Noninvertibility in neural networks

We present and discuss an inherent shortcoming of neural networks used as discrete-time models in system identification, time series processing, and prediction. Trajectories of nonlinear ordinary differential equations (ODEs) can, under reasonable assumptions, be integrated uniquely backward in time. Discrete-time neural network mappings derived from time series, on the other hand, can give rise to multiple trajectories when followed backward in time: they are in principle noninvertible. This fundamental difference can lead to model predictions that are not only slightly quantitatively different, but qualitatively inconsistent with continuous time series. We discuss how noninvertibility arises, present key analytical concepts and some of its phenomenology. Using two illustrative examples (one experimental and one computational), we demonstrate when noninvertibility becomes an important factor in the validity of artificial neural network (ANN) predictions, and show some of the overall complexity of the predicted pathological dynamical behavior. These concepts can be used to probe the validity of ANN time series models, as well as provide guidelines for the acquisition of additional training data. (C) 2000 Elsevier Science Ltd.