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Neural-net-induced Gaussian process (NNGP) regression inherits both the highexpressivity of deep neural networks (deep NNs) as well as the uncertaintyquantification property of Gaussian processes (GPs). We generalize the currentNNGP to first include a larger number of hyperparameters and subsequently trainthe model by maximum likelihood estimation. Unlike previous works on NNGP thattargeted classification, here we apply the generalized NNGP to functionapproximation and to solving partial differential equations (PDEs).Specifically, we develop an analytical iteration formula to compute thecovariance function of GP induced by deep NN with an error-functionnonlinearity. We compare the performance of the generalized NNGP for functionapproximations and PDE solutions with those of GPs and fully-connected NNs. Weobserve that for smooth functions the generalized NNGP can yield the same orderof accuracy with GP, while both NNGP and GP outperform deep NN. For non-smoothfunctions, the generalized NNGP is superior to GP and comparable or superior todeep NN.

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