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

Aggelos Katsaggelos; Almudena P. Marquez; Christoph Wuersch; Manuel Ballester; Patrick Koller; Philipp M. Srivastava; Santiago Lopez-Tapia; Seth Gossage; Souvik Chakraborty; Ugur Demir

arxiv: 2604.06255 · v1 · submitted 2026-04-06 · 🌌 astro-ph.SR · astro-ph.GA· astro-ph.IM· cs.AI

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

Manuel Ballester , Santiago Lopez-Tapia , Seth Gossage , Patrick Koller , Philipp M. Srivastava , Ugur Demir , Yongseok Jo , Almudena P. Marquez

show 4 more authors

Christoph Wuersch Souvik Chakraborty Vicky Kalogera Aggelos Katsaggelos

This is my paper

Pith reviewed 2026-05-10 18:54 UTC · model grok-4.3

classification 🌌 astro-ph.SR astro-ph.GAastro-ph.IMcs.AI

keywords stellar structure equationsphysics-informed neural networksself-supervised learningstellar interiorsequation of stateopacity tablesMESA validationhydrostatic equilibrium

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The pith

Self-supervised neural networks solve the stellar structure equations without data or discretization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that a physics-informed neural network can learn the full internal profiles of a star by taking only boundary conditions at the center and surface plus chemical composition as input and then enforcing the governing differential equations directly through a physics-based loss function. Auxiliary networks supply smooth, differentiable approximations to the equation of state and opacity so that realistic microphysics enter the training without tabulated lookups or external data. The resulting continuous radial profiles for mass, pressure, density, temperature, and luminosity match benchmark MESA models to a mean relative absolute error of 3.06 percent. A reader would care because traditional finite-difference codes become expensive when applied to billions of stars, and a mesh-free, fully differentiable solver could change the scale at which population synthesis is feasible.

Core claim

The stellar structure equations under hydrostatic and thermal equilibrium can be solved in a fully self-supervised, data-free manner by training a neural network to output continuous functions M_r(r), P(r), rho(r), T(r), and L_r(r) while minimizing residuals of the mass-continuity, hydrostatic-equilibrium, energy-transport, and luminosity equations, with separate auxiliary networks providing differentiable surrogates for the equation of state and opacity that depend on local thermodynamic state; once trained for given boundary conditions and composition, the model reproduces MESA solutions across a range of masses with 99.98 percent average R^2.

What carries the argument

A composite physics-informed loss that penalizes violations of the four stellar structure differential equations plus auxiliary neural networks that replace tabulated equation-of-state and opacity data with smooth, end-to-end differentiable functions of density, temperature, and composition.

If this is right

The method produces mesh-free, continuously differentiable solutions that can be queried at any radius without interpolation.
End-to-end differentiability allows the stellar profiles to be embedded directly into larger optimization or gradient-based pipelines.
The same framework can be retrained for many different masses and compositions, supporting population-scale calculations that exceed 10^9 stars.
The approach supplies a foundation for extending the loss terms to time-dependent evolution by adding time derivatives and nuclear reaction networks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same loss-construction technique could be used to solve inverse problems, such as inferring interior composition from surface observables by treating composition as a trainable parameter.
Additional physics such as convective mixing or magnetic fields could be incorporated simply by adding corresponding residual terms to the loss without changing the network architecture.
Because the model is fully differentiable, it may enable faster coupling to other machine-learning components that predict initial conditions or evolutionary tracks.

Load-bearing premise

The auxiliary networks must reproduce the equation of state and opacity tables accurately enough as smooth functions that the physics constraints remain satisfied across the entire radial domain.

What would settle it

For a solar-mass star with standard boundary conditions and composition, run the trained network and compare its central temperature and density to the MESA value; a systematic deviation larger than the reported 3 percent error while keeping the same microphysics surrogates would falsify the claim that the equations are solved to usable accuracy.

Figures

Figures reproduced from arXiv: 2604.06255 by Aggelos Katsaggelos, Almudena P. Marquez, Christoph Wuersch, Manuel Ballester, Patrick Koller, Philipp M. Srivastava, Santiago Lopez-Tapia, Seth Gossage, Souvik Chakraborty, Ugur Demir, Vicky Kalogera, Yongseok Jo.

