Physics-informed neural networks for angular-momentum conservation in computational relativistic spin hydrodynamics

TL;DR

This work develops a physics-informed neural network (PINN) framework to simulate relativistic spin hydrodynamics with rigorous angular-momentum conservation. It formulates a second-order theory including spin density $S^{\mu\nu}$, spin potential $\omega_{\mu\nu}$, and a relaxation equation for the couple-stress $\phi^{\mu\nu}$ driven by the rotation-rate mismatch $\rho^{\mu\nu}$, and enforces local and global angular-momentum constraints within the PINN loss. The authors demonstrate mutual spin–orbit conversion in a rotating cylinder, elucidating how rotational dynamics transfer angular momentum between orbital and spin channels through $\phi^{xy}$, $\varpi^{xy}_\perp$, and $\omega^{xy}$, with the mismatch $\rho^{xy}$ as the key driver. The results validate the PINN approach and offer mechanistic insight into dissipative spin transport in relativistic fluids relevant to heavy-ion phenomenology. This framework provides a drop-in method for angular-momentum-conserving simulations of relativistic spin hydrodynamics and a path toward more detailed studies of spin transport in high-energy systems.

Abstract

Theoretical developments in relativistic spin hydrodynamics, which describes the macroscopic transport of spin angular momentum alongside other fundamental conserved quantities, have progressed rapidly since the experimental observation of the global spin polarization of $Λ$ hyperons in relativistic heavy-ion collision experiments. However, numerical simulations of relativistic spin hydrodynamics remain largely unaddressed due to computational challenges, particularly the accurate numerical conservation of total angular momentum. In this work, we propose the use of physics-informed neural networks (PINNs) for computational relativistic spin hydrodynamics. As a concrete application, we consider a rotating fluid confined within a cylindrical container. We show that angular-momentum conservation can be accurately achieved in the PINNs-based numerical framework. Furthermore, we investigate the spin-orbit conversion induced by the rotational viscous effect, which is the intrinsic dissipative process of relativistic spin hydrodynamics. Our analysis numerically identifies the mismatch between the transverse thermal vorticity and the spin potential as the driving mechanism of the spin-orbit conversion.

Figures (2)

• Figure 1: Time evolution of the net orbital and spin angular momentum for $\gamma=0$ and $\gamma=2$. The left and right panels correspond to Initial Conditions F and S, respectively. Each quantity is normalized by the total angular momentum (the sum of net orbital and spin components) at the initial time $t=0$. In both panels, the green and purple curves represent the net orbital and net spin angular momentum, respectively.
• Figure 2: Spacetime evolution of the $xy$ components for the couple-stress tensor, transverse thermal vorticity, and spin potential at $\gamma=2$. The three columns, from left to right, represent these respective quantities. Initial Condition F is displayed in the top row, while the results for Initial Condition S are shown in the bottom row.