  Despite their widespread success, the application of deep neural networks to
functional data remains scarce today. The infinite dimensionality of functional
data means standard learning algorithms can be applied only after appropriate
dimension reduction, typically achieved via basis expansions. Currently, these
bases are chosen a priori without the information for the task at hand and thus
may not be effective for the designated task. We instead propose to adaptively
learn these bases in an end-to-end fashion. We introduce neural networks that
employ a new Basis Layer whose hidden units are each basis functions themselves
implemented as a micro neural network. Our architecture learns to apply
parsimonious dimension reduction to functional inputs that focuses only on
information relevant to the target rather than irrelevant variation in the
input function. Across numerous classification/regression tasks with functional
data, our method empirically outperforms other types of neural networks, and we
prove that our approach is statistically consistent with low generalization
error. Code is available at: \url{https://github.com/jwyyy/AdaFNN}.
