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 Gaussian processes (GPs). We generalize the current NNGP to first include a larger number of hyperparameters and subsequently train the model by maximum likelihood estimation. Unlike previous works on NNGP that targeted classification, here we apply the generalized NNGP to function approximation and to solving partial differential equations (PDEs). Specifically, we develop an analytical iteration formula to compute the covariance function of GP induced by deep NN with an error-function nonlinearity. We compare the performance of the generalized NNGP for function approximations and PDE solutions with those of GPs and fully-connected NNs. We observe that for smooth functions the generalized NNGP can yield the same order of accuracy with GP, while both NNGP and GP outperform deep NN. For non-smooth functions, the generalized NNGP is superior to GP and comparable or superior to deep NN.
06/22/2018 ∙ by Guofei Pang, et al. ∙ 0 ∙ share
Decoupled fractional Laplacian wave equation can describe the seismic wave propagation in attenuating media. Fourier pseudospectral implementations, which solve the equation in spatial frequency domain, are the only existing methods for solving the equation. For the earth media with curved boundaries, the pseudospectral methods could be less attractive to handle the irregular computational domains. In the paper, we propose a radial basis function collocation method that can easily tackle the irregular domain problems. Unlike the pseudospectral methods, the proposed method solves the equation in physical variable domain. The directional fractional Laplacian is chosen from varied definitions of fractional Laplacian. Particularly, the vector Grünwald-Letnikov formula is employed to approximate fractional directional derivative of radial basis function. The convergence and stability of the method are numerically investigated by using the synthetic solution and the long-time simulations, respectively. The method's flexibility is studied by considering homogeneous and multi-layer media having regular and irregular geometric boundaries.
01/03/2018 ∙ by Yiran Xu, et al. ∙ 0 ∙ share
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