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A geometric physics-informed machine learning inference for the neutron star maximum mass and the inverse problem

Rounak Mukherjee, Ritam Mallick

TL;DR

This work tackles the neutron star maximum mass and the NS–BH mass gap by introducing a physics-informed Transformer that leverages the geometry of the M–R sequence, specifically front-bending (FB) and back-bending (BB). An agnostic ensemble of EoS is generated and evolved via the TOV equations to create M–R sequences, from which FB/BB and a reference point define a finite, informative input space. The model performs a two-step inverse mapping: first predicting the maximum-mass point $(M_{max},R_{max})$, then reconstructing the $c_s^2(\varepsilon)$ profile to recover the underlying EoS; physics-informed losses and uncertainty quantification (Monte Carlo dropout and ensembles) are used throughout. Under current observational constraints, the inferred NS maximum mass is $M_{max}=2.477\,M_\odot$ with a canonical radius around 11.5 km for $1.4\,M_\odot$, while unconstrained predictions allow up to $\sim2.889\,M_\odot$; the study demonstrates that a geometry-driven inverse approach can reduce degeneracy and provide a tractable pathway to constrain dense-matter physics as more NS observations become available.

Abstract

The existence of a distinct mass boundary between the heaviest neutron stars and the lightest black holes remains in question. It is an artefact of our ignorance of the properties of matter at supra-nuclear densities, which exist in the cores of neutron stars. The study addresses these problems with a physics-informed machine learning approach, guided by astrophysical observations. The Transformer model is trained on an agnostically generated ensemble of equations of state. Two geometric parameters are defined on the mass-radius sequence of a neutron star--the front bending and the back bending. The transformer provides a two-step solution: first, the model predicts the maximum mass and radius using the bending parameters. Second, it predicts the square of the sound speed profile, completing the inverse mapping. The prediction is that massive neutron stars form when the sound speed peaks at low density, leading to strong back-bending and an early phase transition to quark matter. Massive stars favour a stiff equation of state at low density, and the density of matter at the star's core is sufficiently small. The maximum mass for a neutron star predicted by the astrophysical constrained transformer model is $2.477$ solar masses, and a minimum radius of about $11.498$ km for a neutron star of $1.4$ solar masses.

A geometric physics-informed machine learning inference for the neutron star maximum mass and the inverse problem

TL;DR

This work tackles the neutron star maximum mass and the NS–BH mass gap by introducing a physics-informed Transformer that leverages the geometry of the M–R sequence, specifically front-bending (FB) and back-bending (BB). An agnostic ensemble of EoS is generated and evolved via the TOV equations to create M–R sequences, from which FB/BB and a reference point define a finite, informative input space. The model performs a two-step inverse mapping: first predicting the maximum-mass point , then reconstructing the profile to recover the underlying EoS; physics-informed losses and uncertainty quantification (Monte Carlo dropout and ensembles) are used throughout. Under current observational constraints, the inferred NS maximum mass is with a canonical radius around 11.5 km for , while unconstrained predictions allow up to ; the study demonstrates that a geometry-driven inverse approach can reduce degeneracy and provide a tractable pathway to constrain dense-matter physics as more NS observations become available.

Abstract

The existence of a distinct mass boundary between the heaviest neutron stars and the lightest black holes remains in question. It is an artefact of our ignorance of the properties of matter at supra-nuclear densities, which exist in the cores of neutron stars. The study addresses these problems with a physics-informed machine learning approach, guided by astrophysical observations. The Transformer model is trained on an agnostically generated ensemble of equations of state. Two geometric parameters are defined on the mass-radius sequence of a neutron star--the front bending and the back bending. The transformer provides a two-step solution: first, the model predicts the maximum mass and radius using the bending parameters. Second, it predicts the square of the sound speed profile, completing the inverse mapping. The prediction is that massive neutron stars form when the sound speed peaks at low density, leading to strong back-bending and an early phase transition to quark matter. Massive stars favour a stiff equation of state at low density, and the density of matter at the star's core is sufficiently small. The maximum mass for a neutron star predicted by the astrophysical constrained transformer model is solar masses, and a minimum radius of about km for a neutron star of solar masses.
Paper Structure (9 sections, 10 equations, 8 figures, 3 tables)

This paper contains 9 sections, 10 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Formalism
  3. Results
  4. Summary and Discussion
  5. Acknowledgments
  6. Data Availability
  7. Architecture of the Physics Informed Transformer Model
  8. Coupled Fourier Equations
  9. Tables

Figures (8)

  • Figure 1: The proposed bending parameters, FB & BB are defined based on the three points, namely ($\mathrm{M_{max},\,R_{max}}$), ($\mathrm{M_{bend},\,R_{bend}}$), and ($\mathrm{M_{ref},\,R_{ref}}$) that are found from the geometry of the M-R sequence. The front bending and back bending are defined as FB = $\cot \Phi$ and BB = $\cot \Theta$. In the figure, a typical M-R sequence with back-bending and front-bending is depicted, along with their characterisation in the left and right panels, respectively.
  • Figure 2: Connecting astrophysical observations with the bending parameters. (Left panel) The probable region of M-Rs with BB that satisfy observations is depicted. (Right panel) The probable region of M-Rs with FB that satisfies observations is shown. The bounded regions are obtained by imposing bounds on the bending parameters based on observations.
  • Figure 3: A schematic representation of the unified transformer model framework showing the two-step network: First, the forward solution to obtain $\mathrm{(M_{max},R_{max})}$ and then the inverse reconstruction of $c_s^2$ from NS observables.
  • Figure 4: The figure shows the $c_s^2$ peak profile as energy density and the M-R sequence coded as $c_s^2$ peak profile. The rise in $c_s^2$ after the CET band dictates the bending properties in the M-R sequence. (Left panel) The $c_{s}^{2}$ vs $\varepsilon$ plot where the energy density is normalised with respect to the nuclear saturation energy density. This figure shows the distribution of the $c_s^2$ peak values in the agnostic data. (Right panel) M-R sequences of the corresponding groups of the square of the speed of sound profiles. The ${c_s}^2$ profiles have been grouped by the location of their maxima, and this shows that bending in the M-R sequence depends on the location of the first maxima of ${c_s}^2$.
  • Figure 5: The figure shows the distribution of the bending parameters and the amount of information carried by them about the maximum mass of NSs. (Left panel) The distribution of the actual bending parameters vs the predicted values, where we used a Fourier fit of $n_{F}\,=\, 120$. (Right panel) The actual vs predicted values of $\mathrm{M_{max}}$, where we used $n_{F}\,=\,120$
  • ...and 3 more figures