Learning the Stellar Structure Equations via Self-supervised Physics-Informed Neural Networks

Stellar astrophysics relies critically on accurate descriptions of the physical conditions inside stars. Traditional solvers such as \texttt{MESA} (Modules for Experiments in Stellar Astrophysics), which employ adaptive finite-difference methods, can become computationally expensive and challenging to scale for large stellar population synthesis ($>10^9$ stars). In this work, we present an self-supervised physics-informed neural network (PINN) framework that provides a mesh-free and fully differentiable approach to solving the stellar structure equations under hydrostatic and thermal equilibrium. The model takes as input the stellar boundary conditions (at the center and surface) together with the chemical composition, and learns continuous radial profiles for mass $M_r(r)$, pressure $P(r)$, density $\rho(r)$, temperature $T(r)$, and luminosity $L_r(r)$ by enforcing the governing structure equations through physics-based loss terms. To incorporate realistic microphysics, we introduce auxiliary neural networks that approximate the equation of state and opacity tables as smooth, differentiable functions of the local thermodynamic state. These surrogates replace traditional tabulated inputs and enable end-to-end training. Once trained for a given star, the model produces continuous solutions across the entire radial domain without requiring discretization or interpolation. Validation against benchmark \texttt{MESA} models across a range of stellar masses yields a Mean Relative Absolute Error of $3.06\%$ and an average $R^2$ score of $99.98\%$. To our knowledge, this is the first demonstration that the stellar structure equations can be solved in a fully self-supervised and data-free fashion employing PINNs. This work establishes a foundation for scalable, physics-informed emulation of stellar interiors and opens the door to future extensions toward time-dependent stellar evolution.