Finding signatures of the nuclear symmetry energy in heavy-ion collisions with deep learning

A deep convolutional neural network (CNN) is developed to study symmetry energy $E_{\rm sym}(\rho)$ effects by learning the mapping between the symmetry energy and the two-dimensional (transverse momentum and rapidity) distributions of protons and neutrons in heavy-ion collisions. Supervised training is performed with labelled data-set from the ultrarelativistic quantum molecular dynamics (UrQMD) model simulation. It is found that, by using proton spectra on event-by-event basis as input, the accuracy for classifying the soft and stiff $E_{\rm sym}(\rho)$ is about 60% due to large event-by-event fluctuations, while by setting event-summed proton spectra as input, the classification accuracy increases to 98%. The accuracy for 5-label (5 different $E_{\rm sym}(\rho)$) classification task are about 58% and 72% by using proton and neutron spectra, respectively. For the regression task, the mean absolute error (MAE) which measures the average magnitude of the absolute differences between the predicted and actual $L$ (the slope parameter of $E_{\rm sym}(\rho)$) are about 20.4 and 14.8 MeV by using proton and neutron spectra, respectively. Fingerprints of the density-dependent nuclear symmetry energy on the transverse momentum and rapidity distributions of protons and neutrons can be identified by convolutional neural network algorithm.