  In inference problems, we often have domain knowledge which allows us to
define summary statistics that capture most of the information content in a
dataset. In this paper, we present a hybrid approach, where such physics-based
summaries are augmented by a set of compressed neural summary statistics that
are optimised to extract the extra information that is not captured by the
predefined summaries. The resulting statistics are very powerful inputs to
simulation-based or implicit inference of model parameters. We apply this
generalisation of Information Maximising Neural Networks (IMNNs) to parameter
constraints from tomographic weak gravitational lensing convergence maps to
find summary statistics that are explicitly optimised to complement angular
power spectrum estimates. We study several dark matter simulation resolutions
in low- and high-noise regimes. We show that i) the information-update
formalism extracts at least $3\times$ and up to $8\times$ as much information
as the angular power spectrum in all noise regimes, ii) the network summaries
are highly complementary to existing 2-point summaries, and iii) our formalism
allows for networks with smaller, physically-informed architectures to match
much larger regression networks with far fewer simulations needed to obtain
asymptotically optimal inference.
