In this talk, I will discuss nonlinear dimension reduction for functional data in a supervised way. In particular, the functional data can be a form of multivariate functional data in which multiple functions defined on different time domains are considered to be one observation. First, I will introduce nested reproducing kernel Hilbert (RKHS) space which provides a general mechanism for nonlinear functional data analysis. Two layers of function spaces are constructed in a nested fashion so that the first space represents the observed functional data, and the second space characterizes nonlinearity of the random functions. Then I will introduce a method of nonlinear functional additive PCA, the detailed procedure for estimation, and its consistency and convergence rate. The simulation studies and applications to real data will be introduced at the end.