  Direction dependent calibration and imaging is a vital part of producing
deep, high fidelity, high-dynamic range radio images with a wide-field
low-frequency array like LOFAR. Currently, state-of-the-art facet-based
direction dependent calibration algorithms rely on the assumption that the
isoplanatic-patch size is much larger than the separation between bright
in-field calibrators. This assumption is often violated due to the dynamic
nature of the ionosphere, and as a result direction dependent errors affect
image quality between calibrators. In this paper we propose a probabilistic
physics-informed model for inferring ionospheric phase screens, providing a
calibration for all sources in the field of view. We apply our method to a
randomly selected observation from the LOFAR Two-Metre Sky Survey archive, and
show that almost all direction dependent effects between bright calibrators are
corrected and that the root-mean-squared residuals around bright sources is
reduced by 32\% on average.
