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Stochastic Solution of Elliptic and Parabolic Boundary Value Problems for the Spectral Fractional Laplacian
Mamikon Gulian, Guofei Pang
We prove and implement stochastic solution (or Feynman-Kac) formulas for boundary value problems involving the spectral fractional Laplacian with nonzero Dirichlet boundary conditi…
fPINNs: Fractional Physics-Informed Neural Networks
Guofei Pang, Lu Lu, George Em Karniadakis
Physics-informed neural networks (PINNs) are effective in solving integer-order partial differential equations (PDEs) based on scattered and noisy data. PINNs employ standard feedf…
Neural-net-induced Gaussian process regression for function approximation and PDE solution
Guofei Pang, Liu Yang, George Em Karniadakis
Neural-net-induced Gaussian process (NNGP) regression inherits both the high expressivity of deep neural networks (deep NNs) as well as the uncertainty quantification property of G…
A new radial basis function collocation method based on the quasi-uniform nodes for 2D fractional wave equation
Yiran Xu, Jingye Li, Guofei Pang +3
We mainly concerned with a decoupled fractional Laplacian wave equation in this paper. A new time-space domain radial basis function (RBF) collocation method is introduced to solve…
Radial basis function collocation method for decoupled fractional Laplacian wave equations
Yiran Xu, Jingye Li, Guofei Pang +2
Decoupled fractional Laplacian wave equation can describe the seismic wave propagation in attenuating media. Fourier pseudospectral implementations, which solve the equation in spa…
What Is the Fractional Laplacian?
Anna Lischke, Guofei Pang, Mamikon Gulian +8
The fractional Laplacian in R^d has multiple equivalent characterizations. Moreover, in bounded domains, boundary conditions must be incorporated in these characterizations in math…