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Deep Learning for Hyperbolic Conservation Laws with Non‐convex Flux

Minbashian, Hadi ; Giesselmann, Jan (2024)
Deep Learning for Hyperbolic Conservation Laws with Non‐convex Flux.
In: PAMM - Proceedings in Applied Mathematics and Mechanics, 2021, 20 (S1)
doi: 10.26083/tuprints-00020142
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Item Type: Article
Type of entry: Secondary publication
Title: Deep Learning for Hyperbolic Conservation Laws with Non‐convex Flux
Language: English
Date: 13 February 2024
Place of Publication: Darmstadt
Year of primary publication: 2021
Place of primary publication: Weinheim
Publisher: Wiley‐VCH
Journal or Publication Title: PAMM - Proceedings in Applied Mathematics and Mechanics
Volume of the journal: 20
Issue Number: S1
Collation: 6 Seiten
DOI: 10.26083/tuprints-00020142
Corresponding Links:
Origin: Secondary publication DeepGreen

In this work, we investigate the capabilities of deep neural networks for solving hyperbolic conservation laws with non‐convex flux functions. The behaviour of solutions to these problems depends on the underlying small‐scale regularization. In many applications concerning phase transition phenomena, the regularization terms consist of diffusion and dispersion which are kept in balance in the limit. This may lead to the development of both classical and non‐classical (or undercompressive) shock waves at the same time which makes the development of approximation schemes that converge towards the appropriate weak solution of these problems challenging. Here, we consider a scalar conservation law with cubic flux function as a toy model and present preliminary results of an ongoing work to study the capabilities of a deep learning algorithm called PINNs proposed in [1] for solving this problem. It consists of a feed‐forward network with a hyperbolic tangent activation function along with an additional layer to enforce the differential equation.

Identification Number: Artikel-ID: e202000347
Status: Publisher's Version
URN: urn:nbn:de:tuda-tuprints-201421
Additional Information:

Special Issue: 7th GAMM Juniors' Summer School on Applied Mathematics and Mechanics (SAMM)

Classification DDC: 500 Science and mathematics > 510 Mathematics
Divisions: 04 Department of Mathematics > Numerical Analysis and Scientific Computing
Date Deposited: 13 Feb 2024 10:38
Last Modified: 13 Feb 2024 10:39
SWORD Depositor: Deep Green
URI: https://tuprints.ulb.tu-darmstadt.de/id/eprint/20142
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