A deep first-order system least squares method for solving elliptic PDEs

DOI10.1016/j.camwa.2022.11.014OpenAlexW4311989058MaRDI QIDQ2679352

Juan Pablo Borthagaray, Francisco M. Bersetche

Publication date: 19 January 2023

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

Full work available at URL: https://arxiv.org/abs/2204.07227

zbMATH Keywords

machine learningelliptic PDEsdeep learninghigh-dimensional PDEs

Mathematics Subject Classification ID

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) Numerical methods for partial differential equations, boundary value problems (65N99)

Uses Software

• Adam
• DGM

Cites Work

• Unnamed Item
• Weak adversarial networks for high-dimensional partial differential equations
• Numerical solution for high order differential equations using a hybrid neural network-optimization method
• The Deep Ritz Method: a deep learning-based numerical algorithm for solving variational problems
• DGM: a deep learning algorithm for solving partial differential equations
• Adaptive two-layer ReLU neural network. I: Best least-squares approximation
• Adaptive two-layer ReLU neural network. II: Ritz approximation to elliptic PDEs
• Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEs
• PFNN: a penalty-free neural network method for solving a class of second-order boundary-value problems on complex geometries
• MIM: a deep mixed residual method for solving high-order partial differential equations
• A mesh-free method using piecewise deep neural network for elliptic interface problems
• ReLU deep neural networks from the hierarchical basis perspective
• A mesh-free method for interface problems using the deep learning approach
• Error bounds for approximations with deep ReLU networks
• Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• Neural algorithm for solving differential equations
• Relu Deep Neural Networks and Linear Finite Elements
• Universal approximation bounds for superpositions of a sigmoidal function
• First-Order System Least Squares for Second-Order Partial Differential Equations: Part I
• Finite Neuron Method and Convergence Analysis
• On the Convergence of Physics Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs
• Approximation by superpositions of a sigmoidal function