Minbashian, Hadi ; Giesselmann, Jan (2021)
Deep Learning for Hyperbolic Conservation Laws with Non‐convex Flux.
In: PAMM - Proceedings in Applied Mathematics and Mechanics, 20 (S1)
doi: 10.1002/pamm.202000347
Artikel, Bibliographie
Dies ist die neueste Version dieses Eintrags.
Kurzbeschreibung (Abstract)
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.
| Typ des Eintrags: | Artikel |
|---|---|
| Erschienen: | 2021 |
| Autor(en): | Minbashian, Hadi ; Giesselmann, Jan |
| Art des Eintrags: | Bibliographie |
| Titel: | Deep Learning for Hyperbolic Conservation Laws with Non‐convex Flux |
| Sprache: | Englisch |
| Publikationsjahr: | 2021 |
| Ort: | Weinheim |
| Verlag: | Wiley‐VCH |
| Titel der Zeitschrift, Zeitung oder Schriftenreihe: | PAMM - Proceedings in Applied Mathematics and Mechanics |
| Jahrgang/Volume einer Zeitschrift: | 20 |
| (Heft-)Nummer: | S1 |
| Kollation: | 6 Seiten |
| DOI: | 10.1002/pamm.202000347 |
| Zugehörige Links: | |
| Kurzbeschreibung (Abstract): | 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. |
| ID-Nummer: | Artikel-ID: e202000347 |
| Zusätzliche Informationen: | Special Issue: 7th GAMM Juniors' Summer School on Applied Mathematics and Mechanics (SAMM) |
| Sachgruppe der Dewey Dezimalklassifikatin (DDC): | 500 Naturwissenschaften und Mathematik > 510 Mathematik |
| Fachbereich(e)/-gebiet(e): | 04 Fachbereich Mathematik 04 Fachbereich Mathematik > Numerik und wissenschaftliches Rechnen |
| Hinterlegungsdatum: | 15 Feb 2024 14:03 |
| Letzte Änderung: | 15 Feb 2024 14:03 |
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| Suche nach Titel in: | TUfind oder in Google |
Verfügbare Versionen dieses Eintrags
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Deep Learning for Hyperbolic Conservation Laws with Non‐convex Flux. (deposited 13 Feb 2024 10:38)
- Deep Learning for Hyperbolic Conservation Laws with Non‐convex Flux. (deposited 15 Feb 2024 14:03) [Gegenwärtig angezeigt]
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