Neural-net-induced Gaussian process regression for function approximation and PDE solution
This work addresses function approximation and PDE solving for computational science and engineering, offering an incremental improvement by extending NNGP to new tasks with enhanced training.
The authors tackled function approximation and PDE solving by generalizing neural-net-induced Gaussian process (NNGP) regression to include more hyperparameters and train via maximum likelihood, applying it beyond classification. They found that for smooth functions, generalized NNGP matches Gaussian process (GP) accuracy and both outperform deep neural networks, while for non-smooth functions, it surpasses GP and is comparable or better than deep neural networks.
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.