Estimates of the Evolucion of Small Perturbations by Radial Spreading (Drain) of a Viscous Ring

2017 ◽  
Keyword(s):  
Small Perturbations ◽  
Radial Spreading

scholarly journals Quasi-stability and continuity of attractors for nonlinear system of wave equations

2021 ◽  
Vol 8 (1) ◽  
pp. 27-45
Author(s):  
M. M. Freitas ◽  
M. J. Dos Santos ◽  
A. J. A. Ramos ◽  
M. S. Vinhote ◽  
M. L. Santos
Keyword(s):  
Wave Equations ◽  
Coupled System ◽  
Global Attractors ◽  
Time Behavior ◽  
Exponential Attractors ◽  
Small Perturbations ◽  
Recent Approach ◽  
Nonlinear Coupled System ◽  
Long Time ◽  
External Forces

Abstract In this paper, we study the long-time behavior of a nonlinear coupled system of wave equations with damping terms and subjected to small perturbations of autonomous external forces. Using the recent approach by Chueshov and Lasiecka in [21], we prove that this dynamical system is quasi-stable by establishing a quasistability estimate, as consequence, the existence of global and exponential attractors is proved. Finally, we investigate the upper and lower semicontinuity of global attractors under autonomous perturbations.

scholarly journals Dynamics of Fourier Modes in Torus Generative Adversarial Networks

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 325
Author(s):  
Ángel González-Prieto ◽  
Alberto Mozo ◽  
Edgar Talavera ◽  
Sandra Gómez-Canaval
Keyword(s):  
Fourier Series ◽  
Generative Adversarial Networks ◽  
Learning Models ◽  
Training Process ◽  
Small Perturbations ◽  
Adversarial Networks ◽  
Novel Method ◽  
Truncated Fourier Series ◽  
Real Flow ◽  
Machine Learning Models

Generative Adversarial Networks (GANs) are powerful machine learning models capable of generating fully synthetic samples of a desired phenomenon with a high resolution. Despite their success, the training process of a GAN is highly unstable, and typically, it is necessary to implement several accessory heuristics to the networks to reach acceptable convergence of the model. In this paper, we introduce a novel method to analyze the convergence and stability in the training of generative adversarial networks. For this purpose, we propose to decompose the objective function of the adversary min–max game defining a periodic GAN into its Fourier series. By studying the dynamics of the truncated Fourier series for the continuous alternating gradient descend algorithm, we are able to approximate the real flow and to identify the main features of the convergence of GAN. This approach is confirmed empirically by studying the training flow in a 2-parametric GAN, aiming to generate an unknown exponential distribution. As a by-product, we show that convergent orbits in GANs are small perturbations of periodic orbits so the Nash equillibria are spiral attractors. This theoretically justifies the slow and unstable training observed in GANs.

Radial Spreading of Localized Corrosion-Induced Selective Leaching on α-Brass in Dilute NaCl Solution

CORROSION ◽  
10.5006/0611 ◽  
2013 ◽  
Vol 69 (5) ◽  
pp. 468-476 ◽  
Author(s):  
Mattias Forslund ◽  
Christofer Leygraf ◽  
Changjian Lin ◽  
Jinshan Pan
Keyword(s):  
Localized Corrosion ◽  
Selective Leaching ◽  
Nacl Solution ◽  
Radial Spreading
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