Neural Network Construction of the Equation of State from Relativistic ab initio Calculations

TL;DR

The work addresses the challenge of constraining the nuclear matter EOS at supra-saturation densities by integrating ab initio RBHF calculations in the full Dirac space with a Swish-activated neural-network ensemble. The authors train thousands of networks on low-density RBHF data, enforce thermodynamic consistency, and select 264 models to extrapolate the EOS to high densities, quantifying uncertainty through model spread. The extrapolated EOS yields a symmetry energy of $E_{sym}(5rho0)=136.0\pm52.8$ MeV, a pressure of $P(5rho0)=346.3\pm97.4$ MeV/fm$^3$, a maximum neutron-star mass of $M_{max}=2.18\pm0.18\,M_\odot$, and a tidal deformability of $\Lambda_{1.4M_\odot}=532\pm34$, compatible with NICER and GW170817 constraints. This demonstrates a general, data-driven framework that couples ab initio dense-matter input with machine learning to probe the high-density EOS, with potential extensions to include chiral EFT inputs for further refinement.

Abstract

Constraining the nuclear matter equation of state (EOS) beyond saturation density is a central goal of nuclear physics and astrophysics. While the relativistic Brueckner-Hartree-Fock (RBHF) theory, an \textit{ab initio,} non-perturbative nuclear many-body theory starting from realistic interactions, accurately describes nuclear matter properties near the saturation density $ρ_0 \approx 0.16$ fm$^{-3}$, its applicability is currently limited to densities up to $3 ρ_0$, necessitating a reliable extrapolation to higher densities. In this work, we employ supervised machine learning to train thousands of fully connected neural networks on low-density RBHF data. By enforcing thermodynamic consistency and smoothness, we finally select a subset of 264 optimal models. These models employ the Swish activation function, which we identify as the most reliable choice for stable extrapolation after extensive testing and comparison. Using these models to extend the EOS over the full density range, we obtain the nuclear matter symmetry energy and then compute the neutron star mass-radius relation and tidal deformability, which are in a great harmony with current astronomical observations. The corresponding extrapolation uncertainty originates from the combined contributions of both the 264 optimal models and the linear regression on nuclear matter EOS, yielding a symmetry energy of $E\mathrm{_{sym}(5ρ_0)=136.0 \pm 52.8 MeV}$, a pressure of $P(5ρ_0) = 346.3 \pm 97.4 \mathrm{MeV/fm^{3}}$, a maximum neutron star mass of $M\mathrm{_{max}=2.18 \pm 0.18} M_{\odot}$, and a tidal deformability of $Λ_{1.4M_\odot} = 532 \pm 34$. This work establishes a general and data-driven framework to explore dense matter EOS by integrating \textit{ab initio} calculations with modern machine learning techniques.

Figures (5)

• Figure 1: Schematic illustration of the feedforward neural network.
• Figure 2: Pressure of neutron star matter as a function of nucleon number density obtained using four activation functions: Swish (red solid line), ReLu (blue dash-dotted line), Sigmoid (green dashed line) and Tanh (orange dotted line). The white area indicates the part of the dataset used for training, whereas the gray-shaded area represents the extrapolation region where the model predictions are evaluated.
• Figure 3: Distribution and selection of 1000 neural networks based on MAPE. The dashed line indicates the 1.5% validation MAPE threshold. Minimum validation MAPE (0.95$\%$) achieved by the neural network with random seed 577 is marked with a purple star.
• Figure 4: Left Panel: Nuclear matter symmetry energy $E_\mathrm{{sym}}$ as a function of density $\rho$. The blue solid lines denote the neural networks satisfying $\mathrm{MAPE_{validation}} \le 1.5 \%$, while the purple solid line denotes the one with the minimum $\mathrm{MAPE_{validation}}$. The green shaded regions represent $95\%$ confidence interval (CI) of the posterior distributions from 2024-Tsang-NatureAstronomy. Right Panel: Neutron star matter pressure $P$ as a function of density $\rho$. The blue shaded region corresponds to the uncertainty obtained from the neural networks with $\mathrm{MAPE_{validation}} \le 1.5\%$, while the red shaded regions correspond to the $95 \%$ CI of $E_{\rm sym}$, derived from the simple linear regression in Equation \ref{['equa1']}. The orange shaded regions correspond to the $95\%$ CI of the posterior distribution from Koehn_2025_PRX.
• Figure 5: Left Panel: Neutron star mass-radius relation. The blue and red bands represent the uncertainties from the neural networks alone and from the combined NN and linear regression, respectively. The horizontal dotted line marks 2$M_{\odot}$. The inner and outer shaded contours indicate the mass-radius constraints from NICER's analysis of PSR J0030+0451 Vinciguerra_2024_APJ, PSR J0740+6620 Salmi_2024_APJ, PSR J0437-4715 Choudhury_2024_APJL and PSR J0614-3329 Lucien_2025_ArXiv. Right Panel: Neutron star tidal deformability $\Lambda$ as a function of mass. Our results (blue and red regions) are compared with the tidal deformabilities for the two neutron stars in the merger event GW170817 as reported in Margherita_2019_PRL (open squares), as well as the value of $\Lambda_{1.4M_{\odot}}$ extracted from GW170817 Abbott_2018_PRL (open circle).