  We introduce a new class of non-linear models for functional data based on
neural networks. Deep learning has been very successful in non-linear modeling,
but there has been little work done in the functional data setting. We propose
two variations of our framework: a functional neural network with continuous
hidden layers, called the Functional Direct Neural Network (FDNN), and a second
version that utilizes basis expansions and continuous hidden layers, called the
Functional Basis Neural Network (FBNN). Both are designed explicitly to exploit
the structure inherent in functional data. To fit these models we derive a
functional gradient based optimization algorithm. The effectiveness of the
proposed methods in handling complex functional models is demonstrated by
comprehensive simulation studies and real data examples.

Machine Learning;Artificial Intelligence;Methodology;Machine Learning