  As it stands, a robust mathematical framework to analyse and study various
topics in deep learning is yet to come to the fore. Nonetheless, viewing deep
learning as a dynamical system allows the use of established theories to
investigate the behaviour of deep neural networks. In order to study the
stability of the training process, in this article, we formulate training of
deep neural networks as a hydrodynamics system, which has a multisymplectic
structure. For that, the deep neural network is modelled using a stochastic
differential equation and, thereby, mean-field type control is used to train
it. The necessary conditions for optimality of the mean--field type control
reduce to a system of Euler-Poincare equations, which has the a similar
geometric structure to that of compressible fluids. The mean-field type control
is solved numerically using a multisymplectic numerical scheme that takes
advantage of the underlying geometry. Moreover, the numerical scheme, yields an
approximated solution which is also an exact solution of a hydrodynamics system
with a multisymplectic structure and it can be analysed using backward error
analysis. Further, nonlinear stability yields the condition for selecting the
number of hidden layers and the number of nodes per layer, that makes the
training stable while approximating the solution of a residual neural network
with a number of hidden layers approaching infinity.
