We investigate several norm spaces derived from neural network architectures, including (extended) Barron spaces, variation spaces, Radon-BV spaces, and spectral Barron spaces. This talk systematically explores the relationships among these spaces, develops new analytical tools across different frameworks, and examines their applications in partial differential equations, as well as in related inverse problems and regularization methods. This is a joint series of works in collaboration with Mourad Choulli (U. Lorraine), Yuanyuan Li (CUHK), Peter Mathé (WIAS), Sergei V. Pereverzev (RICAM) and Hiroshi Takase (Kyushu U.)
