Learning nonlinear functionals defined on infinite-dimensional function spaces is a fundamental problem in approximation theory and operator learning. Traditional approaches often rely on linear feature representations which may suffer from the curse of dimensionality. In this talk, we present a sparse-aware neural approximation framework that exploits the feature extraction capabilities of convolutional neural networks (CNNs). We first establish a theoretical connection between CNNs and sparse coding, providing stability guarantees and exponential convergence rates for sparse feature recovery. Building on these results, we develop an encoder–decoder architecture in which a CNN extracts sparse representations from finitely sampled input functions, while a deep neural network approximates the target nonlinear functional. Our analysis establishes approximation error bounds governed by the sparsity of the extracted features rather than the full dimension of the underlying dictionary. We further show that random sampling is sufficient for stable sparse recovery under suitable conditions. For function classes with rapidly decaying coefficients or mixed smoothness, the resulting approximation rates substantially overcome the curse of dimensionality.
