Machine and Deep Learning Regression for Compact Object Equations of State

TL;DR

The paper tackles the inverse stellar structure problem: can one infer the dense-matter equation of state (EoS) of neutron- and quark-star matter from mass–radius observations? It develops a data-driven framework that combines polytropic EoS parametrizations (via Read-2009), hadronic and CFL/MIT bag quark-matter models, and the Tolman–Oppenheimer–Volkoff equations to generate extensive $M$–$R$ datasets. Regression with diverse machine learning and deep learning models (including XGBoost and a DNN-3) is used to map $M$–$R$ inputs to a set of energy-density targets $\epsilon(P)$, effectively reconstructing the underlying EoS. The results show strong performance at lower densities and reveal higher variance at high densities, with QS EoS generally easier to reconstruct; the work highlights the potential of data-driven EoS inference and future directions for object classification (neutron-star vs quark-star) based on observational data.

Abstract

A central open problem in nuclear physics is the determination of a physically robust equation of state (EoS) for dense nuclear matter, which directly informs our understanding of the internal composition and macroscopic properties of compact objects such as neutron stars and quark stars. Traditional efforts have relied primarily on theoretical modeling grounded in nuclear and particle physics, with subsequent validation against empirical constraints from heavy ion collisions and, increasingly, multimessenger astrophysical observations. Recent developments, however, have introduced complementary analytical strategies that merge theoretical modeling with advanced data driven methodologies. In particular, Bayesian inference, machine learning, and deep learning have emerged as powerful tools for constraining the EoS and extracting physical insight from complex observational datasets. In this work, we employ state of the art machine learning and deep learning techniques to analyze mass radius relations of compact objects with the aim of reconstructing or inferring their underlying equations of state. The analysis is based on an extensive library of physically consistent, multimodal EoSs for neutron stars and a corresponding set for quark stars, each constructed to satisfy established theoretical and observational constraints. By leveraging the predictive capacity of these computational frameworks, we demonstrate the potential of data-driven approaches to provide refined insights into the behavior of matter at supranuclear densities and to contribute to a more unified understanding of the dense matter EoS.

Figures (14)

• Figure 1: Grids of polytropes for $n=4$ mass density segments and a) $l=2$ available choices $\{1,4\}$ or b) $l=4$ available choices $\{1,2,3,4\}$ for $\Gamma$ values. The green grid corresponds to polytropes that start at the nuclear saturation pressure of HLPS-2 ($1.722$${\rm MeV\ fm}^{-3}$), while the yellow one to polytropes that start at the nuclear saturation pressure of HLPS-3 ($2.816$$\rm MeV \ fm^{-3}$). The red line features (from left to right) the sequence $\{\Gamma:1\rightarrow{}1\rightarrow{}1\rightarrow{}4\}$ in a) and $\{\Gamma:4\rightarrow{}4\rightarrow{}1\rightarrow{}4\}$ in b). The blue line features (from left to right) the sequence $\{\Gamma:4\rightarrow{}4\rightarrow{}1\rightarrow{}1\}$ in a) and the sequence $\{\Gamma:2\rightarrow{}1\rightarrow{}2\rightarrow{}1\}$ in b).
• Figure 2: Plots of the $M-R$ curves of mock Neutron Stars EoSs. a) The $M-R$ curves of all $512$ mock EoSs have derived from HLPS-2 and HLPS-3 'main' EoSs, for all $\Gamma$ combinations in four mass density segments. b) The $M-R$ curves of the $304$ out of $512$ mock EoSs, exceed the pressure of $850$$\rm MeV\ fm^{-3}$. The $M-R$ curves of HLPS-2 (green) and HLPS-3 (yellow) are also included, using gray endings to mark the violation of causality.
• Figure 3: Scanning the stability window region for CFL quark matter with $m_s=95$$\rm MeV$. Keeping $m_s$ value constant, Eq. \ref{['bag_high_constraint']} yields to the equation of a curve, namely the $B_{max}=B_{max}(\Delta)$ curve, as shown with blue color and dashed-line style. The orange dashed-line curve marks the minimum value of $57$$\rm MeV\cdot fm^{-3}$. The coordinates of the green points, which scan the stability region, are valid combinations of $B$ and $\Delta$ values, for $m_s=95$$\rm MeV$.
• Figure 4: Plots of the $M-R$ curves of Quark Stars EoSs. The $381$$M-R$ curves of MIT bag model EoSs are shown in red, while the $510$$M-R$ curves of CFL model EoSs are shown in blue.
• Figure 5: Sampling example of mass and radius data, using $8$ points from each of the M-R curves of the following polytropic EoSs: HLPS-2_ADDDL (blue), HLPS-2_DCDCL (orange), HLPS-3_ADDDL (green) and HLPS-3_DCDCL (red). The respective M-R curves are plotted too. The graphs depict: a) the noise-free basic observation of M-R points for each EoS, b) $1$ random M-R observation per EoS, c) $10$ random M-R observations per EoS and d) $100$ random M-R observations per EoS. Each random observation includes additional observational noise: $\Delta M\sim0.1M_\odot$ and $\Delta R\sim0.5 \ {\rm km}$.