A comparison of PINN approaches for drift-diffusion equations on metric graphs

  • In this paper we focus on comparing machine learning approaches for quantum graphs, which are metric graphs, i.e., graphs with dedicated edge lengths, and an associated differential operator. In our case the differential equation is a drift-diffusion model. Computational methods for quantum graphs require a careful discretization of the differential operator that also incorporates the node conditions, in our case Kirchhoff-Neumann conditions. Traditional numerical schemes are rather mature but have to be tailored manually when the differential equation becomes the constraint in an optimization problem. Recently, physics informed neural networks (PINNs) have emerged as a versatile tool for the solution of partial differential equations from a range of applications. They offer flexibility to solve parameter identification or optimization problems by only slightly changing the problem formulation used for the forward simulation. We compare several PINN approaches for solving theIn this paper we focus on comparing machine learning approaches for quantum graphs, which are metric graphs, i.e., graphs with dedicated edge lengths, and an associated differential operator. In our case the differential equation is a drift-diffusion model. Computational methods for quantum graphs require a careful discretization of the differential operator that also incorporates the node conditions, in our case Kirchhoff-Neumann conditions. Traditional numerical schemes are rather mature but have to be tailored manually when the differential equation becomes the constraint in an optimization problem. Recently, physics informed neural networks (PINNs) have emerged as a versatile tool for the solution of partial differential equations from a range of applications. They offer flexibility to solve parameter identification or optimization problems by only slightly changing the problem formulation used for the forward simulation. We compare several PINN approaches for solving the drift-diffusion on the metric graph.show moreshow less

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Metadaten
Author:Jan Blechschmidt, Jan-Frederik PietschmannORCiDGND, Tom-Christian Riemer, Martin Stoll, Max Winkler
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/102107
Parent Title (English):arXiv
Publisher:arXiv
Type:Preprint
Language:English
Year of first Publication:2022
Release Date:2023/02/20
Issue:arXiv:2205.07195
DOI:https://doi.org/10.48550/arXiv.2205.07195
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Inverse Probleme
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