mathematics of Computation
1 Applied Mathematics and Optimization
1 Automatica
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 Operations Research Letters
1 Chinese Annals of Mathematics. Series B
1 Journal of Computational Mathematics
1 Applied Numerical Mathematics
1 Journal of Complexity
1 Journal of Economic Dynamics & Control
1 MCSS. Mathematics of Control, Signals, and Systems
1 Machine Learning
1 Japan Journal of Industrial and Applied Mathematics
1 Applications of Mathematics
1 Applied Mathematical Modelling
1 SIAM Journal on Applied Mathematics
1 SIAM Review
1 Journal of Mathematical Imaging and Vision
1 Journal of Nonlinear Science
1 Advances in Applied Clifford Algebras
1 Journal of Inverse and Ill-Posed Problems
1 Applied Mathematical Finance
1 Engineering Analysis with Boundary Elements
1 Electronic Journal of Probability
1 Sbornik: Mathematics
1 Finance and Stochastics
1 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations
1 Journal of Evolution Equations
1 Discrete and Continuous Dynamical Systems. Series B
1 Nonlinear Dynamics and Systems Theory
1 Journal of Machine Learning Research (JMLR)
1 Journal of Numerical Mathematics
1 SIAM Journal on Applied Dynamical Systems
1 Quantum Information Processing
1 Journal of Statistical Mechanics: Theory and Experiment
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 Communications in Theoretical Physics
1 Armenian Journal of Mathematics
1 Advances in Applied Mathematics and Mechanics
1 MathematicS In Action
1 Probability Surveys
1 Science China. Mathematics
1 Dynamic Games and Applications
1 Statistics and Computing
1 Communications in Mathematics and Statistics
1 Stochastic and Partial Differential Equations. Analysis and Computations
1 Journal of Computational Dynamics
1 SIAM/ASA Journal on Uncertainty Quantification
1 AMM. Applied Mathematics and Mechanics. (English Edition)
...and 3 more Serials
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Cited in 34 Fields

260 Numerical analysis (65-XX)
242 Computer science (68-XX)
146 Partial differential equations (35-XX)
47 Probability theory and stochastic processes (60-XX)
39 Fluid mechanics (76-XX)
37 Game theory, economics, finance, and other social and behavioral sciences (91-XX)
29 Statistics (62-XX)
23 Calculus of variations and optimal control; optimization (49-XX)
22 Biology and other natural sciences (92-XX)
19 Mechanics of deformable solids (74-XX)
17 Approximations and expansions (41-XX)
17 Systems theory; control (93-XX)
13 Statistical mechanics, structure of matter (82-XX)
12 Operations research, mathematical programming (90-XX)
10 Dynamical systems and ergodic theory (37-XX)
8 Ordinary differential equations (34-XX)
4 General and overarching topics; collections (00-XX)
4 Quantum theory (81-XX)
4 Information and communication theory, circuits (94-XX)
3 Functional analysis (46-XX)
3 Operator theory (47-XX)
3 Optics, electromagnetic theory (78-XX)
2 Linear and multilinear algebra; matrix theory (15-XX)
2 Integral equations (45-XX)
2 Mechanics of particles and systems (70-XX)
2 Geophysics (86-XX)
1 History and biography (01-XX)
1 Algebraic geometry (14-XX)
1 Real functions (26-XX)
1 Several complex variables and analytic spaces (32-XX)
1 Difference and functional equations (39-XX)
1 Harmonic analysis on Euclidean spaces (42-XX)
1 Integral transforms, operational calculus (44-XX)
1 Classical thermodynamics, heat transfer (80-XX)

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