A new approach for the fractional Laplacian via deep neural networks
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The fractional Laplacian has been strongly studied during past decades. 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 and elliptic in a bounded domain D ⊂ℝ^d. 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 can be approximated using deep neural networks. We also check that this approximation does not suffer from the curse of dimensionality.
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