A deep neural network-based numerical method for approximating the fractional Laplacian

[Submitted on 30 Aug 2023]

Download a PDF of the paper titled “A numerical approach for the fractional Laplacian via deep neural networks” by Nicolás Valenzuela

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Abstract: We investigate the solution of the fractional elliptic problem with Dirichlet boundary conditions on a bounded and convex domain D in ℝ^d, where d ≥ 2. Our approach involves using a stochastic gradient descent algorithm to approximate the solution of the fractional problem using Deep Neural Networks. We also provide four numerical examples to test the efficiency of our algorithm, considering different values of α in the range (1,2) and d ≥ 2.