🤖 machine learning

Approximating neutron-star radii using gravitational-wave only measurements with symbolic regression

This paper employs symbolic regression (pySR) trained on Tolman-Oppenheimer-Volkoff solutions to derive a symbolic expression that accurately approximates neutron-star radii using only gravitational-wave measurements, achieving agreement within a few hundred meters across various equations of state and successfully applying the model to GW170817 data.

Original authors: Michał Bejger

Published 2026-09-15

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CC BY 4.0

Original authors: Michał Bejger

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep in the cosmos, two dead stars, each crushed into a sphere smaller than a city, spiral toward each other. As they dance closer, their immense gravity ripples through space-time, sending out gravitational waves that travel across the universe to be caught by detectors on Earth. These ripples carry a secret code: they tell us exactly how heavy the stars are and how much their shapes stretch and squish under the pull of their partner. This stretching is called tidal deformability. However, the waves do not directly reveal the stars' size. To know the radius of a neutron star, astronomers usually have to look at the light coming from its surface, a task that requires powerful telescopes and clear views. But what if the light is hidden, or the telescope cannot see? Scientists have long wanted a way to guess the size of these stars using only the gravitational waves, bridging the gap between the invisible ripples of gravity and the visible glow of light.

A researcher named M. Bejger has developed a new method to solve this puzzle. Instead of relying on complex computer models that act like a "black box"—where you put data in and get an answer out without understanding how—the team used a technique called symbolic regression. Think of this as a search for a simple, written rule that describes how the universe works. The computer was fed thousands of examples of neutron stars, generated by solving the fundamental equations of physics that describe how matter behaves under extreme pressure. These examples included stars with different masses, different sizes, and different internal structures. The computer's job was to find a single, straightforward mathematical sentence that could look at just the mass and the stretching of a star and accurately predict its radius.

The result is a surprisingly simple formula that connects the gravitational wave data to the star's size. When the researchers tested this new rule on the same data used to teach the computer, it worked with high precision. But the true test came when they applied it to stars that were completely different from the ones the computer had seen during its training. They used models based on realistic theories of dense matter, including some that behave in ways the training data did not cover. Even for these unfamiliar stars, the formula held up. The predicted sizes were usually within a few hundred meters of the true size, a remarkable margin of error given the extreme conditions inside these objects. In some cases, the difference was as small as fifty meters.

The method also proved robust when faced with tricky scenarios, such as two different types of stars that happen to have the same weight and the same amount of stretching but are made of different materials. While the formula could not perfectly distinguish between these specific twins without extra information, it still provided a very close estimate of their size. The researchers then took their new rule and applied it to the famous gravitational wave signal from the 2017 collision of two neutron stars, known as GW170817. The sizes they calculated matched well with previous estimates made by other teams using more complicated methods. This suggests that the new approach offers a fast, clear, and reliable way to determine the size of neutron stars using only gravitational waves, allowing astronomers to compare data from the invisible gravitational channel with data from the visible light channel without needing to solve complex equations every time.

The work does not claim to have found the ultimate, perfect description of every possible neutron star, nor does it replace the need for detailed physical modeling in all cases. Instead, it provides a powerful, easy-to-use tool that works across a wide range of scenarios. It shows that even without knowing the exact composition of the matter inside a neutron star, one can still make a very good guess at its size just by knowing how heavy it is and how much it deforms. This capability opens a new door for multi-messenger astronomy, where information from different parts of the universe is combined to build a clearer picture of the most extreme objects in existence. By turning complex physics into a simple, understandable expression, the research makes it easier for scientists to explore the secrets of neutron stars, even when the light from those stars is out of reach.

Technical Summary: Approximating Neutron-Star Radii Using Gravitational-Wave Only Measurements with Symbolic Regression

Problem Statement
Binary neutron-star (BNS) inspirals emit gravitational waves (GWs) that encode information regarding the component masses ( $M$ ) and tidal deformabilities ( $\Lambda$ ). However, GW signals do not directly yield the neutron star (NS) radius ( $R$ ), a parameter typically measured via electromagnetic (EM) observations. To facilitate multi-messenger astronomy, there is a need for an expression that estimates $R$ using only GW-derived data ( $M$ and $\Lambda$ ). While relationships between NS parameters and the Equation of State (EOS) exist, deriving an EOS-independent "universal" relation that maps $M$ and $\Lambda$ to $R$ without relying on complex numerical integration or "black box" machine learning models remains a challenge.

