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Weak adversarial networks for high-dimensional partial differential equations
Abstract
Abstract Solving general high-dimensional partial differential equations (PDE) is a long-standing challenge in numerical mathematics. In this paper, we propose a novel approach to solve high-dimensional linear and nonlinear PDEs defined on arbitrary domains by leveraging their weak formulations. We convert the problem of finding the weak solution of PDEs into an operator norm minimization problem induced from the weak formulation. The weak solution and the test function in the weak formulation are then parameterized as the primal and adversarial networks respectively, which are alternately updated to approximate the optimal network parameter setting. Our approach, termed as the weak adversarial network (WAN), is fast, stable, and completely mesh-free, which is particularly suitable for high-dimensional PDEs defined on irregular domains where the classical numerical methods based on finite differences and finite elements suffer the issues of slow computation, instability and the curse of dimensionality. We apply our method to a variety of test problems with high-dimensional PDEs to demonstrate its promising performance.
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“In the multifield context, which is more generic, one finds a similar value of r but variable tilt of the primordial power spectrum according to simple statistical studies such as [77].”
From Hyperbolic compactification of M-theory and de Sitter quantum gravity (2022) · cites this paper as [77]
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Cites
Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019) · Journal of Computational Physics · doi:10.1016/j.jcp.2018.10.045 · Cited by 20,384
Advances in neural information processing systems 7
(1996) · Computers & Mathematics with Applications · doi:10.1016/s0898-1221(96)90463-0 · Cited by 14,281
TensorFlow: A system for large-scale machine learning
Abadi, M., Barham, P., Chen, J., Chen, Z., Davis, A., Dean, J., Devin, M., Ghemawat, S., Irving, G., Isard, M., et al. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1605.08695 · Cited by 11,822
(2010) · Cited by 8,261
The finite element method. Linear static and dynamic finite element analysis
Zimmermann, T. (1987) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/0045-7825(87)90013-2 · Cited by 5,080
The Finite Element Method: Linear Static and Dynamic Finite Element Analysis
Hughes, T. J. R. (1987) · Cited by 4,852
Crank, J., & Nicolson, P. (1947) · Mathematical Proceedings of the Cambridge Philosophical Society · doi:10.1017/s0305004100023197 · Cited by 3,316
Artificial neural networks for solving ordinary and partial differential equations
Lagaris, I. E., Likas, A., & Fotiadis, D. I. (1998) · IEEE Transactions on Neural Networks · doi:10.1109/72.712178 · Cited by 2,412
DGM: A deep learning algorithm for solving partial differential equations
Sirignano, J., & Spiliopoulos, K. (2018) · Journal of Computational Physics · doi:10.1016/j.jcp.2018.08.029 · Cited by 2,208
Numerical approximation of partial differential equations
Quarteroni, A. M., & Valli, A. (1987) · Mathematics and Computers in Simulation · doi:10.1016/0378-4754(87)90091-7 · Cited by 1,864
Numerical Approximation of Partial Differential Equations
Quarteroni, A. M., & Valli, A. (1994) · Springer series in computational mathematics · doi:10.1007/978-3-540-85268-1 · Cited by 1,761
Solving high-dimensional partial differential equations using deep learning
Han, J., Jentzen, A., & E, W. (2018) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1718942115 · Cited by 1,722
The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems
E, W., & Yu, B. (2018) · Communications in Mathematics and Statistics · doi:10.1007/s40304-018-0127-z · Cited by 1,418
Numerical Partial Differential Equations: Finite Difference Methods
Thomas, J. W. (1995) · Texts in applied mathematics · doi:10.1007/978-1-4899-7278-1 · Cited by 1,356
Artificial Neural Networks for Solving Ordinary and Partial Differential Equations
Lagaris, I. E., Likas, A., & Fotiadis, D. I. (1998) · Cited by 944
Raissi, M., Perdikaris, P. G., & Karniadakis, G. E. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1711.10561 · Cited by 925
E, W., Han, J., & Jentzen, A. (2017) · Communications in Mathematics and Statistics · doi:10.1007/s40304-017-0117-6 · Cited by 822
Crank, J., & Nicolson, P. (1996) · Advances in Computational Mathematics · doi:10.1007/bf02127704 · Cited by 684
Artificial Neural Network Methods for the Solution of Second Order Boundary Value Problems
Anitescu, C., Atroshchenko, E., Alajlan, N., & Rabczuk, T. (2019) · Computers, materials & continua/Computers, materials & continua (Print) · doi:10.32604/cmc.2019.06641 · Cited by 669
