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A numerical approach for the fractional Laplacian via deep neural networks

arXiv 2308.16272 ↗First posted 30 Aug 2023math.AP cs.LG cs.NAcited by 0

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Abstract (arXiv)

We consider the fractional elliptic problem with Dirichlet boundary conditions on a bounded and convex domain D of ℝd, with d ≥ 2. In this paper, we perform a stochastic gradient descent algorithm that approximates the solution of the fractional problem via Deep Neural Networks. Additionally, we provide four numerical examples to test the efficiency of the algorithm, and each example will be studied for many values of α ∈ (1,2) and d ≥ 2.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 12 of its 32 references that are in the map. First posted on arXiv 30 Aug 2023.

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References · 12

  1. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  2. 2017Solving high-dimensional partial differential equations using deep learningcited by 556
  3. 2017Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equationscited by 191
  4. 2017Machine Learning of Linear Differential Equations using Gaussian Processescited by 140
  5. 2019Deep Neural Network Approximation Theorycited by 93
  6. 2022Error estimates for physics informed neural networks approximating the Navier-Stokes equationscited by 63
  7. 2021Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEscited by 57
  8. 2018Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equationscited by 45
  9. 2019Deep learning observables in computational fluid dynamicscited by 38
  10. 2020Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networkscited by 27
  11. 2020Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditionscited by 25
  12. 2018Machine Learning of Space-Fractional Differential Equationscited by 12

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