  We introduce a new class of non-linear function-on-function regression models
for functional data using neural networks. We propose a framework using a
hidden layer consisting of continuous neurons, called a continuous hidden
layer, for functional response modeling and give two model fitting strategies,
Functional Direct Neural Network (FDNN) and Functional Basis Neural Network
(FBNN). Both are designed explicitly to exploit the structure inherent in
functional data and capture the complex relations existing between the
functional predictors and the functional response. We fit these models by
deriving functional gradients and implement regularization techniques for more
parsimonious results. We demonstrate the power and flexibility of our proposed
method in handling complex functional models through extensive simulation
studies as well as real data examples.
