  In this work, we propose a deep neural network method to perform
nonparametric regression for functional data. The proposed estimators are based
on sparsely connected deep neural networks with ReLU activation function. By
properly choosing network architecture, our estimator achieves the optimal
nonparametric convergence rate in empirical norm. Under certain circumstances
such as trigonometric polynomial kernel and a sufficiently large sampling
frequency, the convergence rate is even faster than root-$n$ rate. Through
Monte Carlo simulation studies we examine the finite-sample performance of the
proposed method. Finally, the proposed method is applied to analyze positron
emission tomography images of patients with Alzheimer disease obtained from the
Alzheimer Disease Neuroimaging Initiative database.
