Achieving angular-momentum conservation with physics-informed neural networks in computational relativistic spin hydrodynamics

TL;DR

This work proposes physics-informed neural networks (PINNs) as a numerical solver for relativistic spin hydrodynamics and demonstrates that the total angular momentum is accurately conserved throughout the fluid evolution by imposing the conservation law directly in the loss function as a training target.

Abstract

We propose physics-informed neural networks (PINNs) as a numerical solver for relativistic spin hydrodynamics and demonstrate that the total angular momentum, i.e., the sum of orbital and spin angular momentum, is accurately conserved throughout the fluid evolution by imposing the conservation law directly in the loss function as a training target. This enables controlled numerical studies of the mutual conversion between spin and orbital angular momentum, a central feature of relativistic spin hydrodynamics driven by the rotational viscous effect. We present two physical scenarios with a rotating fluid confined in a cylindrical container: one case in which initial orbital angular momentum is converted into spin angular momentum in analogy with the Barnett effect, and the opposite case in which initial spin angular momentum is converted into orbital angular momentum in analogy with the Einstein-de Haas effect. We investigate these conversion processes governed by the rotational viscous effect by analyzing the spacetime profiles of thermal vorticity and spin potential. Our PINNs-based framework provides the first numerical evidence for spin-orbit angular momentum conversion with fully nonlinear computational relativistic spin hydrodynamics.

Figures (16)

• Figure 1: Schematic illustration of a neural network $f_\psi$, parameterized by trainable parameters $\psi$, approximating the solution $f$ of a partial differential equation
• Figure 2: Conceptual sketch of the rotational viscous effect
• Figure 3: Illustration of fluid dynamics in a 2D spatial disk
• Figure 4: Left: Decrease in the loss function \ref{['Eq:loss']} as the training processn proceeds. Middle: Decrease in the total residual of the governing equations, $\bar{R}^\text{G.E.}_\text{sum} = \sum_i \bar{R}^\text{G.E.}_i$, as the same training process proceeds as in the left panel. Right: Time evolution of the total angular momentum, accurately conserved over the 2D disk; The vertical axis is normalized by its initial value at $t=0$.
• Figure 5: The time evolution of the net orbital and spin angular momentum. All quantities are normalized by the spatially integrated total angular momentum at the initial time $t=0$.