Measurements on dynamical systems, experimental or otherwise, are often subjected to inaccuracies capable of introducing corruption. Removal of this is a problem of fundamental importance in the field of fluid dynamics, for instance in PIV measurements [1]. Research on correcting measurements predominantly considers noise-removal and flow-reconstruction [2]. In this work we propose a form of flow-reconstruction to extract the true, underlying solution to the Navier-Stokes equations from biased data with access to only limited ground-truth observations. We introduce a form of physics-informed convolutional neural network, embedding prior knowledge of the physics in the form of the governing equations [3] and exploiting spatial correlations. We showcase the methodology for three physical systems: linear convection-diffusion, non-linear convection diffusion, and the 2D turbulent Kolmogorov flow. Measurements are subjected to corruption, adding stationary, spatially-varying bias parameterised by frequency and magnitude; showing of the methodology to both frequency and magnitude of the corruption. By conditioning sparse ground-truth observations on the governing equations, we demonstrate a novel method to recover the underlying flow across the entire domain. This work opens opportunities for the extraction of Navier-Stokes solutions from PIV data and the detection of faulty sensors that introduce biases [4].