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Einstein-Boltzmann Solvers (EBSs) are run on a massive scale by the cosmologycommunity when fitting cosmological models to data. We present a new conceptfor speeding up such codes with neural networks. The originality of ourapproach stems from not substituting the whole EBS by a machine learningalgorithm, but only its most problematic and least parallelizable step: theintegration of perturbation equations over time. This approach offers twosignificant advantages: the task depends only on a subset of cosmologicalparameters, and it is blind to the characteristics of the experiment for whichthe output must be computed (for instance, redshift bins). These allow us toconstruct 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 networksaccording to physical considerations and analytical approximations. This allowsus to reduce the depth and training time of the networks compared to abrute-force approach. Indeed, the calculation of the source functions using thenetworks 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 MCMCparameter inference from Planck data. This paves the way for a new project,CosmicNet, aimed at gradually extending the use and the range of validity ofneural networks within EBSs, and saving massive computational time in thecontext of cosmological parameter extraction.

CosmicNet I: Physics-driven implementation of neural networks within Boltzmann-Einstein solvers