Methodology
The author employs Symbolic Regression (SR) to discover an analytical, interpretable expression for the NS radius. The specific workflow involves:

Data Generation: Training and validation datasets were constructed by solving the Tolman-Oppenheimer-Volkoff (TOV) equations for a wide range of piecewise relativistic polytropic EOSs. These datasets include solutions for mass ( $M$ ), radius ( $R$ ), tidal Love number ( $k_2$ ), and tidal deformability ( $\Lambda$ ).
Model Discovery: The study utilizes the pySR library (a Python wrapper for Julia's SymbolicRegression.jl), which combines evolutionary algorithms (genetic programming) with symbolic expression trees. The algorithm searches for the optimal functional form that minimizes the loss function (Mean Square Error) while balancing accuracy against complexity (Pareto front optimization).
Target Variable: Instead of directly regressing $R(M, \Lambda)$ , the method first approximates the tidal Love number $k_2$ as a function of $M$ and $\Lambda$ ( $\tilde{k}_2(M, \Lambda)$ ). The radius is then reconstructed using the physical relation:
$\tilde{R}(M, \Lambda) = \left( \frac{3\Lambda}{2\tilde{k}_2(M, \Lambda)} \right)^{1/5} \frac{GM}{c^2}$
Validation: The derived expression is tested against:
- Piecewise polytropic EOSs outside the training set.
- Realistic, non-polytropic dense-matter EOSs (e.g., WFF1, APR, SLy4, BSk21, BM165, NL3).
- Intersecting NS sequences and sequences with strong phase transitions.
- Posterior distributions from the GW170817 event.

Key Contributions and Results
The study successfully identifies a simple, symbolic expression for $\tilde{k}_2(M, \Lambda)$ that achieves a Mean Square Error (MSE) of $\approx 2.7 \times 10^{-6}$ on validation sets. The resulting approximation for the radius is:
$\tilde{k}_2(M, \Lambda) = 0.0351 (\log_{10} \Lambda)^2 \times \left( \frac{M}{M + 2.05} - 0.109 \right) + 0.0137$
(Where $M$ is in solar masses $M_\odot$ .)

Key findings include:

Accuracy: The approximation yields average radius differences ( $\Delta R$ ) of a few hundred meters compared to ground truth values. For realistic EOSs, the distribution of the average absolute error $|\Delta R|$ peaks around 50 m, with the tail of the distribution reaching 200 m. However, specific EOSs exhibit higher errors; for instance, the mean absolute error for the BM165 EOS is 299.8 m, and maximum absolute errors can reach up to 452.6 m for certain sequences.
Generalizability: The model demonstrates robustness when applied to non-polytropic EOSs and sequences partially or fully outside the training parameter space, including those with strong first-order phase transitions.
GW170817 Application: When applied to the mass and tidal deformability posteriors of GW170817, the inferred radius distributions are comparable in precision to those reported by the LIGO-Virgo Collaboration using more complex, EOS-dependent methods.
Interpretability: Unlike neural networks, the resulting formula is an explicit, computationally straightforward algebraic expression, satisfying the requirements for Explainable Artificial Intelligence (XAI).

Significance and Claims
The paper positions this work as a computationally inexpensive and interpretable machine learning solution for multi-messenger astronomy. Its primary significance lies in:

Bridging Channels: It provides a direct link between GW-only measurements and EM observables (radius), enabling direct comparison without requiring prior assumptions about the specific EOS.
Efficiency: The method offers a "simplest inference method possible," allowing for immediate radius estimation from $(M, \Lambda)$ pairs without the need to integrate TOV equations or combine multiple relations based on binary parameters.
Pre-processing Utility: The author suggests the model can serve as a pre-processing step to constrain parameter spaces for more refined estimation methods.

The author notes limitations, specifically that the approximation relies on point estimates of $M$ and $\Lambda$ and may struggle at the intersection of different EOS sequences where additional information (e.g., derivatives like $d\Lambda/dM$ ) might be required for higher precision. Future work is proposed to incorporate such differential aspects and physically inspired constraints.

Technical Summary: Approximating Neutron-Star Radii Using Gravitational-Wave Only Measurements with Symbolic Regression

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