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The medium effects of nucleons and equation of state (EOS) are two important issues in low-intermediate energy nuclear physics. In laboratory, the medium effects of nucleons and EOS can be learned by comparing the heavy ion collisions (HICs) with the transport model simulations. Up to now, the main challenges are to find sensitive probes for learning the medium effects of nucleons, such as the effective mass for proton and neutron, and to improve the reliability of the extraction of the medium effects of nucleons and EOS by comparing the transport model calculations with experimental data. To improve the reliability of the comparisons, it is necessary to use the same conditions as those used in experiments, such as angle cut or energy bins for measured particles or fragments, and centrality of the reaction. The angle cut and energy bins can be easily realized in the calculations. However, the centrality of HICs can not be used directly as an input in the simulations, since the centrality of HICs does not correspond to a specific value of the impact parameter. Another, the impact parameter used in the transport model simulations can not be directly measured in experiments. Thus, reconstructing the impact parameter distribution from the centrality of HIC observables is necessary for simulating HICs and comparing with data in a reasonable way. In this work, we first investigate the fluctuation mechanism in low-intermediate energy HICs. Our calculations show that the initial statistical fluctuations and random nucleon-nucleon scattering are two origins of inherent fluctuations. The fluctuation mechanism leads the observables to be distributed over a wide range even with the same inputs, such as a fixed value of impact parameter, beam energy, EOS, nucleon-nucleon scattering cross section, etc. On the other hand, a fixed or certain value of observables maps to the input variables over a wide range. If a machine learning is used to map the values of observables to the distribution of input physics variables, a novel algorithm should be proposed. For model-independently reconstructing impact parameter distributions with two-dimensional observables, we propose a method by combining an unsupervised machine learning K-means algorithm with Bayesian method. After fitting the conditional probability density distributions of the reduced multiplicity M0 and the total transverse momentum of the light fragment ptott0 at given impact parameters, the probability density function of the impact parameters at given observable is obtained. The events are automatically sorted by using the K-means algorithm, and then a Bayesian method is used to reconstruct the impact parameter distributions for the events selected by certain values of observables. Furthermore, we use the convolutional neural network (CNN) and forward neural network (FNN) to distinguish two Skyrme parameter sets, such as SkM* and SLy4, from the data generated by ImQMD. The two parameter sets have different signs of the effective mass splitting. The two-dimensional observables, such as the longitudinal and transverse kinetic momentum distributions of neutron and proton, and neutron/proton energy spectra are selected to train the CNN and FNN, respectively. Our calculations show that CNN can distinguish two sets from two-dimensional observables with an accuracy of 81% for 132Sn+124Sn system at the beam energy of 200 MeV/nucleon. FNN can distinguish two sets from neutron/proton energy spectra with an accuracy better than 93%. These studies accumulate the experience on the determination of effective mass splitting distribution with the data in the future. |
