We consider the problem of nonparametric identification for a multi-dimensional functional autoregression yt = f(yt−1, …,yt−d) + et on the basis of N observations of yt. In the case when the unknown nonlinear function f belongs to the Barron class, we propose an estimation algorithm which provides approximations of f with expected L2 accuracy O(N1/4ln1/4N). We also show that this approximation rate cannot be significantly improved.