  Information on natural phenomena and engineering systems is typically
contained in data. Data can be corrupted by systematic errors in models and
experiments. In this paper, we propose a tool to uncover the spatiotemporal
solution of the underlying physical system by removing the systematic errors
from data. The tool is the physics-constrained convolutional neural network
(PC-CNN), which combines information from both the systems governing equations
and data. We focus on fundamental phenomena that are modelled by partial
differential equations, such as linear convection, Burgers equation, and
two-dimensional turbulence. First, we formulate the problem, describe the
physics-constrained convolutional neural network, and parameterise the
systematic error. Second, we uncover the solutions from data corrupted by large
multimodal systematic errors. Third, we perform a parametric study for
different systematic errors. We show that the method is robust. Fourth, we
analyse the physical properties of the uncovered solutions. We show that the
solutions inferred from the PC-CNN are physical, in contrast to the data
corrupted by systematic errors that does not fulfil the governing equations.
This work opens opportunities for removing epistemic errors from models, and
systematic errors from measurements.
