Learning the Stellar Structure Equations via Self-supervised Physics-Informed Neural Networks
Authors: Manuel Ballester, Santiago Lopez-Tapia, Seth Gossage, Patrick Koller, Philipp M. Srivastava, Ugur Demir, Yongseok Jo, Almudena P. Marquez, Christoph Wuersch, Souvik Chakraborty, Vicky Kalogera, Aggelos Katsaggelos
Organizations: SkAI Institute (NSF–Simons AI Institute for the Sky), Chicago, IL, USA · Department of Electrical and Computer Engineering, Northwestern University, Chicago, IL, USA · CIERA, Northwestern University, Chicago, IL, USA · Department of Physics and Astronomy, Northwestern University, Chicago, IL, USA · Department of Mathematics, University of Cadiz, Cadiz, Spain · OST Eastern Switzerland University of Applied Sciences, Switzerland · Indian Institute of Technology (IIT) Delhi, New Delhi, India
Stellar astrophysics relies critically on accurate descriptions of the physical conditions inside stars. Traditional solvers such as 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 (>109 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 Mr(r), pressure P(r), density ρ(r), temperature T(r), and luminosity Lr(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 MESA models across a range of stellar masses yields a Mean Relative Absolute Error of 3.06% and an average R2 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.
Figures & tables
Figure 1: Schematic of the Physics-Informed Neural Network (PINN) framework for stellar structure modeling. The network maps the normalized enclosed mass ( M^r ) to the stellar state variables (pressure, radius, temperature, and luminosity). The training objective combines a physics-based loss CPDE , defined as the L2 norm of the equation residuals at collocation points, and a boundary-condition term CBC . An optional data-driven term CDD can be included in supervised settings, but is omitted in the fully self-supervised formulation considered in this work.
Figure 2: Architecture of the auxiliary network used to learn thermodynamic closures. The same architecture is employed for both the EOS (predicting ρ ) and opacity ( κ ), enabling smooth and differentiable approximations of tabulated microphysics.
Figure 3: Stellar profile predictions for relatively low- and high-mass stars. Each subfigure compares the ground-truth solution obtained from the classical MESA solver (blue) with the PINN prediction (red) for the normalized luminosity, pressure, radius, density, and temperature as functions of enclosed mass.
Figure 4: Stellar profile predictions for relatively low- and high-mass stars. Each subfigure compares the ground-truth solution obtained from the classical MESA solver (blue) with the PINN prediction (red) for the normalized luminosity, pressure, radius, density, and temperature as functions of enclosed mass.
Figure 5: Fully supervised model with limited data. Performance of the same architecture but now (a) trained purely on MESA data without enforcing the governing equations, and (b) trained with on MESA data and the governing equations. These models behave as intelligent interpolator rather than solvers.
Figure 6: Performance across stellar masses. Evaluation of the PINN model for stars with initial masses ranging from 0.4 to 9.9M⊙ . The plot shows the MRAE for each physical quantity as well as the total error.
Figure 7: Hertzsprung–Russell diagram. Comparison between the proposed PINN model (dashed lines) and MESA simulations (solid lines) for stars with initial masses between 0.6 and 20M⊙ . The x-axis shows log(Teff) and the y-axis log(L/L⊙) . While the global trends are captured, noticeable noise and deviations highlight the limitations of the current model in time-dependent settings.
Physics-informed neural networks (PINNs) offer a unified framework for solving both forward and inverse problems of differential equations, yet their performance and physical consistency strongly depend on how governing laws are incorporated. In this work, we present a systematic comparison of different thermodynamic structure-informed neural networks by incorporating various thermodynamics formulations, including Newtonian, Lagrangian, and Hamiltonian mechanics for conservative systems, as well as the Onsager variational principle and extended irreversible thermodynamics for dissipative systems. Through comprehensive numerical experiments on representative ordinary and partial differential equations, we quantitatively evaluate the impact of these formulations on accuracy, physical consistency, noise robustness, and interpretability. The results show that Newtonian-residual-based PINNs can reconstruct system states but fail to reliably recover key physical and thermodynamic quantities, whereas structure-preserving formulation significantly enhances parameter identification, thermodynamic consistency, and robustness. These findings provide practical guidance for principled design of thermodynamics-consistency model, and lay the groundwork for integrating more general nonequilibrium thermodynamic structures into physics-informed machine learning.
Guojie Li, Liu Hong
School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China.
Physics-informed neural networks (PINNs) have emerged as a promising mesh-free paradigm for solving partial differential equations, yet adoption in science and engineering is limited by slow training and modest accuracy relative to modern numerical solvers. We introduce the Sequential Correction Algorithm for Learning Efficient PINN (Scale-PINN), a learning strategy that bridges modern physics-informed learning with numerical algorithms. Scale-PINN incorporates the iterative residual-correction principle, a cornerstone of numerical solvers, directly into the loss formulation, marking a paradigm shift in how PINN losses can be conceived and constructed. This integration enables Scale-PINN to achieve unprecedented convergence speed across PDE problems from different physics domain, including reducing training time on a challenging fluid-dynamics problem for state-of-the-art PINN from hours to sub-2 minutes while maintaining superior accuracy, and enabling application to representative problems in aerodynamics and urban science. By uniting the rigor of numerical methods with the flexibility of deep learning, Scale-PINN marks a significant leap toward the practical adoption of PINNs in science and engineering through scalable, physics-informed learning. Codes are available at https://github.com/chiuph/SCALE-PINN.
Pao-Hsiung Chiu, Jian Cheng Wong, Chin Chun Ooi +3
Institute of High Performance Computing, A*STAR, Singapore. · Centre for Frontier AI Research, A*STAR, Singapore. · School of Mechanical Engineering, Tianjin University, China. +1
We propose a self-supervised physics-informed neural network (PINN) framework that adaptively balances physics-based and data-driven supervision for scientific machine learning under data scarcity. Unlike prior PINNs that rely on fixed or heuristic weighting of physics residuals and data loss, our approach introduces a learnable blending neuron that dynamically adjusts the relative contribution of each term based on their uncertainties. This mechanism enables stable training and improved generalization without manual tuning. To further enhance efficiency, we integrate a transfer learning strategy that reuses representations from related domains and adapts them to new physical systems with limited data. We validate the framework for the prediction of heat transfer in liquid-metal miniature heat sinks using only 87 CFD datapoints, where the adaptive PINN achieves an error <8%, outperforming shallow neural networks, kernel methods, and physics-only baselines. Our framework provides a general recipe for embedding physics adaptively into neural networks, offering a robust and reproducible approach for data-scarce problems across various scientific domains, including fluid dynamics and material modeling.