  We propose a new approach, called as functional deep neural network (FDNN),
for classifying multi-dimensional functional data. Specifically, a deep neural
network is trained based on the principle components of the training data which
shall be used to predict the class label of a future data function. Unlike the
popular functional discriminant analysis approaches which rely on Gaussian
assumption, the proposed FDNN approach applies to general non-Gaussian
multi-dimensional functional data. Moreover, when the log density ratio
possesses a locally connected functional modular structure, we show that FDNN
achieves minimax optimality. The superiority of our approach is demonstrated
through both simulated and real-world datasets.
