Towards asteroseismology of neutron stars with physics-informed neural networks

TL;DR

This paper demonstrates, for a simplified NS oscillation problem, that physics-informed neural networks can solve the associated generalized eigenvalue problem to obtain accurate eigenfrequencies and eigenfunctions. The authors implement two PINN-based strategies: a coarse s-PINN for interval identification and two eigenfrequency solvers (bisection with s-PINN and a dedicated f-PINN with $\sigma$ as a network parameter). They show that the f-PINN achieves higher frequency accuracy at the expense of longer computation times, while the bisection approach is faster but slightly less precise, and both rely on careful boundary-condition enforcement and hyperparameter tuning. The work highlights the potential of PINNs to extend to more complex, non-spherical, rotating, or magnetized neutron star configurations, offering a mesh-free framework that naturally incorporates physics constraints. Overall, this study provides a principled path toward integrating more physics into NS asteroseismology via PINNs, enabling flexible exploration of eigenfrequencies and mode structures with differentiable, boundary-aware solutions.

Abstract

The study of the gravitational wave signatures of neutron star oscillations may provide important information of their interior structure and Equation of State (EoS) at high densities. We present a novel technique based on physically informed neural networks (PINNs) to solve the eigenvalue problem associated with normal oscillation modes of neutron stars. The procedure is tested in a simplified scenario, with an analytical solution, that can be used to test the performance and the accuracy of the method. We show that it is possible to get accurate results of both the eigenfrequencies and the eigenfunctions with this scheme. The flexibility of the method and its capability of adapting to complex scenarios may serve in the future as a path to include more physics into these systems.

Figures (7)

• Figure 1: The figure shows the values of $\eta_1$ at the outer boundary $r=1$ with respect to the frequency. The frequencies that correspond to $\eta_1 \, _{\:r=1}=0$ are the wanted eigenfrequencies.
• Figure 2: The cartoon depicts the s-PINN with one input node, $r$, and two output nodes, $\tilde{\eta}_1$ and $\tilde{\eta}_2$. The network consists of two hidden-layers with 256 neurons each.
• Figure 3: The radial profiles of $\eta_1$ (upper figure) and $\eta_2$ (lower figure) for the first frequency, $\sigma_0 = 3.34$, for 256 neurons and different thresholds for the loss, using the f-PINN.
• Figure 4: The radial profiles of $\eta_1$ (upper plot) and $\eta_2$ (lower plot) for the higher-order harmonic, $\sigma_5=20.22$, for 256 neurons and different thresholds for the loss, using the f-PINN.
• Figure 5: Upper panel: Mean square error of the variables $\eta$ across all eigenfrequencies for various values of the loss function, $\mathcal{L}_f$, using the f-PINN. Bottom panel: Relative error of the eigenfrequencies for these same loss values. The network has 256 neurons.