scholarly journals Vanishing viscosity approximation for linear transport equations on finite star-shaped networks

2021 ◽  
Author(s):  
Francesca R. Guarguaglini ◽  
Roberto Natalini
Keyword(s):  
Transport Equations ◽  
Viscosity Approximation ◽  
Vanishing Viscosity ◽  
Linear Transport

scholarly journals Efficient approximation of solutions of parametric linear transport equations by ReLU DNNs

2021 ◽  
Vol 47 (1) ◽  
Author(s):  
Fabian Laakmann ◽  
Philipp Petersen
Keyword(s):  
Neural Networks ◽  
Deep Neural Networks ◽  
Initial Conditions ◽  
Activation Function ◽  
Transport Equations ◽  
High Dimensional ◽  
Linear Transport ◽  
Approximation Rates ◽  
Curse Of Dimension ◽  
Efficient Approximation

AbstractWe demonstrate that deep neural networks with the ReLU activation function can efficiently approximate the solutions of various types of parametric linear transport equations. For non-smooth initial conditions, the solutions of these PDEs are high-dimensional and non-smooth. Therefore, approximation of these functions suffers from a curse of dimension. We demonstrate that through their inherent compositionality deep neural networks can resolve the characteristic flow underlying the transport equations and thereby allow approximation rates independent of the parameter dimension.

scholarly journals POD‐identification reduced order model of linear transport equations for control purposes

2019 ◽  
Vol 90 (8) ◽  
pp. 375-388 ◽  
Author(s):  
S. Kelbij Star ◽  
Francesco Belloni ◽  
Gert Van den Eynde ◽  
Joris Degroote
Keyword(s):  
Transport Equations ◽  
Reduced Order Model ◽  
Order Model ◽  
Reduced Order ◽  
Linear Transport
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