Approximation theory traditionally looks at the structure of function spaces under addition; but approximation spaces under composition are underexamined. Studying approximation spaces under composition may quantitatively explain the outperformance of neural networks, reveal links to dynamical systems, and suggest related architectures.

(Edit: Following the work of Weinan E, Chao Ma, Lei Wu, Ronald DeVore, Gitta Kutyniok, et al.)