The regression of a functional response on a set of scalar predictors can be a challenging task, especially if there is a large number of predictors, or the relationship between those predictors and the response is nonlinear. In this work, we propose a solution to this problem: a feed-forward neural network (NN) designed to predict a functional response using scalar inputs. First, we transform the functional response to a finite-dimensional representation and construct an NN that outputs this representation. Then, we propose to modify the output of an NN via the objective function and introduce different objective functions for network training. The proposed models are suited for both regularly and irregularly spaced data, and a roughness penalty can be further applied to control the smoothness of the predicted curve. The difficulty in implementing both those features lies in the definition of objective functions that can be back-propagated. In our experiments, we demonstrate that our model outperforms the conventional function-on-scalar regression model in multiple scenarios while computationally scaling better with the dimension of the predictors.

翻译：函数型响应变量对一组标量预测变量的回归是一项具有挑战性的任务，尤其是在预测变量数量众多或两者关系呈非线性时。本文提出一种解决方案：设计一种利用标量输入预测函数型响应的前馈神经网络（NN）。首先，我们将函数型响应转换为有限维表示，并构建一个输出该表示的神经网络。随后，我们提出通过目标函数修改神经网络的输出，并引入不同的目标函数用于网络训练。所提出的模型适用于规则与不规则间隔的数据，并可进一步应用粗糙度惩罚项来控制预测曲线的平滑度。实现这两项功能的难点在于定义可反向传播的目标函数。实验表明，我们的模型在多种场景下优于传统标量-函数回归模型，且随预测变量维度的增加其计算扩展性更佳。