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.

翻译：我们提出了一类基于神经网络的非线性函数对函数回归模型，用于函数型数据分析。我们构建了一个包含连续神经元的隐藏层（称为连续隐藏层）框架，用于函数型响应建模，并给出了两种模型拟合策略：函数直接神经网络（FDNN）和函数基神经网络（FBNN）。这两种方法都明确针对函数型数据的内在结构而设计，能够捕捉函数型预测变量与函数型响应之间存在的复杂关系。我们通过推导函数梯度来拟合这些模型，并应用正则化技术以获得更简洁的结果。通过广泛的仿真研究和实际数据案例，我们展示了所提方法在处理复杂函数模型方面的强大能力与灵活性。