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As it stands, a robust mathematical framework to analyse and study varioustopics in deep learning is yet to come to the fore. Nonetheless, viewing deeplearning as a dynamical system allows the use of established theories toinvestigate the behaviour of deep neural networks. In order to study thestability of the training process, in this article, we formulate training ofdeep neural networks as a hydrodynamics system, which has a multisymplecticstructure. For that, the deep neural network is modelled using a stochasticdifferential equation and, thereby, mean-field type control is used to trainit. The necessary conditions for optimality of the mean--field type controlreduce to a system of Euler-Poincare equations, which has the a similargeometric structure to that of compressible fluids. The mean-field type controlis solved numerically using a multisymplectic numerical scheme that takesadvantage of the underlying geometry. Moreover, the numerical scheme, yields anapproximated solution which is also an exact solution of a hydrodynamics systemwith a multisymplectic structure and it can be analysed using backward erroranalysis. Further, nonlinear stability yields the condition for selecting thenumber of hidden layers and the number of nodes per layer, that makes thetraining stable while approximating the solution of a residual neural networkwith a number of hidden layers approaching infinity.

Multisymplectic Formulation of Deep Learning Using Mean--Field Type Control and Nonlinear Stability of Training Algorithm