CosmicNet. Part I. Physics-driven implementation of neural networks within Einstein-Boltzmann Solvers

Einstein-Boltzmann Solvers (EBSs) are run on a massive scale by the cosmology community when fitting cosmological models to data. We present a new concept for speeding up such codes with neural networks. The originality of our approach stems from not substituting the whole EBS by a machine learning algorithm, but only its most problematic and least parallelizable step: the integration of perturbation equations over time. This approach offers two significant advantages: the task depends only on a subset of cosmological parameters, and it is blind to the characteristics of the experiment for which the output must be computed (for instance, redshift bins). These allow us to construct a fast and highly re-usable network. In this proof-of-concept paper, we focus on the prediction of CMB source functions, and design our networks according to physical considerations and analytical approximations. This allows us to reduce the depth and training time of the networks compared to a brute-force approach. Indeed, the calculation of the source functions using the networks is fast enough so that it is not a bottleneck in the EBS anymore. Finally, we show that their accuracy is more than sufficient for accurate MCMC parameter inference from Planck data. This paves the way for a new project, CosmicNet, aimed at gradually extending the use and the range of validity of neural networks within EBSs, and saving massive computational time in the context of cosmological parameter extraction.