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：函数响应在标量预测变量上的回归是一项具有挑战性的任务，尤其在预测变量数量众多或预测变量与响应之间存在非线性关系时。本文提出了一种解决方案：一种前馈神经网络（NN），旨在通过标量输入预测函数响应。首先，我们将函数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。然后，我们提出通过目标函数修改网络的输出，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采样的数据，并可进一步应用粗糙度惩罚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能够进行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的标量-函数回归模型，并且在计算上更好地适应预测变量的维度扩展。