Deep learning approaches to extract nuclear deformation parameters from initial-state information in heavy-ion collisions

Abstract

The deformation of heavy nuclei leaves characteristic imprints on the initial conditions of relativistic heavy-ion collisions. However, event-by-event fluctuations make the quantitative extraction of this information challenging. This study examines the identifiability of the quadrupole ($β_2$) and hexadecapole ($β_4$) deformation parameters from nucleon configurations sampled from a deformed Woods-Saxon distribution commonly used in initial-state modeling of heavy-ion collisions. As a baseline, we first establish an upper bound on the "intrinsic identifiability" of deformation information at the most microscopic level by constructing permutation-invariant point-cloud networks under controlled multi-event grouping. We then extend the analysis to the more realistic initial entropy-density profiles generated by the TRENTo model, where both standard regression and simulation-based inference (SBI) with conditional normalizing flows are employed to reconstruct the deformation parameters from ensembles of event images supplemented with global attributes. Multi-event averaging is found to be essential in this setting for suppressing stochastic fluctuations and revealing the underlying deformation information. While standard regression efficiently captures the central trends of deformation through point estimates, SBI provides calibrated posterior distributions, offering a more complete and robust characterization of uncertainty. Collectively, our results demonstrate that deformation information is effectively encoded in the initial state and becomes increasingly identifiable with sufficient ensemble averaging, laying a solid foundation for future extensions toward more complete dynamical modeling and final-state observables.

Figures (25)

• Figure 1: The structure of the point-cloud neural network used for inferring nuclear deformation parameters.
• Figure 2: The structure of the neural network for regression and SBI from initial entropy density profiles. We use the MLP head for regression and the RealNVP head for SBI.
• Figure 3: Typical deformed Woods--Saxon distributions for representative deformation parameters of $(\beta_{2},\beta_{4})$ in the $x$--$z$ plane.
• Figure 4: Training and validation performance of the point-cloud model for nucleon configurations at different group sizes $N$.
• Figure 5: Predicted versus true $\beta_{2}$ (top) and $\beta_{4}$ (bottom) for nucleon-configuration regression at $N=1$. The diagonal denotes perfect agreement.