Mathematical problems of artificial intelligence and artificial neural networks

We propose a general topological approach to the analysis of artificial neural networks using simplicial complexes and the approximation of continuous mappings with simplicial ones. The essential properties of numerical instability in such problems were identified. It is associated with ill-posed problems in Hilbert space and regularization methods typically applied to Big Data processing. We formulated the criteria of artificial neural network accuracy and applicability and included some implementation examples based on the interpolation theory. Advancing P.L. Chebyshev's ideas about the best approximation may be an entry point to various mathematical research on artificial neural network training dataset optimization.