Recurrent neural networks as optimal mesh refinement strategies

DOI10.1016/j.camwa.2021.05.018OpenAlexW3169729079MaRDI QIDQ2047561

Jan Bohn, Michael Feischl

Publication date: 20 August 2021

Published in: Computers \& Mathematics with Applications (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1909.04275

zbMATH Keywords

neural networksadaptive mesh refinementpartial differential equationsoptimal convergence

Mathematics Subject Classification ID

Boundary value problems for second-order elliptic equations (35J25) Error bounds for boundary value problems involving PDEs (65N15) Stability and convergence of numerical methods for boundary value problems involving PDEs (65N12) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs (65N50)

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Uses Software

• Evolino

Cites Work

• Unnamed Item
• Unnamed Item
• Axioms of adaptivity
• Counting symmetry classes of dissections of a convex regular polygon
• Stochastic recursive algorithms for optimization. Simultaneous perturbation methods
• Convergence of a standard adaptive nonconforming finite element method with optimal complexity
• Adaptive finite element methods with convergence rates
• Convergence and complexity of arbitrary order adaptive mixed element methods for the Poisson equation
• Optimal adaptive nonconforming FEM for the Stokes problem
• Quasi-optimal convergence rates for adaptive boundary element methods with data approximation. I: Weakly-singular integral equation
• On existence and uniqueness properties for solutions of stochastic fixed point equations
• Error bounds for approximations with deep ReLU networks
• Quasi-optimal convergence rates for adaptive boundary element methods with data approximation. II: Hyper-singular integral equation
• Adaptive boundary element methods with convergence rates
• Decay rates of adaptive finite elements with Dörfler marking
• A natural adaptive nonconforming FEM of quasi-optimal complexity
• Optimality of a standard adaptive finite element method
• Quasi-optimal Convergence Rate for an Adaptive Boundary Element Method
• The Adaptive Nonconforming FEM for the Pure Displacement Problem in Linear Elasticity is Optimal and Robust
• Convergence and optimality of adaptive mixed finite element methods
• A Convergent Nonconforming Adaptive Finite Element Method with Quasi-Optimal Complexity
• Quasi-Optimality of Adaptive Nonconforming Finite Element Methods for the Stokes Equations
• Quasioptimal cardinality of AFEM driven by nonresidual estimators
• Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method
• Optimality of a Standard Adaptive Finite Element Method for the Stokes Problem
• Efficiency and Optimality of Some Weighted-Residual Error Estimator for Adaptive 2D Boundary Element Methods
• The completion of locally refined simplicial partitions created by bisection
• Adaptive FEM with Optimal Convergence Rates for a Certain Class of Nonsymmetric and Possibly Nonlinear Problems
• Instance optimality of the adaptive maximum strategy