2020/07/06 by Pietro Vertechi, Vertechi, Pietro, Mattia G. Bergomi +1
Computer Science · Materials Science · Physics and Astronomy · #18A20 #47L05 #FOS: Computer and information sciences #I.2.6 #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Machine Learning in Materials Science #Model Reduction and Neural Networks #Neural Networks and Applications
paper · pdf · doi:10.48550/arxiv.2007.02777
openalex publication_date 2020/07/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Using tools from topology and functional analysis, we provide a framework where artificial neural networks, and their architectures, can be formally described. We define the notion of machine in a general topological context and show how simple machines can be combined into more complex ones. We explore finite- and infinite-depth machines, which generalize neural networks and neural ordinary differential equations. Borrowing ideas from functional analysis and kernel methods, we build complete, normed, infinite-dimensional spaces of machines, and we discuss how to find optimal architectures and parameters -- within those spaces -- to solve a given computational problem. In our numerical experiments, these kernel-inspired networks can outperform classical neural networks when the training dataset is small.