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A data-driven approach for star formation parameterization using symbolic regression

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Symbolic regression on kiloparsec galaxy maps finds star formation equations that beat all tested analytic models, with the 100 Myr fits converging to a single scaling relation.

desk verdict A well-constructed symbolic regression study that recovers a top-down-like SF law at 100 Myr, but the test-set validation is more optimistic than it looks because the split is pixel-level. read the letter →

arxiv 2505.04681 v1 pith:XTJVMXRY submitted 2025-05-07 astro-ph.GA astro-ph.IM

Diane M. Salim , Matthew E. Orr , Blakesley Burkhart , Rachel S. Somerville , Miles Cramner This is my paper · ORCID
classification astro-ph.GAastro-ph.IM
keywords starformationsymbolicregressionKennicutt-SchmidtrelationFIRE-2simulationsinterstellarmediumgalaxyevolutionscalingrelationsmachinelearning
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper sets out to replace the hand-assumed star formation recipes used in galaxy simulations with equations that are discovered from data. It applies symbolic regression to 750 pc pixel maps of seven simulated Milky Way-mass galaxies and searches for the formula that best predicts the star formation rate surface density, $\Sigma_{\rm SFR}$, from eight local variables. The discovered equations beat every analytic star formation model the authors compare against, and at the 100 Myr averaging timescale the top equations all converge to nearly the same scaling relation in $\Sigma_{\rm gas}$, $\Sigma_*$, and $\sigma_{\rm gas,z}$. The authors read this convergence as evidence that the Kennicutt–Schmidt relation is governed top-down, by feedback-regulated disk equilibrium, and that replacing empirical subgrid recipes with such an equation could improve large-volume galaxy simulations and semi-analytic models.

What carries the argument

The central machinery is symbolic regression, a genetic algorithm that searches over equations by representing them as expression trees and evolving them through mutation and crossover, returning a Pareto front that trades equation complexity against fit. The paper runs the search in logarithmic space, using a loss that combines mean squared error with a quantile loss meant to preserve scatter, and selects only equations whose training loss is lower than the best analytic star formation model. The load-bearing output is the converged 100 Myr scaling relation, whose persistent structure — the combination of $\log \Sigma_{\rm gas}$ with $(\log \sigma_{\rm gas,z} + \log \Sigma_*)^{0.86}$ — is the object that carries the paper's physical interpretation.

What would settle it

Take the converged 100 Myr equation and apply it to 750 pc pixels of observed galaxies, or of an independent simulation suite with a different feedback prescription, and compare predicted versus measured $\Sigma_{\rm SFR}$; if the relation fails on galaxies with different star formation histories while still fitting FIRE-2, the convergence is a property of the training data rather than a universal star formation law.

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Extended reading notes

Core claim

The central claim is that a closed-form, data-driven description of star formation exists at kiloparsec scales and that the search finds it. On the 100 Myr averaging timescale, every selected equation converges to $$\log \Sigma_{\rm SFR} \approx \log \Sigma_{\rm gas} + (\log \sigma_{\rm gas,z} + \log \Sigma_*)^{0.86} - 6.1,$$ with only small variations in the exponents and constant across the selected equations. On the 10 Myr timescale, the equations are less stable and capture less variance, which the paper attributes to stochastic feedback that cannot be represented as a deterministic function of instantaneous local variables. Because $\Sigma_*$ and $\sigma_{\rm gas,z}$ appear alongside $\Sigma_{\rm gas}$ in the converged relation, the paper interprets the result as favoring 'top-down' feedback-regulated models over 'bottom-up' local-cloud models of the Kennicutt–Schmidt relation, and notes that the same variables appear in the dynamical-equilibrium pressure used by those theories.

Load-bearing premise

The entire derivation assumes the FIRE-2 simulations are a faithful, unbiased picture of real star formation and that, at 750 parsec resolution, the star formation rate in a pixel is fully determined by the eight local quantities measured in that same pixel; if star formation history, sub-cloud physics, or the pixels excluded for not having recent star formation carry information, every fitted equation is biased.

