Abstract：Neural networks have proven their versatility in approximating continuous functions, but their capabilities extend far beyond. In this talk, we delve into the realm of functional deep neural networks, which offer a promising approach for approximating nonlinear functionals defined on Lp spaces. By investigating the convergence rates of approximation and generalization errors under different regularity conditions, we gain insights into the theoretical properties of these networks. This analysis contributes to a deeper understanding of functional neural networks and opens up new possibilities for their effective application in diverse domains such as dynamic system identification, mean field control, and functional data analysis.