Core Concepts
Neural networks are used to approximate partial differential equations with local phenomena, introducing adaptive finite element interpolated neural networks.
Abstract
Introduction to the use of neural networks for PDE approximation.
Challenges in approximating PDEs with sharp gradients and singularities.
Methodology of ℎ-adaptive finite element interpolated neural networks.
Numerical analysis confirming effectiveness in capturing sharp gradients and singularities.
Comparison with traditional methods like PINNs and FEINNs.
Stats
The method relies on the interpolation of a neural network onto a finite element space.
Automatic mesh adaptation is performed based on error indicators till accuracy is reached.
Quotes
"The proposed methodology can be applied to indefinite and nonsymmetric problems."
"The training relies on a gradient-based optimization of a loss function based on the norm of the finite element residual."