Physics-informed Neural Networks Achieve Angular-Momentum Conservation in Relativistic Spin Hydrodynamics Simulations

The accurate modelling of spin angular momentum is crucial for understanding the behaviour of matter in extreme conditions, such as those created in heavy-ion collisions. Hidefumi Matsuda, Koichi Hattori, and Koichi Murase, from Zhejiang University and Osaka University, address the significant computational challenges associated with simulating relativistic spin hydrodynamics. Their research introduces a novel approach utilising physics-informed neural networks (PINNs) to ensure the accurate numerical conservation of angular momentum, a long-standing problem in the field. By applying this framework to a rotating fluid, the team demonstrates precise angular-momentum conservation and investigates the process of spin conversion driven by rotational viscous effects, identifying a key relationship between thermal vorticity and spin potential. This work represents a substantial step forward in the development of robust and reliable simulations of relativistic spin hydrodynamics.

This work addresses a longstanding challenge in the field, the accurate numerical conservation of total angular momentum, which has hindered progress in simulating relativistic spin hydrodynamics. The team demonstrated this innovative approach by modelling a rotating fluid contained within a cylindrical space, proving that PINNs can maintain precise angular-momentum conservation throughout the simulation. This represents a substantial step forward in the ability to model extreme states of matter.

The research establishes a novel numerical framework leveraging PINNs, a machine learning technique, to solve the governing equations of relativistic spin hydrodynamics. Unlike conventional finite-volume methods which struggle with independent angular momentum conservation, PINNs incorporate physical laws directly into the network’s training process. The study meticulously defines a total loss function that penalises violations of hydrodynamic equations, boundary conditions, and crucially, both local and global angular momentum conservation. This ensures the simulation adheres to fundamental physical principles, delivering a robust and reliable computational tool for investigating complex fluid dynamics.

Experiments show that the PINNs-based framework accurately captures the spin-orbit conversion induced by the rotational viscous effect, a key dissipative process within relativistic spin hydrodynamics. Through detailed numerical analysis, the scientists identified the mismatch between transverse thermal vorticity and the spin potential as the primary driver of this conversion. This discovery provides crucial insight into the interplay between spin and orbital angular momentum, furthering understanding of the behaviour of fluids under extreme conditions. The team’s approach offers a powerful method for studying the intricate relationship between these fundamental quantities.

This breakthrough opens new avenues for exploring the behaviour of matter in extreme astrophysical environments, such as those found in relativistic heavy-ion collisions and the interiors of neutron stars. By accurately simulating the transport of spin angular momentum, this research provides a vital tool for interpreting experimental observations and refining theoretical models. The ability to model spin-orbit conversion with precision will be instrumental in understanding the polarization of particles produced in heavy-ion collisions, and ultimately, in probing the fundamental properties of the quark-gluon plasma. The study’s success in conserving angular momentum within the PINNs framework demonstrates the potential of machine learning techniques to overcome traditional computational limitations in relativistic hydrodynamics. This innovative approach promises to accelerate research in this field, enabling scientists to tackle increasingly complex simulations and unlock new insights into the behaviour of matter at its most fundamental level. The work paves the way for more accurate and comprehensive models of extreme physical phenomena, with implications for both nuclear physics and astrophysics.

PINNs Conserve Angular Momentum in Hydrodynamics

The research team pioneered a novel approach to simulating relativistic spin hydrodynamics, a complex field describing the transport of spin angular momentum, by employing physics-informed neural networks (PINNs). Traditional finite-volume methods struggle to accurately conserve angular momentum alongside other conserved quantities, as angular momentum conservation often isn’t explicitly solved for in these systems. To overcome this limitation, scientists developed a PINN-based framework that directly incorporates angular momentum conservation into the neural network’s training process, ensuring consistent preservation of this crucial physical property. This innovative method achieves accurate numerical conservation of total angular momentum, a significant advancement for modelling relativistic fluids.

