Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control

DOI10.1137/20M1366587zbMath1486.65238arXiv2105.14094OpenAlexW3182298826MaRDI QIDQ5005011

Justin Dong, Mark Ainsworth

Publication date: 4 August 2021

Published in: SIAM Journal on Scientific Computing (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2105.14094

zbMATH Keywords

neural networkpartial differential equationsa posteriori error estimate

Mathematics Subject Classification ID

Artificial neural networks and deep learning (68T07) Variational methods applied to PDEs (35A15) 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) PDEs on graphs and networks (ramified or polygonal spaces) (35R02)

Related Items (7)

Unlocking the secrets of locking: finite element analysis in planar linear elasticity ⋮ Galerkin neural network approximation of singularly-perturbed elliptic systems ⋮ Neural control of discrete weak formulations: Galerkin, least squares \& minimal-residual methods with quasi-optimal weights ⋮ Exponential ReLU neural network approximation rates for point and edge singularities ⋮ ReLU neural network Galerkin BEM ⋮ Multi-level neural networks for accurate solutions of boundary-value problems ⋮ DNN-MG: a hybrid neural network/finite element method with applications to 3D simulations of the Navier-Stokes equations

Uses Software

• Adam
• DGM
• hp-VPINNs

Cites Work

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• Weak adversarial networks for high-dimensional partial differential equations
• The Deep Ritz Method: a deep learning-based numerical algorithm for solving variational problems
• DGM: a deep learning algorithm for solving partial differential equations
• Topological properties of the set of functions generated by neural networks of fixed size
• Adaptive activation functions accelerate convergence in deep and physics-informed neural networks
• Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• Relu Deep Neural Networks and Linear Finite Elements
• Neural‐network‐based approximations for solving partial differential equations
• Solving high-dimensional partial differential equations using deep learning
• The Mathematical Theory of Finite Element Methods
• Approximation by superpositions of a sigmoidal function

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