Final-state centrality in relativistic heavy-ion simulations from neural networks

Technical Summary: Final-state centrality in relativistic heavy-ion simulations from neural networks

Problem Statement
Relativistic heavy-ion collision experiments determine event centrality using final-state observables, typically the charged-particle multiplicity ($N_{ch}$). However, performing this definition in state-of-the-art hybrid hydrodynamic simulations is computationally prohibitive. A single hybrid event (initial conditions through hydrodynamic evolution and hadronic transport) can cost up to 4 CPU-hours. Consequently, Bayesian analyses and model comparisons often approximate centrality using initial-state quantities (e.g., initial entropy or energy), which are computationally cheap but do not perfectly correlate with the final-state multiplicity used in experiments. This discrepancy is negligible in wide centrality classes (e.g., 10% width) but becomes significant in narrow classes (e.g., 1% width), which are increasingly required for high-precision data from the LHC's high-luminosity runs. The mismatch introduces spurious correlations and biases when comparing models to experimental data, particularly for observables dependent on multiplicity.

Methodology
The authors propose using neural networks (NN) as surrogate models to predict final-state observables from initial conditions, enabling the generation of millions of events at a fraction of the cost of full simulations.

• Training Data: The networks were trained on a limited set of $\sim$600 fully simulated hybrid events for Pb–Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV and Au–Au collisions at $\sqrt{s_{NN}} = 200$ GeV. These events were generated using a hybrid framework (TRENTo initial conditions, free-streaming, MUSIC hydrodynamics, iS3D particlization, and SMASH hadronic transport) with parameters fixed at the maximum a posteriori values from the JETSCAPE Bayesian analysis.
• Network Architecture and Strategy:
  • The network predicts the event-by-event charged-particle multiplicity ($N_{ch}$) and mean transverse momentum ($\langle p_T \rangle$).
  • Input: Initial entropy density (or energy density) profiles.
  • Decomposition Strategy: To handle the wide dynamic range of $N_{ch}$ (spanning three orders of magnitude), the authors decompose the target observable $y$ into a smooth trend $g(S)$ dependent on total initial entropy $S$, and a residual deviation $d$. The network is trained only to predict the residual $d$, while the trend is fixed analytically. This prevents the trend from dominating the training and forces the network to learn the geometric fluctuations distinguishing events of equal entropy.
  • Simultaneous Regression: A single network architecture predicts both $N_{ch}$ and $\langle p_T \rangle$ simultaneously.
  • Validation: Five-fold cross-validation was used, with a separate test set (25% of events) never seen during training.
• Application: Once trained, the networks were used to evaluate $N_{ch}$ and $\langle p_T \rangle$ for $1.5 \times 10^6$ initial conditions per system. This process took approximately 15 GPU-hours per system, compared to an estimated $10^6$–$10^7$ CPU-hours for full hybrid simulations of the same size.

Key Results

1. Prediction Accuracy: The trained networks predict $N_{ch}$ and $\langle p_T \rangle$ with relative errors of 1–2% for centralities up to 80%. The correlation coefficient for the residual deviations is $r = 0.90$–$0.96$. Errors rise to 8–10% in the most peripheral events (80–100%) where the training sample is sparse.
2. Centrality Definition Mismatch: Comparing centrality classes defined by initial entropy ($S$) versus predicted final multiplicity ($N_{ch}$):
   • In wide classes (5–10%), 96–97% of events are assigned to the same class by both definitions.
   • In narrow classes (1% width), this agreement drops to 83–95%, with the remaining events migrating primarily to adjacent classes.
   • This confirms that initial-state proxies are insufficient for defining narrow centrality classes required for high-precision correlation studies.
3. Transverse Momentum Fluctuations:
   • The model reproduces STAR data for Au–Au $\langle p_T \rangle$ fluctuations across the full centrality range.
   • For Pb–Pb, the model agrees with ALICE data at mid-central and peripheral centralities but overpredicts fluctuations by 20–30% in the most central classes. The authors note this may be due to the centrality estimator mismatch (V0M vs. mid-rapidity $N_{ch}$) or $p_T$ range differences.
   • The relative fluctuations $\sigma_{p_T}/\langle p_T \rangle$ are similar for both systems at the same centrality in the mid-central/peripheral region, suggesting the hydrodynamic response to initial geometric fluctuations is weakly dependent on collision energy in this regime.
4. Correlation Coefficients:
   • $\rho(v_2^2, \langle p_T \rangle)$: This correlation is largely insensitive to the centrality definition (whether based on $S$ or $N_{ch}$) and is similar for both collision systems.
   • $\rho(N_{ch}, \langle p_T \rangle)$: This correlation is highly sensitive to the centrality definition. When centrality is defined by $N_{ch}$, the correlation is small and positive. When defined by initial entropy, the correlation is substantially larger and positive. This is because defining centrality by $N_{ch}$ artificially restricts the range of $N_{ch}$ within a class, suppressing its correlation with other variables.

Significance and Claims
The paper demonstrates that neural networks trained on a modest number of hybrid events (hundreds) can serve as accurate emulators for final-state observables, making it computationally feasible to generate the millions of events required to define centrality using final-state estimators rather than initial-state proxies.

The authors claim that:

• Necessity of Final-State Definitions: For meaningful comparisons between hydrodynamic models and experimental data, especially in narrow centrality classes and for observables involving multiplicity, the model's event selection must emulate the experimental estimator (final-state multiplicity) as closely as possible. Initial-state definitions are inadequate for these specific purposes.
• Impact on Physics Observables: The choice of centrality definition significantly alters the magnitude and centrality dependence of correlations like $\rho(N_{ch}, \langle p_T \rangle)$, which are used to extract fundamental properties like the speed of sound of the QGP.
• Feasibility for Bayesian Analysis: This approach can be applied at every design point of Bayesian analyses (where typically only hundreds of events are simulated per point) to provide large event samples and consistent centrality definitions. The authors note that while the method is promising, the cost of training networks across the full parameter space (where residual structures might change) and the sufficiency of a few hundred events across all parameter regions remain open questions for future work.

The study does not claim to have solved the discrepancy between the model and ALICE data in the most central Pb–Pb collisions but attributes part of the difference to the centrality estimator mismatch, highlighting the importance of the proposed methodology for future precision studies.