Optimal Dirichlet control of partial differential equations on networks

Stoll, Martin; Winkler, Max

doi:10.1553/etna_vol54s392

Cited by 3 publications

(2 citation statements)

References 42 publications

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“…Here we focus on metric graph where each edge can be associated with an interval thus allowing the definition of differential operators (quantum graphs) [3,4]. Numerical methods for such graphs have gained recent interest [5,6,7] both for simulation and optimal control. As the structure of the network can possibly become rather complex efficient schemes such as domain decomposition methods [8] are often needed.…”

Section: Introductionmentioning

confidence: 99%

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

Blechschmidt¹,

Pietschman²,

Riemer³

et al. 2022

Preprint

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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 the drift-diffusion on the metric graph.

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Section: Introductionmentioning

confidence: 99%

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

Blechschmidt¹,

Pietschman²,

Riemer³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

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“…Only recently more general investigations appeared directed to developing the finite element method for elliptic and parabolic equations on graphs [45][46][47].…”

Section: Introductionmentioning

confidence: 99%