  In this paper we present a foundational study on a constrained method that
defines learning problems with Neural Networks in the context of the principle
of least cognitive action, which very much resembles the principle of least
action in mechanics. Starting from a general approach to enforce constraints
into the dynamical laws of learning, this work focuses on an alternative way of
defining Neural Networks, that is different from the majority of existing
approaches. In particular, the structure of the neural architecture is defined
by means of a special class of constraints that are extended also to the
interaction with data, leading to "architectural" and "input-related"
constraints, respectively. The proposed theory is cast into the time domain, in
which data are presented to the network in an ordered manner, that makes this
study an important step toward alternative ways of processing continuous
streams of data with Neural Networks. The connection with the classic
Backpropagation-based update rule of the weights of networks is discussed,
showing that there are conditions under which our approach degenerates to
Backpropagation. Moreover, the theory is experimentally evaluated on a simple
problem that allows us to deeply study several aspects of the theory itself and
to show the soundness of the model.
