In this paper we discuss several aspects of the mathematical problem of non linear black-box identification. We explain that the quality of the identification procedure always results from a certain trade-off between the expressive power of the model structure we try to identify, and the error variance of the estimates. The richer is the model structure, the lesser is the bias. The richer si the model structure, the larger is the error variance. Thus a first step si to choose a good model structure, i.e., capable of both good and parcomonious approximation -- this is a function approximation problem, and we can take advantage of recent advances in functional analysis for this purpose. We pay particular attention to so-called spatially adaptive approximants, i.e., approximants which concentrate expressiveness in the regions where the system is less regular.

Keywords : nonparametric identification, nonlinear systems, wavelet estimators, neural networks.