Asteroseismology of solar-like oscillators: emulating individual mode frequencies with a branching neural network

TL;DR

This work tackles the challenge of inferring stellar fundamental properties from solar-like oscillations by marrying a neural emulator with rigorous Bayesian inference. The authors introduce PITCHFORK, a branching multilayer perceptron that rapidly predicts both classical observables and 35 individual radial-mode frequencies from a MESA+GYRE model grid, augmented by PCA for dimensionality reduction. They implement a vectorised multivariate Gaussian likelihood that accounts for observational noise, emulator uncertainty, and a flexible Gaussian-process-based surface term, enabling fully marginalised posterior distributions for parameters such as $M_{ ext{ini}}$, $Z_{ ext{ini}}$, $Y_{ ext{ini}}$, $\alpha_{\text{MLT}}$, $\tau$, and surface-term coefficients. Validation through hare-and-hounds exercises and benchmark stars (the Sun and 16 Cygni A/B) demonstrates accurate recovery of fundamental properties and realistic uncertainty propagation, while highlighting current limitations in mode-frequency precision and the need for ensemble emulators and non-radial-mode treatment in future work. Overall, this framework offers a scalable, statistically robust path to exploit detailed asteroseismic data for precise stellar parameter inference and systematics treatment ahead of upcoming data rich in individual mode frequencies.

Abstract

Accurately measuring stellar ages and internal structures is challenging, but the inclusion of asteroseismic observables can substantially improve precision. However, the curse of dimensionality means this comes at a high computational cost when using standard interpolation methods across grids of stellar models. Furthermore, without a rigorous treatment of random uncertainties in grid-based modelling, it is not possible to address systematic errors in stellar models. We present PITCHFORK -- a multilayer perceptron neural network with a branching architecture capable of rapid emulation of both classical stellar observables and individual asteroseismic oscillation modes of solar-like oscillators. PITCHFORK can predict the classical observables $T_{\text{eff}}$, $L$, and $\left[\mathrm{Fe}/\mathrm{H}\right]$ with precisions of $5.88\,\text{K}$, $0.014\,\text{L}_{\odot}$, and $0.001\,\text{dex}$, respectively, and can predict 35 individual radial mode frequencies with a uniform precision of $0.02$ per cent. PITCHFORK is coupled to a vectorised Bayesian inference pipeline to return well-sampled and fully marginalised posterior distributions. We validate our rigorous treatment of the random uncertainties -- including the asteroseismic surface effect -- in an extensive hare-and-hounds exercise. We also demonstrate our ability to infer the stellar properties of benchmark stars -- namely, the Sun and the binary stars 16 Cygni A and B. This work demonstrates a computationally scalable and statistically robust framework for stellar parameter inference of solar-like oscillators using individual asteroseismic mode frequencies. This provides a foundation for the treatment of systematics in preparation for the imminent abundance of asteroseismic data from future missions.

Figures (17)

• Figure 1: Top: hexbin plot showing counts of model grid points across the HR-diagram. Bottom: distributions of model input parameters used.
• Figure 2: The structure of a single neuron. Inputs are combined with weights via a dot product, with a bias term applied. The result is passed through an activation function which scales the neuron output. The neuron output is passed on to the next neuron as an input.
• Figure 3: Pitchfork prediction precision for the classical observables. Top: hexbin plot showing mean percentage error averaged across the classical observables over the HR-diagram. Bottom: distributions of test set residuals for each classical observable.
• Figure 4: Pitchfork prediction precision for the individual mode frequencies. Top: hexbin plot showing mean percentage error averaged across all individual mode frequencies (radial orders $(6\leq n\leq40)$) over the HR-diagram. Bottom: distributions of test set residuals on each individual mode frequency, with radial order indicated in the top right.
• Figure 5: Examples of the mode frequency component of the different covariances matrices used in defining the multivariate Gaussian likelihood function. (a): the observational noise component $\boldsymbol\Sigma_{\text{obs}}$. (b): the Pitchfork component, $\boldsymbol{\Sigma}_{\psi}$, from emulation error. (c): the surface correction component $\boldsymbol{\Sigma}_{\text{surf}}$ from the Gaussian process squared exponential kernel. (d): the combined covariance matrix $\boldsymbol{\Sigma}$.