Error assessment of an adaptive finite elements -- neural networks method for an elliptic parametric PDE
DOI10.1016/j.cma.2024.116784OpenAlexW4391285825MaRDI QIDQ6202970
Alexandre Caboussat, Marco Picasso, Maude Girardin
Publication date: 26 March 2024
Published in: Computer Methods in Applied Mechanics and Engineering (Search for Journal in Brave)
Full work available at URL: https://doi.org/10.1016/j.cma.2024.116784
Artificial neural networks and deep learning (68T07) Monte Carlo methods (65C05) Error bounds for boundary value problems involving PDEs (65N15) 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)
Cites Work
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Data driven approximation of parametrized PDEs by reduced basis and neural networks
- A posteriori error estimators for convection-diffusion equations
- Approximation and estimation bounds for artificial neural networks
- A posteriori error estimation in finite element analysis
- Non-intrusive reduced order modeling of nonlinear problems using neural networks
- Provable approximation properties for deep neural networks
- The Deep Ritz Method: a deep learning-based numerical algorithm for solving variational problems
- Numerical solution of the parametric diffusion equation by deep neural networks
- A theoretical analysis of deep neural networks and parametric PDEs
- Active training of physics-informed neural networks to aggregate and interpolate parametric solutions to the Navier-Stokes equations
- Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEs
- Optimal approximation of piecewise smooth functions using deep ReLU neural networks
- Nonparametric regression using deep neural networks with ReLU activation function
- Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders
- 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
- PROPER ORTHOGONAL DECOMPOSITION AND ITS APPLICATIONS—PART I: THEORY
- Certified Reduced Basis Methods for Parametrized Partial Differential Equations
- A simple error estimator and adaptive procedure for practical engineerng analysis
- Deep learning in high dimension: Neural network expression rates for generalized polynomial chaos expansions in UQ
- Optimization Methods for Large-Scale Machine Learning
- Error estimates for DeepONets: a deep learning framework in infinite dimensions
- Physics Informed Neural Networks (PINNs) For Approximating Nonlinear Dispersive PDEs
- Neural network approximation
- A New Certified Hierarchical and Adaptive RB-ML-ROM Surrogate Model for Parametrized PDEs
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