Estimation of the mean function of functional data via deep neural networks

In this work, we propose a deep neural networks-based method to perform non-parametric regression for functional data. The proposed estimators are based on sparsely connected deep neural networks with rectifier linear unit (ReLU) activation function. We provide the convergence rate of the proposed deep neural networks estimator in terms of the empirical norm. Through Monte Carlo simulation studies, we examine the finite sample performance of the proposed method. Finally, the proposed method is applied to analyse positron emission tomography images of patients with Alzheimer's disease obtained from the Alzheimer Disease Neuroimaging Initiative database.