A functional sliced inverse regression (FSIR) technique is considered for fitting the model
where is the response, is a functional regressor (a random function in ), is an unknown regression function, are unknown functions, and is an error term. Consistency of regularized FSIR estimates for is demonstrated. It is proposed to estimate the function by a multilayer perceptron technique. Consistency of the proposed training algorithm for such perceptrons is shown. Applications to real data are considered.