The knockoffs is a recently proposed powerful framework that effectively controls the false discovery rate (FDR) for variable selection. However, none of the existing knockoff solutions are directly suited to handle multivariate or high-dimensional functional data, which has become increasingly prevalent in various scientific applications. In this paper, we propose a novel functional model-X knockoffs selection framework tailored to sparse high-dimensional functional models, and show that our proposal can achieve the effective FDR control for any sample size. Furthermore, we illustrate the proposed functional model-X knockoffs selection procedure along with the associated theoretical guarantees for both FDR control and asymptotic power using examples of commonly adopted functional linear additive regression models and the functional graphical model. In the construction of functional knockoffs, we integrate essential components including the correlation operator matrix, the Karhunen-Lo\`eve expansion, and semidefinite programming, and develop executable algorithms. We demonstrate the superiority of our proposed methods over the competitors through both extensive simulations and the analysis of two brain imaging datasets.

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