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The Generalized Mapping Regressor (GMR) neural network is able to solve for inverse problems even when multiple solutions are given. In this case, it does not only identify these solutions (even if infinite, e.g. contours), but also specifies to which branch of the underlying mapping it belongs. It is also able to model mapping with discontinuities. The basic idea is the transformation of the mapping problem in a pattern recognition problem in a higher dimensional space (where the function branches are represented by clusters). Training is given by a multiresolution quantization represented by a pool of neurons whose number is determined by the training set. Then, neurons are linked each other by using some kind of local principal component analysis (LPCA). This phase is the most important and original. Other techniques (e.g. SVM's, mixture-of-experts) could work a priori on the same problems, but are not able to understand automatically when to stop the data quantization. This linking phase can be viewed as a reconstruction phase in which the correct clusters are recovered. The production phase uses a Gaussian kernel interpolation technique. Some examples conclude the paper.