用自监督神经网络直接求解恒星结构方程，无需网格和数据。
Learning the Stellar Structure Equations via Self-supervised Physics-Informed Neural Networks
- 构建物理信息神经网络，输入边界条件与化学成分，输出连续的恒星内部参数分布。
- 相比传统方法误差仅3.06%，决定系数达99.98%，精度接近基准模型。
- 适合需要快速模拟海量恒星演化或研究物理机制的天体物理学者。
恒星天体物理学依赖于对恒星内部物理状态的精确描述。传统求解器如 \texttt{MESA} 采用自适应有限差分法，当进行大规模恒星种群合成（超过10^9颗）时计算成本高且难以扩展。本文提出一种自监督物理信息神经网络（PINN）框架，以无网格、全可微方式求解恒星结构方程（静力平衡与热平衡）。模型输入为恒星边界条件（中心与表面）及化学成分，通过物理损失项强制满足控制方程，学习质量 $M_r(r)$、压强 $P(r)$、密度 $ρ(r)$、温度 $T(r)$、光度 $L_r(r)$ 的连续径向分布。为引入真实微观物理，引入辅助神经网络，将状态方程和消光表近似为局部热力学状态的光滑可微函数，替代传统查表方式，实现端到端训练。训练完成后，模型可在整个径向域生成连续解，无需离散化或插值。在多种恒星质量范围内的基准 \texttt{MESA} 模型验证中，平均相对绝对误差为 3.06%，平均 $R^2$ 得分为 99.98%。据我们所知，这是首个实现完全自监督、无数据条件下求解恒星结构方程的 PINN 示例。该工作为可扩展的物理引导恒星内部模拟奠定基础，并为未来拓展至时变恒星演化打开新路径。
原文摘要 · Abstract (English)
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 $ρ(r)$, 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.
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