With the rapid development of deep learning in various fields of science and
technology, such as speech recognition, image classification, and natural
language processing, recently it is also widely applied in the functional data
analysis (FDA) with some empirical success. However, due to the infinite
dimensional input, we need a powerful dimension reduction method for functional
learning tasks, especially for the nonlinear functional regression. In this
paper, based on the idea of smooth kernel integral transformation, we propose a
functional deep neural network with an efficient and fully data-dependent
dimension reduction method. The architecture of our functional net consists of
a kernel embedding step: an integral transformation with a data-dependent
smooth kernel; a projection step: a dimension reduction by projection with
eigenfunction basis based on the embedding kernel; and finally an expressive
deep ReLU neural network for the prediction. The utilization of smooth kernel
embedding enables our functional net to be discretization invariant, efficient,
and robust to noisy observations, capable of utilizing information in both
input functions and responses data, and have a low requirement on the number of
discrete points for an unimpaired generalization performance. We conduct
theoretical analysis including approximation error and generalization error
analysis, and numerical simulations to verify these advantages of our
functional net.

利用平滑核积分变换的思想，我们提出了一种功能深度神经网络，其具有高效和完全依赖数据的降维方法。我们的功能网络的架构包括核嵌入步骤、投影步骤和表达深度 ReLU 神经网络，利用平滑核嵌入使其具有离散不变性、高效性和对噪声观测的鲁棒性，能够利用输入函数和响应数据的信息，对离散点数量的要求较低，具有良好的泛化性能。我们进行了理论分析和数值模拟来验证我们功能网络的这些优势。