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This paper introduces a convolutional neural network (CNN) based framework for learning nonlinear functionals by extracting sparse features from function samples. The approach leverages universal discretization methods to achieve stable recovery from discrete samples, using both deterministic and random sampling. Theoretical analysis demonstrates that this sparsity-aware approach improves approximation rates and reduces sample complexity, particularly for functions with fast frequency decay or mixed smoothness, thus mitigating the curse of dimensionality in functional learning.
Sparsity could be the key to unlocking efficient neural networks for learning operators on infinite-dimensional function spaces, sidestepping the curse of dimensionality.
Deep neural networks have emerged as powerful tools for learning operators defined over infinite-dimensional function spaces. However, existing theories frequently encounter difficulties related to dimensionality and limited interpretability. This work investigates how sparsity can help address these challenges in functional learning, a central ingredient in operator learning. We propose a framework that employs convolutional architectures to extract sparse features from a finite number of samples, together with deep fully connected networks to effectively approximate nonlinear functionals. Using universal discretization methods, we show that sparse approximators enable stable recovery from discrete samples. In addition, both the deterministic and the random sampling schemes are sufficient for our analysis. These findings lead to improved approximation rates and reduced sample sizes in various function spaces, including those with fast frequency decay and mixed smoothness. They also provide new theoretical insights into how sparsity can alleviate the curse of dimensionality in functional learning.
