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 equation. This work follows the lines of Huré 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 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.