  We consider a deterministic optimal control problem with a maximum running
cost functional, in a finite horizon context, and propose deep neural network
approximations for Bellman's dynamic programming principle, corresponding also
to some first-order Hamilton-Jacobi-Bellman equations. This work follows the
lines of Hur\'e et al. (SIAM J. Numer. Anal., vol. 59 (1), 2021, pp. 525-557)
where algorithms are proposed in a stochastic context. However, we need to
develop a completely new approach in order to deal with the propagation of
errors in the deterministic setting, where no diffusion is present in the
dynamics. Our analysis gives precise error estimates in an average norm. The
study is then illustrated on several academic numerical examples related to
front propagations models in the presence of obstacle constraints, showing the
relevance of the approach for average dimensions (e.g. from $2$ to $8$), even
for non-smooth value functions.
