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.

翻译：我们提出了一类基于神经网络的函数型数据非线性模型。深度学习在非线性建模领域取得了巨大成功，但在函数型数据场景下的研究尚显不足。我们提出了框架的两种变体：一种采用连续隐藏层的函数型神经网络，称为直接函数型神经网络（FDNN）；另一种结合基展开与连续隐藏层，称为基函数型神经网络（FBNN）。两种模型均明确设计用于挖掘函数型数据的固有结构。为拟合这些模型，我们推导出基于函数梯度的优化算法。通过全面的模拟研究与实际数据案例，验证了所提方法在处理复杂函数型模型方面的有效性。