Spectral properties and time asymptotic behaviour of linear transport equations in slab geometry

2001 ◽  
Vol 24 (10) ◽  
pp. 689-711 ◽  
Author(s):  
Khalid Latrach ◽  
Abdelkader Dehici
Keyword(s):  
Asymptotic Behaviour ◽  
Spectral Properties ◽  
Transport Equations ◽  
Slab Geometry ◽  
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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