Physics-Informed Neural Network for Solving the Diffusion Equation in the Expanding QCD Medium

TL;DR

This paper demonstrates that Physics-Informed Neural Networks can efficiently solve the diffusion equation for charm-quark densities in an expanding quark-gluon plasma, with the heavy quarks treated as kinetically thermalized. By embedding the hydrodynamic velocity fields from MUSIC into the PINN loss, the authors achieve accurate, mesh-free solutions that match traditional RK4 baselines and support event-by-event analyses. The approach enables rapid computation of charm-quark densities crucial for charmonium regeneration studies in fluctuating media, and it shows promise for extending to joint spatial-momentum descriptions via a future Fokker–Planck formulation. Overall, PINNs offer a robust, physics-guided framework for heavy-quark transport in complex QCD backgrounds.

Abstract

We employ Physics-Informed Neural Networks (PINNs) to solve the diffusion of heavy quarks within the expanding hot QCD medium generated in relativistic heavy-ion collisions. Due to the strong coupling between heavy quarks and the bulk medium, the evolution of heavy quarks can be effectively characterized by a diffusion equation. This approach assumes the instantaneous kinetic thermalization of heavy quarks following their production in nuclear collisions. The local density of heavy quarks is intrinsically coupled to the velocity profile of the hot QCD medium. By incorporating the fluid velocity profiles provided by a hydrodynamic model directly into the diffusion equation, we utilize the deep neural network (DNN) to efficiently determine the heavy-quark evolution. Furthermore, this work provides a valuable reference for the application of deep learning techniques to the treatment of non-thermalized heavy-quark dynamics. The rapid calculation of heavy-quark diffusion using DNNs further facilitates the study of heavy-quark coalescence within a large ensemble of fluctuating hot media.

Figures (6)

• Figure 1: The fluid velocity distributions at the central rapidity region for Pb-Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV generated by MUSIC package, with an impact parameter of $b = 6$ fm, are presented. The direction and color of the arrows represent the local velocity vectors and their corresponding magnitudes, respectively. The four panels illustrate the temporal evolution of the velocity distribution at distinct proper time points.
• Figure 2: Schematic architecture of the PINN utilized to resolve the charm quark density $\rho(\tau, x, y)$. The model incorporates the governing diffusion equation, initial condition and boundary conditions as physical constraints.
• Figure 3: The total loss function $\mathcal{L}(\theta)$ of the PINNs varying with the number of training epochs.
• Figure 4: Charm quark density $\rho_{T}(\tau, x, y)$ at mid-rapidity in 5.02 TeV Pb-Pb collisions with an impact parameter $b=6.0$ fm. The upper and lower panels display the results obtained from the fourth-order Runge-Kutta (RK4) method and the PINN framework, respectively. The color scale indicates the local density magnitude.
• Figure 5: Temporal evolution of the charm quark spatial density $\rho_{T}(\tau, x, y=0)$ at mid-rapidity in 5.02 TeV Pb-Pb collisions with an impact parameter $b=6.0$ fm. Numerical solutions obtained via the fourth-order Runge-Kutta (RK4) method are represented by discrete points, while the PINN predictions are shown as solid lines.