A deep Fourier residual method for solving PDEs using neural networks

DOI10.1016/j.cma.2022.115850OpenAlexW4312222006MaRDI QIDQ2683430

Ignacio Muga, Jamie M. Taylor, David Pardo

Publication date: 10 February 2023

Published in: Computer Methods in Applied Mechanics and Engineering (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2210.14129

zbMATH Keywords

neural networksdeep learningnumerical PDEsFourier methods

Mathematics Subject Classification ID

Biology and other natural sciences (92-XX) Numerical analysis (65-XX)

Uses Software

• DiffSharp
• Adam
• DeepXDE
• DGM
• DeepONet

Cites Work

• Unnamed Item
• Unnamed Item
• The finite cell method: a review in the context of higher-order structural analysis of CAD and image-based geometric models
• The finite cell method for three-dimensional problems of solid mechanics
• Distributed Lagrange multipliers based on fictitious domain method for second order elliptic problems
• Functional analysis, Sobolev spaces and partial differential equations
• On goal-oriented error estimation for elliptic problems: Application to the control of pointwise errors
• Deep neural networks motivated by partial differential equations
• DGM: a deep learning algorithm for solving partial differential equations
• A machine-learning minimal-residual (ML-MRes) framework for goal-oriented finite element discretizations
• Solving PDEs by variational physics-informed neural networks: an a posteriori error analysis
• When and why PINNs fail to train: a neural tangent kernel perspective
• On quadrature rules for solving partial differential equations using neural networks
• The finite cell method with least squares stabilized Nitsche boundary conditions
• Conservative physics-informed neural networks on discrete domains for conservation laws: applications to forward and inverse problems
• Automatically adaptive, stabilized finite element method via residual minimization for heterogeneous, anisotropic advection-diffusion-reaction problems
• An energy approach to the solution of partial differential equations in computational mechanics via machine learning: concepts, implementation and applications
• Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• A fictitious domain approach with spread interface for elliptic problems with general boundary conditions
• A finite element based deep learning solver for parametric PDEs
• A physics-informed variational DeepONet for predicting crack path in quasi-brittle materials
• Spectral Methods
• Large-Scale Machine Learning with Stochastic Gradient Descent
• IMMERSED BOUNDARY METHODS
• The immersed boundary method
• Universal approximation bounds for superpositions of a sigmoidal function
• DeepXDE: A Deep Learning Library for Solving Differential Equations
• On the Convergence of Physics Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs
• Computing with hp-ADAPTIVE FINITE ELEMENTS
• Approximation by superpositions of a sigmoidal function

This page was built for publication: A deep Fourier residual method for solving PDEs using neural networks