Motivated by the abundance of functional data such as time series and images, there has been a growing interest in integrating such data into neural networks and learning maps from function spaces to R (i.e., functionals). In this paper, we study the approximation of functionals on reproducing kernel Hilbert spaces (RKHS's) using neural networks. We establish the universality of the approximation of functionals on the RKHS's. Specifically, we derive explicit error bounds for those induced by inverse multiquadric, Gaussian, and Sobolev kernels. Moreover, we apply our findings to functional regression, proving that neural networks can accurately approximate the regression maps in generalized functional linear models. Existing works on functional learning require integration-type basis function expansions with a set of pre-specified basis functions. By leveraging the interpolating orthogonal projections in RKHS's, our proposed network is much simpler in that we use point evaluations to replace basis function expansions.

翻译：受时间序列和图像等函数型数据大量涌现的驱动，将此类数据整合至神经网络并学习从函数空间到实数（即泛函）的映射日益受到关注。本文研究利用神经网络对再生核希尔伯特空间（RKHS）上泛函的逼近问题。我们建立了RKHS上泛函逼近的普适性，具体推导了由逆多重二次核、高斯核和索伯列夫核诱导的泛函的显式误差界限。此外，我们将研究成果应用于函数型回归，证明神经网络可精确逼近广义函数线性模型中的回归映射。现有函数型学习工作需借助一组预指定的基函数进行积分型基函数展开。通过利用RKHS中的插值正交投影，我们提出的网络更为简洁——采用点评估替代基函数展开。