Solving the QCD effective kinetic theory with neural networks

TL;DR

The paper tackles the expensive evaluation of the eight-dimensional collision integral in the QCD effective kinetic theory (EKT) by training a neural-network surrogate to learn the local mapping from the distribution function to its collisional time derivative $\partial_t f_{\mathrm{coll}}$. It leverages conformal and rest-frame symmetries to reduce data requirements, discretizes momentum space with a coarse grid to accelerate training, and uses Monte Carlo evaluations to generate training targets. The authors demonstrate accurate reproduction of isotropic and anisotropic evolutions, achieving orders-of-magnitude speed-ups (from days to minutes) while providing uncertainty estimates via an ensemble of networks. This surrogate enables feasible event-by-event, 3D pre-equilibrium simulations in heavy-ion collisions and offers a scalable pathway to include more realistic QCD dynamics, such as quarks and electromagnetic probes, in future work.

Abstract

Event-by-event QCD kinetic theory simulations are hindered by the large numerical cost of evaluating the high-dimensional collision integral in the Boltzmann equation. In this work, we show that a neural network can be used to obtain an accurate estimate of the collision integral in a fraction of the time required for the ordinary Monte Carlo evaluation of the integral. We demonstrate that for isotropic and anisotropic distribution functions, the network accurately predicts the time evolution of the distribution function, which we verify by performing traditional evaluations of the collision integral and comparing several moments of the distribution function. This work sets the stage for an event-by-event modeling of the pre-equilibrium initial stages in heavy-ion collisions.

Figures (7)

• Figure 1: Schematic overview of the use case of our neural network. It takes the distribution function (with thermal equilibrium subtracted) as input and provides a prediction for the collision kernel $\mathcal{C}$.
• Figure 2: Randomly chosen samples of our training data for the isotropic case. On the left, we show the input data $p^3(f(p) - f_{\mathrm{eq}}(p))$ and on the right, we show the sum of the collision kernels $p^3C(f[p])$. The tick marks in the $y$-axis have a spacing $8\times 10^{-4}T^3$ in the inputs and $2.8\times 10^{-3}~T^4$ for the outputs. Panels with a larger number of tick marks correspond to distributions that are further from equilibrium.
• Figure 3: Training (lighter) and validation (darker) loss during each epoch of the training. The left panel shows the best (blue) and worst (orange) trainings for the $C_{12}$ kernel. The right panel shows the same but for $C_{22}$.
• Figure 4: Three different evolutions. Left plots show the distribution functions at different times and right plots show their corresponding collision kernel with the same color. Dashed lines are the results from the Monte Carlo simulation, while the solid lines with error bars are the evolution obtained from the neural networks. The functional form of the initial distribution in the top two panels have not been used in the training of the network.
• Figure 5: Number density as a function of time. Solid lines correspond to the neural network prediction, meanwhile, colored dashed lines are the EKT prediction at the same time in the evolution. Colors correspond to the same of the ones plotted in Figure \ref{['fig:distributions_1d']}.