Experiments employed a rotating fluid contained within a cylindrical geometry to rigorously test the PINN framework. The study defined a total loss function encompassing hydrodynamic equations, boundary conditions, and both local and global angular momentum conservation, expressed mathematically as ∇μJμxy(x) = 0 and ∫ d3xJtxy(t, x) = const. This loss function penalizes any violation of these conditions during the neural network’s training, effectively forcing the solution to adhere to the laws of physics. Researchers restricted the fluid dynamics to a two-dimensional disk, focusing on angular momentum in the z-direction, defined as Jtxy= xΘty−yΘtx+Σtxy, to simplify the computational setup and enhance precision.

The team investigated the spin-orbit conversion induced by the rotational viscous effect, a fundamental process in relativistic spin hydrodynamics, using a second-order viscous relativistic spin hydrodynamics formulation in (3 + 1) dimensions. The energy-momentum tensor and spin current tensor were defined, satisfying conservation laws of energy-momentum and total angular momentum, and the Landau frame and pseudo-gauge were adopted for simplification. Analysis numerically identified a mismatch between transverse thermal vorticity and the spin potential as the driving force behind this spin conversion, revealing a key mechanism in the dynamics of spinning relativistic fluids. This detailed investigation, enabled by the precise PINN methodology, provides new insights into the interplay between spin and orbital angular momentum in extreme conditions.

PINNs Conserve Angular Momentum in Relativistic Fluids

Scientists achieved accurate conservation of total angular momentum using physics-informed neural networks (PINNs) in computational relativistic spin hydrodynamics. The work addresses a significant computational challenge in simulating the macroscopic transport of spin angular momentum alongside other conserved quantities. Researchers employed PINNs to model a rotating fluid contained within a cylindrical container, demonstrating the framework’s ability to maintain angular-momentum conservation throughout the simulation. This breakthrough delivers a novel approach to numerically solving the complex equations governing relativistic spin fluids.

Experiments revealed the crucial role of the rotational viscous effect in inducing spin conversion, a fundamental process within relativistic spin hydrodynamics. Analysis numerically identified a mismatch between transverse thermal vorticity and the spin potential as the driving force behind this spin conversion. Specifically, the team measured the xy-components of the couple-stress tensor, transverse thermal vorticity, and spin potential, observing a close correlation between the spatial structure of thermal vorticity and the early-time evolution of the couple-stress tensor for Initial Condition F. These measurements confirm the interplay between these quantities in mediating angular momentum exchange.

Further tests, utilising Initial Condition S, showed a pronounced peak in the couple-stress tensor appearing around r= 0.5 at t≈0.02, correlating with the spatial distribution of the spin potential. The initial growth of the couple-stress tensor was driven by a significant mismatch between the spin potential and vanishing transverse thermal vorticity. Data shows that the rotation-rate mismatch enhances the couple-stress tensor, acting as the mediator for angular momentum exchange between orbital and spin components. The study’s results, obtained from a converged model trained with the Adam optimizer and a learning rate between 10−3 and 10−5, validate the PINNs-based methodology.

Time evolution analysis of net orbital and spin angular momentum for γ= 0 and γ= 2 demonstrated mutual conversion between these components. This work, supported by the JSPS KAKENHI under Grant Nos. JP22H01216, JP23H05439, and JP23K13102, and performed on Supercomputer Yukawa-21, provides insights into the dissipative mechanisms governing relativistic spin hydrodynamics.

Spin-Orbit Angular Momentum Interconversion Demonstrated

This work presents a novel computational framework grounded in physics-informed neural networks for simulating relativistic spin hydrodynamics. Researchers successfully addressed the significant challenge of maintaining total angular momentum conservation within these complex simulations, a crucial requirement for accurate modelling. Through investigation of the Barnett and Einstein, de Haas effects, the study numerically demonstrates the interconversion of orbital and spin angular momentum, furthering understanding of this fundamental physical process. Analysis reveals that the conversion between spin and orbital angular momentum is driven by a mismatch between transverse thermal vorticity and the spin potential, which in turn influences the couple-stress tensor responsible for angular momentum exchange. These findings validate the efficacy of the physics-informed neural network methodology and offer new insights into the dissipative mechanisms governing relativistic spin hydrodynamics. The authors acknowledge limitations inherent in numerical calculations and suggest future research could explore more complex physical scenarios to further refine the model.