scholarly journals The model reduction of the Vlasov-Poisson-Fokker-Planck system to the Poisson-Nernst-Planck system via the Deep Neural Network Approach

2021 ◽  
Author(s):  
Jae Yong Lee ◽  
Jin Woo Jang ◽  
Hyung Ju Hwang
Keyword(s):  
Neural Network ◽  
Boundary Condition ◽  
Model Reduction ◽  
Deep Neural Network ◽  
Specular Reflection ◽  
Classical Solutions ◽  
Diffusion Limit ◽  
Flux Boundary Condition ◽  
Bounded Interval ◽  
Fokker Planck

The model reduction of a mesoscopic kinetic dynamics to a macroscopic continuum dynamics has been one of the fundamental questions in mathematical physics since Hilbert's time. In this paper, we consider a diagram of the diffusion limit from the Vlasov-Poisson-Fokker-Planck (VPFP) system on a bounded interval with the specular reflection boundary condition to the Poisson-Nernst–Planck (PNP) system with the no-flux boundary condition. We provide a Deep Learning algorithm to simulate the VPFP system and the PNP system by computing the time-asymptotic behaviors of the solution and the physical quantities. We analyze the convergence of the neural network solution of the VPFP system to that of the PNP system via the Asymptotic-Preserving (AP) scheme. Also, we provide several theoretical evidence that the Deep Neural Network (DNN) solutions to the VPFP and the PNP systems converge to the a priori classical solutions of each system if the total loss function vanishes.

On the quenching set for a fast diffusion equation: regional quenching

2005 ◽  
Vol 135 (3) ◽  
pp. 585-602
Author(s):  
R. Ferreira ◽  
A. de Pablo ◽  
F. Quirós ◽  
J. D. Rossi
Keyword(s):  
Boundary Condition ◽  
Diffusion Equation ◽  
Blow Up ◽  
Single Point ◽  
Fast Diffusion ◽  
Fast Diffusion Equation ◽  
Flux Boundary Condition ◽  
Bounded Interval ◽  
Quenching Set ◽  
Type Boundary

We study positive solutions of a very fast diffusion equation, ut = (um−1ux)x, m < 0, in a bounded interval, 0 < x < L, with a quenching-type boundary condition at one end, u (0, t) = (T − t)1/(1 − m) and a zero-flux boundary condition at the other, (um −1ux)(L, t) = 0. We prove that for m ≥ −1 regional quenching is not possible: the quenching set is either a single point or the whole interval. Conversely, if m < −1 single-point quenching is impossible, and quenching is either regional or global. For some lengths the above facts depend on the initial data. The results are obtained by studying the corresponding blow-up problem for the variable v = um −1.

On the quenching set for a fast diffusion equation: regional quenching

2005 ◽  
Vol 135 (3) ◽  
pp. 585-602
Author(s):  
R. Ferreira ◽  
A. de Pablo ◽  
F. Quirós ◽  
J. D. Rossi
Keyword(s):  
Boundary Condition ◽  
Diffusion Equation ◽  
Blow Up ◽  
Single Point ◽  
Fast Diffusion ◽  
Fast Diffusion Equation ◽  
Flux Boundary Condition ◽  
Bounded Interval ◽  
Quenching Set ◽  
Type Boundary

We study positive solutions of a very fast diffusion equation, ut = (um−1ux)x, m < 0, in a bounded interval, 0 < x < L, with a quenching-type boundary condition at one end, u (0, t) = (T − t)1/(1 − m) and a zero-flux boundary condition at the other, (um −1ux)(L, t) = 0. We prove that for m ≥ −1 regional quenching is not possible: the quenching set is either a single point or the whole interval. Conversely, if m < −1 single-point quenching is impossible, and quenching is either regional or global. For some lengths the above facts depend on the initial data. The results are obtained by studying the corresponding blow-up problem for the variable v = um −1.

scholarly journals Numerical quenching versus blow-up for a nonlinear parabolic equation with nonlinear boundary outflux

2020 ◽  
pp. 151-171
Author(s):  
Kouakou Cyrille N'dri ◽  
Kidjegbo Augustin Toure ◽  
Gozo Yoro
Keyword(s):  
Boundary Condition ◽  
Original Problem ◽  
Nonlinear Boundary ◽  
Blow Up ◽  
Heat Equations ◽  
Quenching Rate ◽  
Flux Boundary Condition ◽  
Bounded Interval ◽  
Nonlinear Parabolic ◽  
Quenching Time

In this paper, we study numerical approximations for positive solutions of a semilinear heat equations, $u_{t}=u_{xx}+u^{p}$, in a bounded interval $(0,1)$, with a nonlinear flux boundary condition at the boundary $u_{x}(0,t)=0$, $u_{x}(1,t)=-u^{-q}(1,t)$. By a semi-discretization using finite difference method, we get a system of ordinary differential equations which is expected to be an approximation of the original problem. We obtain some conditions under which the positive solution of our system quenches or blows up in a finite time and estimate its semidiscrete blow-up and quenching time. We also estimate the semidiscrete blow-up and quenching rate. Finally, we give some numerical results to illustrate our analysis.

scholarly journals Specular Reflections Detection and Removal Based on Deep Neural Network for Endoscope Images

2021 ◽  
Author(s):  
Tianyu Liu ◽  
chongyu wang ◽  
Junyu Chang ◽  
Liangjing Yang
Keyword(s):  
Neural Network ◽  
Deep Neural Network ◽  
Specular Reflection ◽  
New Approach ◽  
Learning Framework ◽  
Specular Reflections ◽  
Clinical Purpose ◽  
Novel Model ◽  
Opening Up ◽  
Surgical Videos

Specular reflections have always been undesirable when processing endoscope vision for clinical purpose. Scene afflicted with strong specular reflection could result in visual confusion for the operation of surgical robot. In this paper, we propose a novel model based on deep learning framework, known as Surgical Fix Deep Neural Network (SFDNN). This model can effectively detect and fix the reflection points in different surgical videos hence opening up a whole new approach in handling undesirable specular reflections.

scholarly journals Specular Reflections Detection and Removal Based on Deep Neural Network for Endoscope Images

2021 ◽  
Author(s):  
Tianyu Liu ◽  
chongyu wang ◽  
Junyu Chang ◽  
Liangjing Yang
Keyword(s):  
Neural Network ◽  
Deep Neural Network ◽  
Specular Reflection ◽  
New Approach ◽  
Learning Framework ◽  
Specular Reflections ◽  
Clinical Purpose ◽  
Novel Model ◽  
Opening Up ◽  
Surgical Videos

Specular reflections have always been undesirable when processing endoscope vision for clinical purpose. Scene afflicted with strong specular reflection could result in visual confusion for the operation of surgical robot. In this paper, we propose a novel model based on deep learning framework, known as Surgical Fix Deep Neural Network (SFDNN). This model can effectively detect and fix the reflection points in different surgical videos hence opening up a whole new approach in handling undesirable specular reflections.

Deep Neural Network Classification of I-123 Ioflupane SPECT

2020 ◽  
Author(s):  
David T. Wang ◽  
Brady Williamson ◽  
Thomas Eluvathingal ◽  
Bruce Mahoney ◽  
Jennifer Scheler
Keyword(s):  
Neural Network ◽  
Deep Neural Network ◽  
Neural Network Classification
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