The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems [PDF]
Communications in Mathematics and Statistics, 2018 We propose a deep learning based method, the Deep Ritz Method, for numerically solving variational problems, particularly the ones that arise from partial differential equations. The Deep Ritz method is naturally nonlinear, naturally adaptive and has the potential to work in rather high dimensions.
Weinan E, Bing Yu
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SIAM Journal on Scientific Computing, 2022 In this paper, we propose a novel method for solving high-dimensional spectral fractional Laplacian equations. Using the Caffarelli-Silvestre extension, the $d$-dimensional spectral fractional equation is reformulated as a regular partial differential equation of dimension $d+1$.
Yiqi Gu, Michael K. Ng
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Neutrino-induced Muon Fluxes from Neutralino Annihilations in the Sun and in the Earth [PDF]
, 1995 The flux of neutrino-induced muons at the surface of the Earth is calculated from injection of neutralino annihilation products in the core of the Sun and the Earth.
Bahcall +9 more
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Solving Elliptic Problems with Singular Sources Using Singularity Splitting Deep Ritz Method
SIAM Journal on Scientific Computing, 2023 28 ...
Tianhao Hu, Bangti Jin, Zhi Zhou
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Observations and models for needle-tissue interactions [PDF]
, 2009 The asymmetry of a bevel-tip needle results in the needle naturally bending when it is inserted into soft tissue. In this study we present a mechanics-based model that calculates the deflection of the needle embedded in an elastic medium.
Misra, Sarthak +4 more
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A Deep Double Ritz Method (D2rm) for Solving Partial Differential Equations Using Neural Networks
Computer Methods in Applied Mechanics and Engineering, 2022 Residual minimization is a widely used technique for solving Partial Differential Equations in variational form. It minimizes the dual norm of the residual, which naturally yields a saddle-point (min-max) problem over the so-called trial and test spaces. In the context of neural networks, we can address this min-max approach by employing one network to
Uriarte, Carlos +3 more
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Error Estimates for the Deep Ritz Method with Boundary Penalty
Proceedings of Mathematical and Scientific Machine Learning, 2021 We estimate the error of the Deep Ritz Method for linear elliptic equations. For Dirichlet boundary conditions, we estimate the error when the boundary values are imposed through the boundary penalty method. Our results apply to arbitrary sets of ansatz functions and estimate the error in dependence of the optimization accuracy, the approximation ...
Müller, Johannes, Zeinhofer, Marius
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New modeling of the Vostok ice flow line and implication for the glaciological chronology of the Vostok ice core [PDF]
, 2004 International audienceWe have used new spaceborne (elevation) and airborne (ice thickness) data to constrain a 2D1/2 model of snow accumulation and ice flow along the Ridge B‐Vostok station ice flow line (East Antarctica). We show that new evaluations of
Arnaud +30 more
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The confinement effect in spherical inhomogeneous quantum dots and stability of excitons
AIP Advances, 2017 We investigate in this work the quantum confinement effect of exciton in spherical inhomogeneous quantum dots IQDs. The spherical core is enveloped by two shells. The inner shell is a semiconductor characterized by a small band-gap.
F. Benhaddou, I. Zorkani, A. Jorio
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Some results on thermal stress of layered plates and shells by using Unified Formulation [PDF]
, 2013 This work presents some results on two-dimensional modelling of thermal stress problems in multilayered structures. The governing equations are written by referring to the Unified Formulation (UF) introduced by the first author.
Argyris J. +19 more
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