The computational cost of fast non-LTE synthesis is one of the challenges that limits the development of 2D and 3D inversion codes. It also makes the interpretation of observations of lines formed in the chromosphere and transition region a slow and computationally costly process, which limits the inference of the physical properties on rather small fields of view. Having access to a fast way of computing the deviation from the LTE regime through the departure coefficients could largely alleviate this problem. We propose to build and train a graph network that quickly predicts the atomic level populations without solving the non-LTE problem. We find an optimal architecture for the graph network for predicting the departure coefficients of the levels of an atom from the physical conditions of a model atmosphere. A suitable data set with a representative sample of potential model atmospheres is used for training. This data set has been computed using existing non-LTE synthesis codes. The graph network has been integrated into existing synthesis and inversion codes for the particular case of Ca II. We demonstrate orders-of-magnitude gain in computing speed. We analyze the generalization capabilities of the graph network and demonstrate that it produces good predicted departure coefficients for unseen models. We implement this approach in Hazel2 and show how the inversions nicely compare with those obtained with standard non-LTE inversion codes. Our approximate method opens up the possibility of extracting physical information from the chromosphere on large fields of view with time evolution.