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Optimally weighted loss functions for solving PDEs with neural networks
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van der Meer, Remco, Oosterlee, Cornelis W. and Borovykh, Anastasia (2022) Optimally weighted loss functions for solving PDEs with neural networks. Journal of Computational and Applied Mathematics, 405 . 113887. doi:10.1016/j.cam.2021.113887 ISSN 0377-0427.
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WRAP-Optimally-weighted-loss-functions-solving-PDEs-Neural-Networks-2021.pdf - Accepted Version - Requires a PDF viewer. Available under License Creative Commons Attribution Non-commercial No Derivatives 4.0. Download (1334Kb) | Preview |
Official URL: https://doi.org/10.1016/j.cam.2021.113887
Abstract
Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks (Raissi et al., 2007). We introduce a generalization for these methods that manifests as a scaling parameter which balances the relative importance of the different constraints imposed by partial differential equations. A mathematical motivation of these generalized methods is provided, which shows that for linear and well-posed partial differential equations, the functional form is convex. We then derive a choice for the scaling parameter that is optimal with respect to a measure of relative error. Because this optimal choice relies on having full knowledge of analytical solutions, we also propose a heuristic method to approximate this optimal choice. The proposed methods are compared numerically to the original methods on a variety of model partial differential equations, with the number of data points being updated adaptively. For several problems, including high-dimensional PDEs the proposed methods are shown to significantly enhance accuracy.
| Item Type: | Journal Article | ||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Social Sciences > Warwick Business School | ||||||||
| Library of Congress Subject Headings (LCSH): | Neural networks (Computer science), Differential equations, Partial, Mathematical optimization | ||||||||
| Journal or Publication Title: | Journal of Computational and Applied Mathematics | ||||||||
| Publisher: | Elseivier Science BV | ||||||||
| ISSN: | 0377-0427 | ||||||||
| Official Date: | 15 May 2022 | ||||||||
| Dates: |
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| Volume: | 405 | ||||||||
| Article Number: | 113887 | ||||||||
| DOI: | 10.1016/j.cam.2021.113887 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||
| Date of first compliant deposit: | 27 January 2022 | ||||||||
| Date of first compliant Open Access: | 2 December 2022 |
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