  Neural networks have been achieving high generalization performance on many
tasks despite being highly over-parameterized. Since classical statistical
learning theory struggles to explain this behavior, much effort has recently
been focused on uncovering the mechanisms behind it, in the hope of developing
a more adequate theoretical framework and having a better control over the
trained models. In this work, we adopt an alternate perspective, viewing the
neural network as a dynamical system displacing input particles over time. We
conduct a series of experiments and, by analyzing the network's behavior
through its displacements, we show the presence of a low kinetic energy
displacement bias in the transport map of the network, and link this bias with
generalization performance. From this observation, we reformulate the learning
problem as follows: finding neural networks which solve the task while
transporting the data as efficiently as possible. This offers a novel
formulation of the learning problem which allows us to provide regularity
results for the solution network, based on Optimal Transport theory. From a
practical viewpoint, this allows us to propose a new learning algorithm, which
automatically adapts to the complexity of the given task, and leads to networks
with a high generalization ability even in low data regimes.
