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 $\alpha \in (1,2)$ and $d \geq 2$.

翻译：考虑定义在有界凸区域$D \subset \mathbb{R}^d$（$d \geq 2$）上、带Dirichlet边界条件的分数阶椭圆问题。本文采用随机梯度下降算法，通过深度神经网络逼近分数阶问题的解。此外，我们提供了四个数值算例验证算法的有效性，每个算例均针对$\alpha \in (1,2)$和$d \geq 2$的多组参数进行研究。