Berg, J., & Nyström, K. (2018) · Neurocomputing · doi:10.1016/j.neucom.2018.06.056 · Cited by 657
Neural‐network‐based approximations for solving partial differential equations
Dissanayake, M. W. M. G., & Phan‐Thien, N. (1994) · Communications in Numerical Methods in Engineering · doi:10.1002/cnm.1640100303 · Cited by 622
Neural algorithm for solving differential equations
Lee, H., & Kang, I. S. (1990) · Journal of Computational Physics · doi:10.1016/0021-9991(90)90007-n · Cited by 436
Adversarial uncertainty quantification in physics-informed neural networks
Yang, Y., & Perdikaris, P. (2019) · Journal of Computational Physics · doi:10.1016/j.jcp.2019.05.027 · Cited by 414
Beck, C., E, W., & Jentzen, A. (2019) · Journal of Nonlinear Science · doi:10.1007/s00332-018-9525-3 · Cited by 308
Accelerating Eulerian Fluid Simulation With Convolutional Networks
Tompson, J., Schlachter, K., Sprechmann, P., & Perlin, K. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1607.03597 · Cited by 280
The numerical solution of linear ordinary differential equations by feedforward neural networks
Meade, A. J., & Fernandez, A. A. (1994) · Mathematical and Computer Modelling · doi:10.1016/0895-7177(94)90095-7 · Cited by 256
Solution of nonlinear ordinary differential equations by feedforward neural networks
Meade, A. J., & Fernandez, A. A. (1994) · Mathematical and Computer Modelling · doi:10.1016/0895-7177(94)00160-x · Cited by 178
Solving for high-dimensional committor functions using artificial neural networks
Khoo, Y., Lu, J., & Ying, L. (2018) · Research in the Mathematical Sciences · doi:10.1007/s40687-018-0160-2 · Cited by 165
A deep learning solution approach for high-dimensional random differential equations
Nabian, M. A., & Meidani, H. (2019) · Probabilistic Engineering Mechanics · doi:10.1016/j.probengmech.2019.05.001 · Cited by 143
Rudd, K., & Ferrari, S. (2014) · Neurocomputing · doi:10.1016/j.neucom.2014.11.058 · Cited by 113
Asymptotic Expansion as Prior Knowledge in Deep Learning Method for High Dimensional BSDEs
Fujii, M., Takahashi, A., & Takahashi, M. (2017) · Asia-Pacific Financial Markets · doi:10.1007/s10690-019-09271-7 · Cited by 91
1990 IJCNN International Joint Conference on Neural Networks
(1990) · doi:10.1109/ijcnn.1990 · Cited by 69
Neural Networks Trained to Solve Differential Equations Learn General RepresentationsPreprint
Magill, M., Qureshi, F. Z., & de Haan, H. W. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1807.00042 · Cited by 50
Structured trainable networks for matrix algebra
Wang, L., & Mendel, J. M. (1990) · 2022 International Joint Conference on Neural Networks (IJCNN) · doi:10.1109/ijcnn.1990.137705 · Cited by 47
VLSI implementation of locally connected neural network for solving partial differential equations
Yentis, R., & Zaghloul, M. E. (1996) · IEEE Transactions on Circuits and Systems I Fundamental Theory and Applications · doi:10.1109/81.526685 · Cited by 45
Han, J., Jentzen, A., & Weinan, E. (2017) · Cited by 40
Gobovic, D., & Zaghloul, M. E. (2002) · IEEE International Symposium on Circuits and Systems proceedings · doi:10.1109/iscas.1994.409600 · Cited by 15
Beck, C., Weinan, E., & Jentzen, A. (2017) · Repository for Publications and Research Data (ETH Zurich) · doi:10.3929/ethz-b-000314918 · Cited by 14
Neural network-based discretization of nonlinear differential equations
Suzuki, Y. (2017) · Neural Computing and Applications · doi:10.1007/s00521-017-3249-4 · Cited by 6
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Rackauckas, C., Ma, Y., Martensen, J., Warner, C., Zubov, K., Supekar, R., Skinner, D., Ramadhan, A., & Edelman, A. (2020) · doi:10.21203/rs.3.rs-55125/v1 · Cited by 643
Hyperbolic compactification of M-theory and de Sitter quantum gravity
De Luca, G. B., Silverstein, E., & Torroba, G. (2022) · SciPost Physics · doi:10.21468/scipostphys.12.3.083 · Cited by 39
Metadata
- Authors
- Zang, Y., Bao, G., Ye, X., & Zhou, H.
- ORCID
- Zang, Yaohua
- Venue
- Journal of Computational Physics
- Publisher
- Elsevier BV
- Published
- 2020-06
- Volume
- 411
- Pages
- 109409
- ISSN
- 0021-9991
- Cited by
- 444
- References
- 40
- Language
- en
- Licence
- www.elsevier.com/tdm/userlicense/1.0/ (tdm)
- www.elsevier.com/legal/tdmrep-license (tdm)
- www.elsevier.com/open-access/userlicense/1.0/ (am, 295 d)
- Funders
- China Scholarship Council (201806320325)
- Office of Naval Research (N00014-18-1-2852)
- Foundation for Innovative Research Groups of the National Natural Science Foundation of China
- National Science Foundation (DMS-1830225, DMS-1925263, CMMI-1745382, DMS-1818886, DMS-1620345, DMS-1620342)
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