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Residual deep neural networks (ResNets) are mathematically described asinteracting particle systems. In the case of infinitely many layers the ResNetleads to a system of coupled system of ordinary differential equations known asneural differential equations. For large scale input data we derive amean--field limit and show well--posedness of the resulting description.Further, we analyze the existence of solutions to the training process by usingboth a controllability and an optimal control point of view. Numericalinvestigations based on the solution of a formal optimality system illustratethe theoretical findings.

Continuous limits of residual neural networks in case of large input data