A numerical approach for the fractional Laplacian via deep neural networks
This work addresses computational challenges in fractional PDEs for researchers in applied mathematics and scientific computing, but it appears incremental as it applies existing deep learning methods to a specific problem.
The paper tackles solving fractional elliptic problems with Dirichlet boundary conditions using deep neural networks, achieving efficient approximations through a stochastic gradient descent algorithm and testing across multiple numerical examples with varying parameters.
We consider the fractional elliptic problem with Dirichlet boundary conditions on a bounded and convex domain $D$ of $\mathbb{R}^d$, with $d \geq 2$. In this paper, we perform a stochastic gradient descent algorithm that approximates the solution of the fractional problem via Deep Neural Networks. Additionally, we provide four numerical examples to test the efficiency of the algorithm, and each example will be studied for many values of $α\in (1,2)$ and $d \geq 2$.