A Numerical Approach for the Fractional Laplacian via Deep Neural Networks

In this paper we consider the linear fractional elliptic problem with Dirichlet boundary conditions on a bounded and convex domain D of , with \(d \ge 2\) . This work is devoted to the approximations of such a PDE solutions via deep neural networks, even if the solution does not have an analytical expression. We have used that the PDE solution has a stochastic representation, and therefore we can use Monte Carlo approximations to generate train data related to the solution. With such a train data we perform a stochastic gradient descent (SGD) like algorithm to find a deep neural network that approximates the PDE solution in the whole domain D. The DNN found will be considered to have a fixed number of hidden layers and neurons per hidden layer. Additionally, we provide four numerical examples with different settings on the boundary condition and source term to test the efficiency of the approximation given by our algorithm, and each example will be studied for many values of \(\alpha \in (1,2)\) and \(d \ge 2\) .