The nuclear equation of state (EoS) delineates the thermodynamic relationship between nucleon energy and nuclear matter density, temperature, and isospin asymmetry. This relationship is essential for validating existing nuclear theoretical models, investigating the nature of nuclear forces, and understanding the structure of compact stars, neutron star mergers, and related astrophysical phenomena. Heavy-ion collision experiments combined with transport models serve as a pivotal method to explore the high-density behavior of the EoS. With the rapid development of next-generation high-current heavy-ion accelerators and advanced detection technologies, the variety, volume, and precision of data generated from heavy-ion collision experiments have significantly improved. Effectively utilizing and analyzing these experimental datasets to extract critical insights into the EoS represents one of the central challenges in contemporary heavy-ion physics research. Bayesian analysis, a statistical approach, can extract reliable physical information by comparing experimental data with theoretical calculations and quantifying parameter uncertainties, thereby gaining widespread attention. In determining the range of EoS parameters using Bayesian inference, Monte Carlo sampling is employed to extract observables from final-state particle information simulated by transport models under various EoS parameters. However, the complexity of transport model calculations significantly hinders data generation efficiency and limits exploration of the full parameter space.

This study aims to a more efficient approach to simulate transport models, particularly one that leverages modern computational techniques to accelerate the process.

Here, a machine learning-based approach was proposed to develop a transport model emulator capable of significantly reducing computation time. We evaluate three machine learning algorithms—Gaussian processes, multi-task neural networks, and random forests—to train emulators based on the UrQMD transport model. The selected observables include protons' directed flow, elliptical flow, and nuclear stopping extracted from the final state of Au+Au collisions at E_lab=0.25 GeV/nucleon under different EoS parameters (incompressibility K₀, effective mass m^*, and in-medium correction factor F for nucleon-nucleon elastic cross sections). A total of 150 parameter sets of the UrQMD model are run, with K₀=180 MeV, 220 MeV, 260 MeV, 300 MeV, 340 MeV, 380 MeV, m^*/m=0.6, 0.7, 0.8, 0.9, 0.95, and F=0.6, 0.7, 0.8, 0.9, 1.0. For each case, 2×10⁵ events with a reduced impact parameter b₀<0.45 are simulated to ensure negligible statistical errors. The results from these 150 parameter sets are used to train the emulators via the three machine learning algorithms. Additionally, 20 parameter sets of the UrQMD model with randomly chosen K₀, m^*, and F are run, and the resulting observables are used to test emulator performance.

The results obtained from Gaussian processes and multi-task neural networks align with those calculated by the UrQMD model, indicating high accuracy and suitability for use in Bayesian analysis. However, when predicting v11 and v20 with a reduced impact parameter b₀<0.25, some data points predicted by random forests exhibit significant errors, suggesting that random forests are relatively less effective in predicting observables. To further compare the prediction performance of the three emulators, we use the coefficient of determination R² as the evaluation index. The R2 values for Gaussian processes, multi-task neural networks, and random forests in the test set are 0.95, 0.93, and 0.85, respectively. These results demonstrate that both Gaussian processes and multi-task neural networks achieve high accuracy when simulating UrQMD model data and can effectively accelerate the calculation process. However, for complex tasks involving a large number of parameters and observables, the efficiency and accuracy of Gaussian processes may decline. Thus, relying solely on Gaussian processes may be insufficient. In such cases, multi-task neural networks exhibit greater adaptability, better handling of complex datasets, and enhanced capability to learn information within parameter spaces.

In summary, Gaussian processes generally perform well within Bayesian frameworks as transport model emulators, particularly for moderate-sized datasets, while multi-task neural networks may be a more suitable choice for complex tasks involving numerous parameters and observables. In practical applications, the most appropriate emulator should be selected based on specific task requirements and data characteristics.