The rapid development of measuring devices allows for easier collection of physical information about, e.g., fluid velocity flows. However, such data is usually corrupted by a significant amount of noise generated, for instance, by measurement errors or interference of other physical processes. Moreover, the data could be sparse in time and space. Reconstructing higher spatial resolution and decreasing the noise level of velocity flow fields is essential for many practical applications. In this talk, we present a physics informed data driven method which allows one to estimate the “true” state of the flow field from noisy observations, provided that the underlying field is described by Navier-Stokes equations. We discuss a theoretical convergence result for the case of noisy and incomplete measurements and validate it with the numerical simulation. Furthermore, we propose to estimate high-resolution flow fields by projecting the original HR noisy observations onto a lower resolution grid. Such averaging is lossy and, therefore, without using a prior knowledge about the physical process, one cannot uniquely reconstruct the high dimensional information from its spatial averages. Although it is counterintuitive, we demonstrate that this approach produces the high-resolution ground truth estimates with lower error compared to using the original observations directly. Finally, we show that our algorithm follows a self-supervised paradigm and does not require the high-resolution data for training.