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

TL;DR

This paper introduces a generalized neural-net-induced Gaussian process (NNGP) model that combines deep neural network expressivity with Gaussian process uncertainty quantification, applied to function approximation and PDE solving.

Contribution

It extends NNGP to include more hyperparameters, trains via maximum likelihood, and develops an analytical covariance formula for deep NNs with error-function nonlinearities.

Findings

For smooth functions, NNGP matches GP accuracy and outperforms deep NN.

For non-smooth functions, NNGP surpasses GP and is comparable or better than deep NN.

The generalized NNGP effectively solves PDEs with high accuracy.

Abstract

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.