A new approach for the fractional Laplacian via deep neural networks

. The fractional Laplacian has been strongly studied during past decades, see e.g. [9]. In this paper we present a different approach for the associated Dirichlet problem, using recent deep learning techniques. In fact, recently certain parabolic PDEs with a stochastic representation have been understood via neural networks, overcoming the so-called curse of dimensionality . Among these equations one can find parabolic ones in R d , see [20], and elliptic in a bounded domain D ⊂ R d , see [15]. In this paper we consider the Dirichlet problem for the fractional Laplacian with exponent α ∈ (1 , 2). We show that its solution, represented in a stochastic fashion by [22], can be approximated using deep neural networks. We also check that this approximation does not suffer from the curse of dimensionality. The spirit of our proof follows the ideas in [15], with important variations due to the nonlocal nature of the fractional Laplacian; in particular, the stochastic representation is given by an α -stable isotropic L´evy process, and not given by a standard Brownian motion.