Recent spectroscopic surveys provide element abundances for large samples of Milky Way stars, from which stellar parameters can be inferred. Stellar ages, among them, are both a notoriously difficult parameter to estimate and a fundamental property for Galactic archaeology studies. We aim to develop a model-driven deep learning approach to age determination, by training neural networks on stellar evolutionary grids. Contrary to the usual data-driven deep learning approach of using prior age estimates as training data, our method has the potential for a wider and less biased range of application. The low computational cost of deep learning methods compared to e.g., bayesian isochrone-fitting allows for a broad analysis of large spectroscopic catalogues. We train multilayer perceptrons on different stellar evolutionary grids to map [M/H], M_G_, (G_BP_-G_RP_) to stellar age tau. We combine Gaia photometry and parallaxes, metallicities and alpha elements from spectroscopic surveys and extinction maps, which are passed through the neural networks to estimate stellar ages. We apply our method to the LAMOST DR10, GALAH DR3 & DR4 and APOGEE DR17 spectroscopic surveys, for which we estimate the ages using the BaSTI tracks, along with other stellar evolutionary models. We leverage this novel technique to study, for the first time, differences in age estimates from several evolutionary grids applied on very large datasets. In addition, we date 13 open clusters and one globular cluster and find a median absolute deviation with literature ages of 0.20Gyr. Along with the stellar ages catalogues from our estimates, we release NEST (Neural Estimator of Stellar Times), a python package to estimate stellar age based on this work and available at https://github.com/star-age/NEST, as well as a web interface https://star-age.github.io. We show that, when using the same evolutionary grid, our method retrieves the same ages as a bayesian approach like SPInS, for only a fraction of the computational cost, with a 60000 speedup factor for a typical star. This model-driven deep learning technique thus opens up the way for broad galactic archeology studies on the largest datasets available today and in the near future with upcoming surveys such as 4MOST.