scholarly journals The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems

2018 ◽  
Vol 6 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Weinan E ◽  
Bing Yu
Keyword(s):  
Deep Learning ◽  
Numerical Algorithm ◽  
Ritz Method ◽  
Variational Problems

scholarly journals Solving Non-Linear Fractional Variational Problems Using Jacobi Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 224 ◽  
Author(s):  
Harendra Singh ◽  
Rajesh Pandey ◽  
Hari Srivastava
Keyword(s):  
Chebyshev Polynomials ◽  
Jacobi Polynomials ◽  
Ritz Method ◽  
Variational Problems ◽  
Algebraic Equations ◽  
The Third ◽  
Fractional Variational Problems ◽  
Nonlinear Algebraic Equations ◽  
Non Linear ◽  
Fourth Kind

The aim of this paper is to solve a class of non-linear fractional variational problems (NLFVPs) using the Ritz method and to perform a comparative study on the choice of different polynomials in the method. The Ritz method has allowed many researchers to solve different forms of fractional variational problems in recent years. The NLFVP is solved by applying the Ritz method using different orthogonal polynomials. Further, the approximate solution is obtained by solving a system of nonlinear algebraic equations. Error and convergence analysis of the discussed method is also provided. Numerical simulations are performed on illustrative examples to test the accuracy and applicability of the method. For comparison purposes, different polynomials such as 1) Shifted Legendre polynomials, 2) Shifted Chebyshev polynomials of the first kind, 3) Shifted Chebyshev polynomials of the third kind, 4) Shifted Chebyshev polynomials of the fourth kind, and 5) Gegenbauer polynomials are considered to perform the numerical investigations in the test examples. Further, the obtained results are presented in the form of tables and figures. The numerical results are also compared with some known methods from the literature.

A Development in the Extended Ritz Method for Solving a General Class of Fractional Variational Problems

2019 ◽  
Vol 15 (2) ◽  
Author(s):  
Ali Lotfi
Keyword(s):  
General Class ◽  
Fractional Derivatives ◽  
Ritz Method ◽  
Approximate Solutions ◽  
Variational Problems ◽  
Polynomial Functions ◽  
Great Flexibility ◽  
Fractional Polynomial ◽  
Fractional Variational Problems ◽  
Free Parameters

Abstract In this paper, based on the idea of the extended Ritz method, we introduce an efficient approximate technique for solving a general class of fractional variational problems. In the discussed problem, the fractional derivatives are considered in the Caputo sense. First, we introduce a family of fractional polynomial functions with a free parameter in the exponent. With the aid of the presented fractional polynomials, we construct a family of functions with free parameters, which provides the extended Ritz method with a great flexibility in searching for the approximate solution of the problem. The approximate solutions satisfy all the initial and the boundary conditions of the problem. The convergence of the method is analytically studied and some test examples are included to demonstrate the superiority of the new technique over the ordinary Ritz method.

Exponentially Accurate Rayleigh–Ritz Method for Fractional Variational Problems

2015 ◽  
Vol 10 (5) ◽  
Author(s):  
Zhi Mao ◽  
Aiguo Xiao ◽  
Dongling Wang ◽  
Zuguo Yu ◽  
Long Shi
Keyword(s):  
Exponential Decay ◽  
Numerical Approximation ◽  
Exponential Convergence ◽  
Ritz Method ◽  
Variational Problems ◽  
Basis Functions ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Fractional Variational Problems ◽  
Sturm Liouville

A high accurate Rayleigh–Ritz method is developed for solving fractional variational problems (FVPs). The Jacobi poly-fractonomials proposed by Zayernouri and Karniadakis (2013, “Fractional Sturm–Liouville Eigen-Problems: Theory and Numerical Approximation,” J. Comput. Phys., 252(1), pp. 495–517.) are chosen as basis functions to approximate the true solutions, and the Rayleigh–Ritz technique is used to reduce FVPs to a system of algebraic equations. This method leads to exponential decay of the errors, which is superior to the existing methods in the literature. The fractional variational errors are discussed. Numerical examples are given to illustrate the exponential convergence of the method.

Deep Learning

2009 ◽  
Author(s):  
Stellan Ohlsson
Keyword(s):  
Deep Learning

Intelligence artificielle : le futur de l’Orthodontie ?

2019 ◽  
Vol 53 (3) ◽  
pp. 281-294
Author(s):  
Jean-Michel Foucart ◽  
Augustin Chavanne ◽  
Jérôme Bourriau
Keyword(s):  
Big Data ◽  
Deep Learning ◽  
Intelligence Artificielle ◽  
Set Up

Nombreux sont les apports envisagés de l’Intelligence Artificielle (IA) en médecine. En orthodontie, plusieurs solutions automatisées sont disponibles depuis quelques années en imagerie par rayons X (analyse céphalométrique automatisée, analyse automatisée des voies aériennes) ou depuis quelques mois (analyse automatique des modèles numériques, set-up automatisé; CS Model +, Carestream Dental™). L’objectif de cette étude, en deux parties, est d’évaluer la fiabilité de l’analyse automatisée des modèles tant au niveau de leur numérisation que de leur segmentation. La comparaison des résultats d’analyse des modèles obtenus automatiquement et par l’intermédiaire de plusieurs orthodontistes démontre la fiabilité de l’analyse automatique; l’erreur de mesure oscillant, in fine, entre 0,08 et 1,04 mm, ce qui est non significatif et comparable avec les erreurs de mesures inter-observateurs rapportées dans la littérature. Ces résultats ouvrent ainsi de nouvelles perspectives quand à l’apport de l’IA en Orthodontie qui, basée sur le deep learning et le big data, devrait permettre, à moyen terme, d’évoluer vers une orthodontie plus préventive et plus prédictive.

Vollautomatische Quantifizierung der Volumina und Funktion des linken Ventrikels in der kardialen MRT: Evaluation eines Deep-Learning basierten Algorithmus

2020 ◽  
Author(s):  
B Böttcher ◽  
E Beller ◽  
A Busse ◽  
F Streckenbach ◽  
M Weber ◽  
...  
Keyword(s):  
Deep Learning

Primäre ZNS-Lymphome: Vollautomatisierte Detektion und Segmentierung auf multiparametrischen MRT-Datensätzen mittels Deep-Learning

2020 ◽  
Author(s):  
L Pennig ◽  
L Lourenco Caldeira ◽  
C Hoyer ◽  
L Görtz ◽  
R Shahzad ◽  
...  
Keyword(s):  
Deep Learning
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