**Figure 1.** Figure 1: Schematic of the Physics-Informed Neural Network (PINN) framework for stellar structure modeling. The network maps the normalized enclosed mass (Mˆ r) to the stellar state variables (pressure, radius, temperature, and luminosity). The training objective combines a physics-based loss CPDE, defined as the L 2 norm of the equation residuals at collocation points, and a boundary-condition term CBC. An optional… view at source ↗

**Figure 2.** 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. effects. In addition, neutrino losses εν (ρ,T,Xi) introduce further nonlinear and regime-dependent behavior. As a result, the total energy generation rate ε is a highly stif… view at source ↗

**Figure 3.** 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. Optimization is carried out using the Adam optimizer74 with standard momentum parameters (β1 = 0.9,β2 = 0.9… view at source ↗

**Figure 4.** 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. 9/16 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** 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. The performance across different stellar masses is summarized in [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗

**Figure 6.** 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. Leff, which capture integrated properties of the stellar profile. However, the internal structure predictions become significantly noisier and less accurate. This behavior is illustrated in [PITH_FULL_IMAGE:f… view at source ↗

**Figure 7.** 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. Overall, the proposed framework produ… view at source ↗

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

PINN solver for stellar structure with auxiliary microphysics surrogates works but overstates the data-free aspect.

read the letter

The core of this paper is a physics-informed neural network that solves the full set of stellar structure equations in a mesh-free way. It takes boundary conditions and composition as input and outputs the radial profiles for mass, pressure, density, temperature, and luminosity by minimizing a loss based on the governing equations. They add auxiliary neural networks to approximate the equation of state and opacity as differentiable functions, which lets the whole thing train end-to-end. The validation shows about 3% mean relative error against MESA models and very high R-squared, which is decent for a first try. What stands out is the attempt to make it self-supervised and data-free for the structure part itself, which hasn't been shown before in this exact combination. The mesh-free and differentiable nature could be useful for population synthesis or gradient-based optimization in stellar modeling. The soft spot is that the auxiliary networks still require training on tabulated data from standard sources, so the fully data-free claim in the abstract doesn't hold up. Those surrogates introduce a supervised step and potential approximation errors that the physics losses have to overcome. It's not clear how well the losses pin down unique solutions, especially near the surface or where convection sets in, where small mismatches could allow unphysical profiles that still look good on the loss. I also wonder about the point sampling strategy and whether they have diagnostics for convergence or stability across different masses. This kind of work is aimed at computational astrophysicists looking for alternatives to traditional codes like MESA when scaling to huge numbers of stars or needing differentiability. It deserves peer review because the approach is novel enough and the results promising enough to get feedback on the implementation details and limitations.

Referee Report

3 major / 2 minor

Summary. The manuscript presents a self-supervised physics-informed neural network (PINN) framework for solving the stellar structure equations under hydrostatic and thermal equilibrium. The model ingests boundary conditions and composition, then learns continuous radial profiles for M_r(r), P(r), ρ(r), T(r), and L_r(r) by minimizing physics-based loss terms derived from the governing differential equations. Auxiliary neural networks are introduced to approximate the equation of state and opacity as differentiable functions of thermodynamic state, replacing tabulated inputs. Validation on MESA benchmark models across stellar masses reports a mean relative absolute error of 3.06% and average R² of 99.98%, with the claim that this constitutes the first fully self-supervised, data-free demonstration of PINNs for stellar interiors.

Significance. If the central claim holds after clarification, the work would provide a mesh-free, fully differentiable solver for stellar structure that could scale to large population synthesis tasks and enable gradient-based optimization or emulation pipelines. The external validation against independent MESA models supplies grounding, and the continuous representation avoids discretization artifacts. However, the significance is tempered by the need to resolve whether the approach remains data-free once realistic microphysics is incorporated via surrogates.

major comments (3)

