Neural-net-induced Gaussian process regression for function approximation and PDE solution

Guofei Pang

doi:10.1016/j.jcp.2019.01.045

Return

Neural-net-induced Gaussian process regression for function approximation and PDE solution

2019-05-01

Guofei Pang

Liu Yang

George Em Karniadakis *

DOI：10.1016/j.jcp.2019.01.045

Original

Save

View PDF

Abstract

En 中文

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. (C) 2019 Elsevier Inc. All rights reserved.

Keywords:

NN-induced Gaussian process

Neural network

Machine learning

Partial differential equation

Uncertainty quantification

AI Summary

Key information extracted from the uploaded paper, including a brief overview, abstract, background, key highlights, visual analysis, and future outlook.