Fig. 1. The neural network framework structure of the deep-learning quasi-particle mass model employed in this studyEach mass model integrates two residual neural networks (ResNets). Each ResNet consists of 8 hidden layers, with each layer containing 32 neurons. The swish activation function is applied at the end of each hidden layer.

Fig. 2. The results obtained from the deep learning method are compared with those calculated from LQCD and the HRG model at zero chemical potential. The red dotted lines represent the results from the DNN, while the black dotted lines correspond to the results from LQCD and HRG. (a, b, c, d) respectively display the curves of s/T3, Δ/T4, χ2B and χ4B as functions of temperature (color online)

Fig. 3. The mass distributions of quasi-partons at different values of μB and T(a) Quasi-light quarks, (b) Quasi-strange quarks, (c) Gluons

Fig. 4. The χ4B as a function of μB at different temperatures (color online)The red solid line represents the isotherm where the minimum value is located. The figure presents 16 sets of training results.

Fig. 5. The cumulant ratio R42 as a function of sNN (color online)The dashed lines in various colors represent the results calculated using the functional renormalization group (fRG) method^[60]. The blue shaded region corresponds to the predictions from the neural network, while the yellow region denotes the results from the transport model (UrQMD)^[59]. The gray and red data points represent the experimental results from BESI and BESII, respectively^[59].