A new approach for the fractional Laplacian via deep neural networks

2022/05/11 by Nicolás Valenzuela, Valenzuela, Nicolás · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2205.05229

openalex publication_date 2022/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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, intensively 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.

Cited by

• Error analysis for the deep Kolmogorov method

Related

• A numerical approach for the fractional Laplacian via deep neural networks
• Fractional Deep Neural Network via Constrained Optimization
• Parametric Complexity Bounds for Approximating PDEs with Neural Networks
• Solving the Poisson Equation with Dirichlet data by shallow ReLU^α-networks: A regularity and approximation perspective
• Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions