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Gamma-ray bursts (GRBs) and double neutron-star merger gravitational wave events are followed by afterglows that shine from X-rays to radio, and these broadband transients are generally interpreted using analytical models. Such models are relatively fast to execute, and thus easily allow estimates of the energy and geometry parameters of the blast wave, through many trial-and-error model calculations. One problem, however, is that such analytical models do not capture the underlying physical processes as well as more realistic relativistic numerical hydrodynamic (RHD) simulations do. Ideally, those simulations are used for parameter estimation instead, but their computational cost makes this intractable. To this end, we present DeepGlow, a highly efficient neural network architecture trained to emulate a computationally costly RHD-based model of GRB afterglows, to within a few percent accuracy. As a first scientific application, we compare both the emulator and a different analytical model calibrated to RHD simulations, to estimate the parameters of a broadband GRB afterglow. We find consistent results between these two models, and also give further evidence for a stellar wind progenitor environment around this GRB source. DeepGlow fuses simulations that are otherwise too complex to execute over all parameters, to real broadband data of current and future GRB afterglows.