Fast prediction of the hydrodynamic QGP evolution in ultra-relativistic heavy-ion collisions using Fourier Neural Operators

TL;DR

This work investigates Fourier Neural Operators (FNOs) as fast, discretization-invariant surrogates for ultra-relativistic QGP hydrodynamics in MC simulations, comparing against MUSIC within the JETSCAPE framework for Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV. The study trains FNOs on diverse initial-condition classes, validates against PDE solutions, and demonstrates strong performance for flow and jet-quenching observables, including spatial super-resolution. Key findings show sub-percent accuracy in energy-density evolution for matched ICs, robust interpolation to intermediate nucleon widths, and significant runtime reductions when using GPU-accelerated inference, enabling faster, high-statistics MC studies. The results suggest that FNOs can dramatically accelerate QGP evolution calculations and potentially facilitate forward-modeling in Bayesian analyses, with future work extending to 3+1D and viscous hydrodynamics and iterative jet-medium feedback.

Abstract

Recent research in machine learning has employed neural networks to learn mappings between function spaces on bounded domains termed ``neural operators''. As such, these operators can provide alternatives to standard numerical methods for partial differential equation (PDE) solutions. In particular, the Fourier Neural Operator (FNO) has been shown to map solutions for classical fluid flow problems with accuracy competitive with traditional PDE solvers and with much greater computing speed. This paper explores the first application of FNOs to model ultra-relativistic hydrodynamic flow of the quark-gluon plasma (QGP) generated in relativistic heavy-ion collisions. The application in ultra-relativistic flow is novel relative to classical flow, due to the hydrodynamic evolution of the QGP occurring in femtometer-scaled explosions characterized by rapid expansion cooling. In this study we investigate the applicability of FNOs as computationally fast alternatives to standard numerical PDE solvers. The FNO predictions are evaluated by comparing to standard PDE solutions, using \MUSIC in the \JETSCAPE Monte Carlo event generator framework. The performance of calculating established experimental observables for flow and jet quenching using FNOs in the MC framework are also reported.

Figures (19)

• Figure 1: Energy density ($\varepsilon$) distributions of the QGP in a central $\sqrt{s_\mathrm{NN}}=200GeV$ MC Au+Au collision at first and last values $\tau$ predicted by the FNO. The color scale indicates the $\varepsilon$ distribution in $\mathrm{GeV}/\mathrm{fm}^3$ calculated by the PDE solver music. The dotted lines plot constant $\varepsilon$ boundaries (analogous to elevation lines in a topographical map) containing percentiles of the total event energy. Values of $\varepsilon$ traced by the dotted lines are printed in white. The location of the percentiles predicted by an FNO are shown in red lines, whose $\varepsilon$ values (relative to the music values) are printed in red.
• Figure 2: Energy density ($\varepsilon$) distributions of the QGP in a central $\sqrt{s_\mathrm{NN}}=200GeV$ MC Au+Au collision, with a nucleon width of 0.8fm, at the first and last values of $\tau$ predicted by the FNO. The color scale indicates the $\varepsilon$ distribution in $\mathrm{GeV}/\mathrm{fm}^3$ calculated by the PDE solver music. The dotted lines plot constant $\varepsilon$ boundaries (analogous to elevation lines in a topographical map) containing percentiles of the total event energy. Values of $\varepsilon$ traced by the dotted lines are printed in white. The location of the percentiles predicted by an FNO are shown in red lines, whose $\varepsilon$ values (relative to the music values) are printed in red.
• Figure 3: Super-resolution: The results of using an FNO trained on central events with $60\times60$ resolution predicting values on a separate MC event with a resolution of $150\times150$. For an explanation of the legends and lines see caption of Fig. \ref{['fig:central_2D']}.
• Figure 4: A vector field plot of the QGP velocities at $\tau=3.5$ in a central event. The values of the maximum length arrows are close to $c$.
• Figure 5: Average values of radial QGP velocity (relative to the $x$, $y$, center of collision in the MC) for 4000 central events at $x$ and $y$ grid points within the radii indicated and during the time evolution in $\tau$ indicated.