Abstract In recent years, there has been growing interest in the field of functional neural networks. They have been proposed and studied with the aim of approximating continuous functionals defined on sets of functions on Euclidean domains. In this paper, we consider functionals defined on sets of functions on spheres. The approximation ability of deep ReLU neural networks is analyzed using an encoder-decoder framework on the unit sphere. An encoder is introduced first to accommodate the infinite-dimensional nature of the functional’s domain. It utilizes spherical harmonics to help us extract the latent finite-dimensional information of functions, which in turn facilitates in the next step of approximation analysis using fully connected neural networks. Moreover, real-world objects are frequently sampled discretely and are often corrupted by noise. Therefore, encoders with discrete inputs and those with discrete and random noise inputs are constructed, respectively. The approximation rates with different encoder structures are provided therein.
Yang et al. (2026) studied this question.