Machine Learning for Conservative-to-Primitive in Relativistic Hydrodynamics

Tobias Dieselhorst,Sebastiano Bernuzzi,William Cook,David Radice

doi:10.3390/sym13112157

Tobias Dieselhorst, Sebastiano Bernuzzi + Show 2 more

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https://doi.org/10.3390/sym13112157

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Abstract

The numerical solution of relativistic hydrodynamics equations in conservative form requires root-finding algorithms that invert the conservative-to-primitive variables map. These algorithms employ the equation of state of the fluid and can be computationally demanding for applications involving sophisticated microphysics models, such as those required to calculate accurate gravitational wave signals in numerical relativity simulations of binary neutron stars. This work explores the use of machine learning methods to speed up the recovery of primitives in relativistic hydrodynamics. Artificial neural networks are trained to replace either the interpolations of a tabulated equation of state or directly the conservative-to-primitive map. The application of these neural networks to simple benchmark problems shows that both approaches improve over traditional root finders with tabular equation-of-state and multi-dimensional interpolations. In particular, the neural networks for the conservative-to-primitive map accelerate the variable recovery by more than an order of magnitude over standard methods while maintaining accuracy. Neural networks are thus an interesting option to improve the speed and robustness of relativistic hydrodynamics algorithms.

Highlights

Relativistic hydrodynamics is often written as a set of conservation laws ∂u ∂t + ∂Fi(u) ∂xi = s, (1)where the state vector u = (D, Si, τ) is composed of the conserved variables: rest-mass density, momentum density and the energy density relative to D, τ := E − D, all measured in the laboratory frame [1] (i = 1, 2, 3)
∂Fi(u) ∂xi where the state vector u = (D, Si, τ) is composed of the conserved variables: rest-mass density, momentum density and the energy density relative to D, τ := E − D, all measured in the laboratory frame [1] (i = 1, 2, 3)
We explored artificial neural networks (NN) for the transformation of conservative quantities to primitives (C2P) in relativistic hydrodynamics

Summary

Introduction

In its simplest form, the EOS is the thermodynamical relation connecting the pressure to the fluid’s rest-mass density and internal energy, p = p(ρ, ). The Helmholtz and Lattimer– Swesty EOS used in various relativistic astrophysics simulations [2,3] return the pressure as function of the rest-mass density (or baryon number density), the temperature and the electron fraction, i.e., p = p(ρ, T, Ye). In this case, Equation (1) can contain additional continuity equations for the rest-mass or number densities of the different species (e.g., for the variable DYe). These highresolution shock-capturing (HRSC) methods are routinely employed for the simulation of relativistic jets [1,7], supernova explosions [8] and binary neutron star mergers in general relativity [9,10,11]

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