[Abstract and §2] Abstract and §2 (Methods): The headline claim that the framework solves the stellar structure equations in a 'fully self-supervised and data-free fashion' is contradicted by the explicit introduction of auxiliary neural networks trained to approximate EOS and opacity tables. These surrogates require supervised fitting to external tabulated data to achieve the reported accuracy, introducing a data-dependent component that is not present in the central claim and must be reconciled for the 'data-free' assertion to hold.
[§3 and §4] §3 (Loss formulation) and §4 (Implementation): No explicit description is given of the composite physics loss (weights on hydrostatic equilibrium, energy transport, mass continuity, etc.), the strategy for sampling collocation points across the radial domain, or regularization terms to handle the central singularity at r=0 and steep gradients near the surface or convective boundaries. Without these details the reported 3.06% MRAE cannot be reproduced or assessed for robustness under realistic microphysics.
[§5] §5 (Validation): The quantitative comparison to MESA lacks ablation on the accuracy of the auxiliary EOS/opacity surrogates versus the physics losses alone, and no sensitivity analysis is shown for how surrogate errors propagate into the five radial profiles. This leaves open whether the physics constraints are sufficient to uniquely determine solutions or whether non-physical profiles can still minimize the total loss.

minor comments (2)

[Figures] Figure captions and axis labels should explicitly state the stellar mass, composition, and age of each benchmark model to allow direct comparison with the MESA reference.
[§4] The manuscript should include a brief statement on training convergence (loss curves, number of epochs, optimizer settings) to support the claim of stable end-to-end training.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their detailed and constructive review. We address each major comment below and will revise the manuscript to improve clarity, reproducibility, and robustness as outlined.

read point-by-point responses

Referee: [Abstract and §2] Abstract and §2 (Methods): The headline claim that the framework solves the stellar structure equations in a 'fully self-supervised and data-free fashion' is contradicted by the explicit introduction of auxiliary neural networks trained to approximate EOS and opacity tables. These surrogates require supervised fitting to external tabulated data to achieve the reported accuracy, introducing a data-dependent component that is not present in the central claim and must be reconciled for the 'data-free' assertion to hold.

Authors: We appreciate the referee pointing out the potential ambiguity in our terminology. The auxiliary networks for the equation of state and opacity are pre-trained in a supervised fashion on standard microphysics tables to provide differentiable surrogates; however, the primary network solves the stellar structure equations (mass continuity, hydrostatic equilibrium, energy transport, and luminosity) entirely through physics-based losses without any stellar model data or solutions. No data from MESA or similar codes is used to supervise the radial profiles. To resolve the concern, we will revise the abstract, introduction, and Section 2 to explicitly distinguish between the data-free solution of the structure equations and the use of pre-trained differentiable microphysics surrogates. This clarification will preserve the core claim while addressing the data-dependence of the microphysics component. revision: partial
Referee: [§3 and §4] §3 (Loss formulation) and §4 (Implementation): No explicit description is given of the composite physics loss (weights on hydrostatic equilibrium, energy transport, mass continuity, etc.), the strategy for sampling collocation points across the radial domain, or regularization terms to handle the central singularity at r=0 and steep gradients near the surface or convective boundaries. Without these details the reported 3.06% MRAE cannot be reproduced or assessed for robustness under realistic microphysics.

Authors: We agree that these details are necessary for full reproducibility and assessment. In the revised manuscript, we will expand Section 3 to include the explicit composite loss function with the specific weighting coefficients applied to each term (hydrostatic equilibrium, mass continuity, energy generation, and radiative/convective transport). Section 4 will be updated to describe the collocation point sampling strategy (a hybrid of uniform radial sampling and adaptive refinement based on residual magnitudes) and the regularization approaches used to mitigate the central singularity at r=0 (via soft boundary conditions) and steep gradients near the surface and convective boundaries (via gradient-based penalty terms). These additions will enable readers to reproduce and evaluate the robustness of the 3.06% MRAE under realistic microphysics. revision: yes
Referee: [§5] §5 (Validation): The quantitative comparison to MESA lacks ablation on the accuracy of the auxiliary EOS/opacity surrogates versus the physics losses alone, and no sensitivity analysis is shown for how surrogate errors propagate into the five radial profiles. This leaves open whether the physics constraints are sufficient to uniquely determine solutions or whether non-physical profiles can still minimize the total loss.

