Two-dimensional active nematics are often modeled using phenomenological continuum theories that describe the dynamics of the nematic director and fluid velocity through partial differential equations (PDEs). While these models provide a statistically accurate description of the experiments, the identification of the relevant terms in the PDEs and their parameters is usually indirect. Here, we adapt a recently developed method to automatically identify optimal continuum models for active nematics directly from the spatio-temporal director and velocity data, via sparse fitting of the coarse-grained fields onto generic low order PDEs. We test the method extensively on computational models, and then apply it to data from experiments on microtubule-based active nematics. Thereby, we identify the optimal models for microtubule-based active nematics, along with the relevant phenomenological parameters. We find that the dynamics of the orientation field are largely governed by its coupling to the underlying flow, with free-energy gradients playing a negligible role. Furthermore, by fitting the flow equation to experimental data, we estimate a key parameter quantifying the `activity' of the nematic.