Results 161 to 170 of about 1,001,967 (186)

Kernel Methods in the Deep Ritz framework: Theory and practice

CoRR
In this contribution, kernel approximations are applied as ansatz functions within the Deep Ritz method. This allows to approximate weak solutions of elliptic partial differential equations with weak enforcement of boundary conditions using Nitsche's ...
H. Kleikamp, T. Wenzel
semanticscholar   +3 more sources

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Deep Ritz Method for Elliptical Multiple Eigenvalue Problems

Journal of Scientific Computing
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Xia Ji, Yuling Jiao, Xiliang Lu, Pengcheng Song, Fengru Wang   +4 more
semanticscholar   +2 more sources

Generalization Error in the Deep Ritz Method with Smooth Activation Functions

Communications in Computational Physics
The manuscript presents an in-depth theoretical analysis of the Deep Ritz method (DRM) and addresses the generalization error inherent in this deep learning paradigm for solving partial differential equations (PDEs), focusing specifically on the Poisson equation.
Janne Siipola
semanticscholar   +4 more sources

Deep Rayleigh-Ritz method for elastic local buckling analysis of cold-formed steel columns

Structures
Yan Lu, Bo Ren, Bin Wu, Ying Yang, Bin Li   +4 more
semanticscholar   +2 more sources

Erratum to “A priori and a posteriori error estimates for the Deep Ritz method applied to the Laplace and Stokes problem” [J. Comput. Appl. Math. 421 (2023) 114845 ]

Journal of Computational and Applied Mathematics
P. Minakowski, T. Richter
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Analysis of Deep Ritz Methods for Semilinear Elliptic Equations

Numerical Mathematics: Theory, Methods and Applications
Summary: In this paper, we propose a method for solving semilinear elliptical equations using a ResNet with ReLU\(^2\) activations. Firstly, we present a comprehensive formulation based on the penalized variational form of the elliptical equations. We then apply the Deep Ritz Method, which works for a wide range of equations.
Chen, Mo, Jiao, Yuling, Lu, Xiliang, Song, Pengcheng, Wang, Fengru, Yang, Jerry Zhijian   +5 more
openaire   +2 more sources

The deep Ritz methods for the elastic contact problems with the Signorini, Tresca-friction, Coulomb-friction, and non-convex non-smooth subgradient-type boundary conditions

Japan Journal of Industrial and Applied Mathematics
Tongtong Wang, Longteng Hu, Guanyu Zhou, Ping Zeng, Yichen Ren   +4 more
semanticscholar   +2 more sources

Neural numerical homogenization based on Deep Ritz corrections

arXiv.org
Numerical homogenization methods aim at providing appropriate coarse-scale approximations of solutions to (elliptic) partial differential equations that involve highly oscillatory coefficients.
M. Elasmi, Felix Krumbiegel, Roland Maier   +2 more
semanticscholar   +1 more source

Deep Domain Decomposition Method for Solving the Variational Inequality Problems

Journal of Big Data and Computing
By integrating physics-informed neural network (PINN) techniques with domain decomposition method, a deep domain decomposition method is presented for solving elliptic variational inequality problems.
Yiyang Wang, Qijian Zhou, S. Deng, Chenliang Li   +3 more
semanticscholar   +1 more source

Error Analysis of the Deep Mixed Residual Method for High-order Elliptic Equations

arXiv.org
This paper presents an a priori error analysis of the Deep Mixed Residual method (MIM) for solving high-order elliptic equations with non-homogeneous boundary conditions, including Dirichlet, Neumann, and Robin conditions.
M. Bai, Jingrun Chen, Rui Du, Z. Sun
semanticscholar   +1 more source