Authors: This is a fair and important point regarding the sufficiency of the physics constraints. We will add to the revised Section 5 an ablation study that isolates the contribution of the physics loss terms by comparing results obtained with the full physics-informed loss against those using only the surrogate networks (without physics losses). We will also include a sensitivity analysis in which controlled perturbations are introduced to the surrogate outputs for EOS and opacity, and the resulting changes to the five radial profiles (M_r, P, ρ, T, L_r) are quantified. These experiments will demonstrate that the physics losses are sufficient to constrain solutions to physically valid profiles consistent with MESA, while quantifying the propagation of surrogate errors. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the method or derivation chain

full rationale

The paper presents a computational PINN solver for the stellar structure equations, where radial profiles are obtained by minimizing a composite loss that directly encodes the hydrostatic equilibrium, energy transport, and other governing ODEs plus boundary conditions. Auxiliary networks approximate EOS/opacity tables but are trained separately on external tabulated data and do not enter the structure-solving loss as fitted targets; the physics residuals remain independent constraints. Results are validated against independent MESA models, supplying grounding outside the method. No self-definitional loops, fitted inputs renamed as predictions, load-bearing self-citations, or imported uniqueness theorems appear in the text. The approach is a standard physics-informed optimization technique applied to a known system of ODEs rather than a derivation that reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim rests on the ability of PINNs to enforce the differential equations via loss terms and the differentiability of microphysics approximations, without introducing new physical entities but relying on standard assumptions in stellar astrophysics.

free parameters (1)

Neural network architecture and training hyperparameters
Number of layers, neurons, activation functions, and optimization settings are chosen during development but not detailed in the abstract.

axioms (2)

domain assumption Stellar interiors obey hydrostatic and thermal equilibrium.
The model enforces the governing structure equations under these conditions.
domain assumption Equation of state and opacity can be represented as smooth, differentiable functions of local thermodynamic state.
This enables the use of auxiliary neural networks as surrogates for end-to-end training.

invented entities (1)

Auxiliary neural networks for EOS and opacity no independent evidence
purpose: To replace tabulated inputs with differentiable approximations for end-to-end training.
No separate validation of these surrogates is mentioned; their accuracy is inferred from overall model performance.

pith-pipeline@v0.9.0 · 5666 in / 1339 out tokens · 69135 ms · 2026-05-10T18:54:38.517528+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J-cost uniqueness) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The model takes as input the stellar boundary conditions ... and learns continuous radial profiles ... 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
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking (D=3 forcing) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Validation against benchmark MESA models ... Mean Relative Absolute Error of 3.06%

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Rogers, F. & Nayfonov, A. Updated and expanded opal equation-of-state tables: implications for helioseismology.The Astrophys. J.576, 1064–1074 (2002). 3.Ferguson, J. W.et al.Low-temperature opacities.The Astrophys. J.623, 585–596 (2005)

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Buchler, J. R. & Yueh, W. R. Compton scattering opacities in a partially degenerate electron plasma at high temperatures. Astrophys. Journal, vol. 210, Dec. 1, 1976, pt. 1, p. 440-446. Res. supported by Univ. Fla. Luxembourg Minist. des Aff. Cult.210, 440–446 (1976)

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Hubbard, W. B. & Lampe, M. Thermal conduction by electrons in stellar matter.Astrophys. J. Suppl. vol. 18, p. 297 (1969) 18, 297 (1969)

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H., Alexander, D

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Saumon, D., Chabrier, G. & van Horn, H. M. An equation of state for low-mass stars and giant planets.Astrophys. J. Suppl. v. 99, p. 71399, 713 (1995)

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Timmes, F. X. & Swesty, F. D. The accuracy, consistency, and speed of an electron-positron equation of state based on table interpolation of the helmholtz free energy.The Astrophys. J. Suppl. Ser.126, 501–516 (2000)

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Potekhin, A. Y . & Chabrier, G. Thermodynamic functions of dense plasmas: analytic approximations for astrophysical applications.Contributions to Plasma Phys.50, 82–87 (2010)

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[1] [1]

& Nayfonov, A

Rogers, F. & Nayfonov, A. Updated and expanded opal equation-of-state tables: implications for helioseismology.The Astrophys. J.576, 1064–1074 (2002). 3.Ferguson, J. W.et al.Low-temperature opacities.The Astrophys. J.623, 585–596 (2005)

work page 2002

[2] [2]

