The DOI matches its Crossref record; 40 of the 40 references it cites were found in a registry or an index.
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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Universal Differential Equations for Scientific Machine Learning
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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- Zang, Y., Bao, G., Ye, X., & Zhou, H. (2020). Weak adversarial networks for high-dimensional partial differential equations. Journal of Computational Physics. https://doi.org/10.1016/j.jcp.2020.109409
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