jats:titleAbstract</jats:title>jats:pGenerating high‐resolution flow fields is of paramount importance for various applications in engineering and climate sciences. This is typically achieved by solving the governing dynamical equations on high‐resolution meshes, suitably nudged toward available coarse‐scale data. To alleviate the computational cost of such downscaling process, we develop a physics‐informed deep neural network (PI‐DNN) that mimics the mapping of coarse‐scale information into their fine‐scale counterparts of continuous data assimilation (CDA). Specifically, the PI‐DNN is trained within the theoretical framework described by Foias et al. (2014, <jats:ext-link xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="https://doi.org/10.1070/rm2014v069n02abeh004891">https://doi.org/10.1070/rm2014v069n02abeh004891</jats:ext-link>) to generate a surrogate of the theorized jats:italicdetermining form</jats:italic> map from the coarse‐resolution data to the fine‐resolution solution. We demonstrate the PI‐DNN methodology through application to 2D Rayleigh‐Bénard convection, and assess its performance by contrasting its predictions against those obtained by dynamical downscaling using CDA. The analysis suggests that the surrogate is constrained by similar conditions, in terms of spatio‐temporal resolution of the input, as the ones required by the theoretical determining form map. The numerical results also suggest that the surrogate's downscaled fields are of comparable accuracy to those obtained by dynamically downscaling using CDA. Consistent with the analysis of Farhat, Jolly, and Titi (2015, <jats:ext-link xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="https://doi.org/10.48550/arxiv.1410.176">https://doi.org/10.48550/arxiv.1410.176</jats:ext-link>), temperature observations are not needed for the PI‐DNN to predict the fine‐scale velocity, pressure and temperature fields.</jats:p>