Jets from supermassive black holes at the centers of active galaxies are the most powerful and persistent sources of electromagnetic radiation in the Universe. To infer the physical conditions in the otherwise out-of-reach regions of extragalactic jets, we usually rely on fitting their spectral energy distributions (SEDs). The calculation of radiative models for the jet's non-thermal emission usually relies on numerical solvers of coupled partial differential equations. In this work, we use machine learning to tackle the problem of high computational complexity to significantly reduce the SED model evaluation time, which is necessary for SED fittings carried out with Bayesian inference methods. We computed the SEDs based on the synchrotron self-Compton model for blazar emission using the radiation code We used them to train neural networks (NNs) to explore whether they can replace the original code, which is computationally expensive. We find that a NN with gated recurrent unit neurons (GRUN) can effectively replace the leptonic code for this application, while it can be efficiently coupled with Markov chain Monte Carlo (MCMC) and nested sampling algorithms for fitting purposes. We demonstrate this approach through an application to simulated data sets, as well as a subsequent application to observational data. We present a proof-of-concept application of NNs to blazar science as the first step in a list of future applications involving hadronic processes and even larger parameter spaces. We offer this tool to the community through a public repository.