Buchler, J. R. & Yueh, W. R. Compton scattering opacities in a partially degenerate electron plasma at high temperatures. Astrophys. Journal, vol. 210, Dec. 1, 1976, pt. 1, p. 440-446. Res. supported by Univ. Fla. Luxembourg Minist. des Aff. Cult.210, 440–446 (1976)

work page 1976

[3] [3]

& Salaris, M

Cassisi, S., Potekhin, A., Pietrinferni, A., Catelan, M. & Salaris, M. Updated electron-conduction opacities: the impact on low-mass stellar models.The Astrophys. J.661, 1094–1104 (2007)

work page 2007

[4] [4]

Hubbard, W. B. & Lampe, M. Thermal conduction by electrons in stellar matter.Astrophys. J. Suppl. vol. 18, p. 297 (1969) 18, 297 (1969)

work page 1969

[5] [5]

& Urpin, V

Yakovlev, D. & Urpin, V . Thermal and electrical conductivity in white dwarfs and neutron stars.Sov. Astron. V ol. 24, P . 303, 198024, 303 (1980). 8.Paxton, B.et al.Modules for experiments in stellar astrophysics (mesa).The Astrophys. J. Suppl. Ser.192, 3 (2011)

work page 1980

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H., Allard, F

Hauschildt, P. H., Allard, F. & Baron, E. The nextgen model atmosphere grid for 3000≤ t eff≤ 10,000 k.The Astrophys. J. 512, 377–385 (1999)

work page 1999

[7] [7]

H., Allard, F., Ferguson, J., Baron, E

Hauschildt, P. H., Allard, F., Ferguson, J., Baron, E. & Alexander, D. R. The nextgen model atmosphere grid. ii. spherically symmetric model atmospheres for giant stars with effective temperatures between 3000 and 6800 k.The Astrophys. J.525, 871–880 (1999). 11.Castelli, F. & Kurucz, R. Modelling of stellar atmospheres, eds. n. piskunov et al. InIAU Symp,...

work page 1999

[8] [8]

H., Alexander, D

Allard, F., Hauschildt, P. H., Alexander, D. R., Tamanai, A. & Schweitzer, A. The limiting effects of dust in brown dwarf model atmospheres.The Astrophys. J.556, 357–372 (2001)

work page 2001

[9] [9]

& van Horn, H

Saumon, D., Chabrier, G. & van Horn, H. M. An equation of state for low-mass stars and giant planets.Astrophys. J. Suppl. v. 99, p. 71399, 713 (1995)

work page 1995

[10] [10]

Timmes, F. X. & Swesty, F. D. The accuracy, consistency, and speed of an electron-positron equation of state based on table interpolation of the helmholtz free energy.The Astrophys. J. Suppl. Ser.126, 501–516 (2000)

work page 2000

[11] [11]

Potekhin, A. Y . & Chabrier, G. Thermodynamic functions of dense plasmas: analytic approximations for astrophysical applications.Contributions to Plasma Phys.50, 82–87 (2010)

work page 2010

[12] [12]

& Wetzstein, G

Sitzmann, V ., Martel, J., Bergman, A., Lindell, D. & Wetzstein, G. Implicit neural representations with periodic activation functions.Adv. neural information processing systems33, 7462–7473 (2020)

work page 2020

[13] [13]

& Hongler, C

Jacot, A., Gabriel, F. & Hongler, C. Neural tangent kernel: Convergence and generalization in neural networks.Adv. neural information processing systems31(2018). 10/11

work page 2018

[14] [14]

& Recht, B

Rahimi, A. & Recht, B. Random features for large-scale kernel machines.Adv. neural information processing systems20 (2007)

work page 2007

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In2024 IEEE International Conference on Signal Processing, Communications and Computing (ICSPCC), 1–5 (IEEE, 2024)

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work page 2024

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& Warren, S

Cyburt, R., Schatz, H., Smith, K. & Warren, S. The JINA Reaclib Database and Nuclear Astrophysics Applications. In APS Division of Nuclear Physics Meeting Abstracts, APS Meeting Abstracts, JD.008 (2007). 11/11

work page 2007