  This paper proposes a multivariate nonlinear function-on-function regression
model, which allows both the response and the covariates can be
multi-dimensional functions. The model is built upon the multivariate
functional reproducing kernel Hilbert space (RKHS) theory. It predicts the
response function by linearly combining each covariate function in their
respective functional RKHS, and extends the representation theorem to
accommodate model estimation. Further variable selection is proposed by adding
the lasso penalty to the coefficients of the kernel functions. A block
coordinate descent algorithm is proposed for model estimation, and several
theoretical properties are discussed. Finally, we evaluate the efficacy of our
proposed model using simulation data and a real-case dataset in meteorology.