Editorial extensions

If this is right

  • The 100 Myr equation can be used directly as a subgrid star formation recipe in large-volume cosmological simulations, replacing the empirical Kennicutt–Schmidt power law, and on FIRE-2-like test pixels it should outperform all analytic recipes examined here.
  • If the convergence is physical rather than a fitting artifact, the resolved Kennicutt–Schmidt scatter seen in real galaxies is partly determined by $\Sigma_*$ and $\sigma_{\rm gas,z}$, so observations that ignore those variables will miss real information.
  • Simulations that do not resolve the vertical structure of the disk cannot supply $\sigma_{\rm gas,z}$ and $\Sigma_*$ reliably, so the discovered equation imposes a resolution requirement on the data it is applied to.
  • The instability of the 10 Myr equations implies that a deterministic, instantaneous star formation law is not the right target for stochastic short-timescale star formation; averaging over roughly a cloud lifetime is needed before the relation stabilizes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural test the paper leaves open is whether the same converged 100 Myr equation reappears when the pipeline is trained on an independent hydrodynamical simulation with different feedback physics; if it does, the relation is likely a property of real star-forming disks rather than of FIRE-2's subgrid model.
  • The authors note the quantile-loss term contributed only 10 to 15 percent of the total loss; a distribution-matching loss such as KL divergence is an obvious extension that could fix the observed underestimation of scatter in the feature-space planes.
  • The paper's own resolution caveat points to a sharp experiment: rerun on 200 pc or finer maps; if the equation's form shifts toward the bottom-up variables (free-fall time, Mach number, virial parameter), the claimed top-down dominance is resolution-dependent.
  • Because the relation includes the stellar surface density as a proxy for disk potential, it may also connect to observed dynamical-equilibrium pressure correlations in nearby galaxy surveys; testing on those maps would tell whether the equation extrapolates beyond simulated galaxies.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This manuscript applies the symbolic regression package PySR to 750 pc pixel maps from seven FIRE-2 Latte galaxies, aiming to discover closed-form expressions for the star formation rate surface density averaged over 10 Myr and 100 Myr. The input features are reduced from 76 candidates to eight via XGBoost and SHAP analysis supplemented by physical arguments; the outputs are evaluated on a held-out 20% test set and compared with XGBoost, the Kennicutt-Schmidt K98 relation, three bottom-up models (KDM12, SFK15, SAF20), and one top-down model (FG13). The paper reports that the found equations outperform all analytic models on the test set, that the 100 Myr equations converge to a scaling relation of the form log Sigma_SFR ~ log Sigma_gas + (log sigma_gas,z + log Sigma_*)^0.86 - 6.1, and interprets this as support for a top-down, feedback-regulated origin of the Kennicutt-Schmidt relation. An appendix presents a synthetic recovery test based on FG13 as validation of the pipeline.

Significance. If the central claims hold, this is a useful contribution: it provides interpretable, data-driven parameterizations of star formation from a modern high-resolution simulation suite, with potential applications as subgrid recipes in large-volume simulations and semi-analytic models. The pipeline itself is transferable to other datasets. The manuscript has real strengths: the use of a held-out test set, a synthetic recovery test in Appendix D demonstrating that the method can rediscover a known relation while ignoring a pure-noise variable, an explicit complexity/loss trade-off analysis, and a candid discussion of caveats in Section 4.1. However, the significance of the headline claims depends on whether the reported test-set gains reflect genuine generalization; the current validation strategy does not yet establish that.

major comments (3)
  1. [Section 2.1.1 and Table 3] The central claim that the PySR equations "statistically outperform" the analytic models rests on a random pixel-level 80/20 split, but pixels drawn from the same galaxy and snapshot are strongly spatially correlated at 750 pc. The test set is therefore not independent of the training set, and the modest margins (10 Myr: 0.413 vs 0.402 for SAF20; 100 Myr: 0.555 vs 0.420 for FG13) could be inflated by this leakage. The synthetic FG13 recovery test in Appendix D is encouraging but does not address this issue, since its synthetic data inherit the same pixel-level structure. I request a leave-one-galaxy-out or leave-one-snapshot-out cross-validation, with per-fold R^2, sigma_plane, and found equations reported; without it, the generalization claim in the abstract and Section 5 is not established.
  2. [Sections 3.1 and 3.2] The model-selection step is not fully out-of-sample. In Section 3.2 the baseline loss is computed on the test set, while the selection criterion applied to PySR candidates is the final training loss; this leaks test information into the choice of the top four equations. In addition, Section 3.1 states that the analytic baselines have their normalizations or efficiencies fitted to the test set. Fitting the baselines to the test set is conservative in that it favors the analytic models, but using the test set to set the selection threshold weakens the claim that the reported test-set R^2 values are a clean out-of-sample comparison. Please use a separate validation set, or an internal cross-validation loop, for both the baseline loss and the selection of equations, and reserve the test set for the final comparison only.
  3. [Tables 3 and 4] No uncertainties are reported for R^2, sigma_plane, or the fitted exponents and constants. At the 10 Myr timescale the difference between the best PySR equation and SAF20 is 0.413 vs 0.402, which is too small to interpret without error bars. Please add resampling-based uncertainties (for example, bootstrap over pixels or over snapshots) or repeated PySR runs with different random seeds, and report the scatter of the coefficients across those runs. This is needed to support the quantitative claim of statistical outperformance over the analytic models.
minor comments (5)
  1. [Table 4 and Figures 6-9] The equations use "log" without specifying the base, while expressions such as e^{log f_gas} imply a natural logarithm and the text states training was performed in log10 space; please define the convention and verify that all equations are internally consistent.
  2. [Equation 5] Equation 5 uses alpha_i but the surrounding text only lists quantile values 0.1, 0.25, 0.5, 0.75, and 0.9; please state explicitly that alpha_i takes these values and that N_alpha = 5.
  3. [Figure 4 caption] The caption refers to "Equation 2.1.2" for the quantile loss, which should be Equation 5.
  4. [Section 2.2 and Figure 3] The pixel selection criteria are described only qualitatively as "both recent star formation and gas," with a Sigma_gas cut shown in Figure 3; please give the exact thresholds and the final pixel counts after selection.
  5. [Throughout] There are typographical errors, including "for that for that" in Section 2.2 and "casual" for "causal" in Section 4.1.2; a careful proofread is needed.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the SR equations are empirical fits to FIRE-2 outputs, validated by a synthetic FG13 recovery test, and the central claims do not reduce to their inputs.

full rationale

The paper's pipeline is an empirical regression: y = Sigma_SFR (10/100 Myr) is a simulation-derived target, and the eight input variables are separately extracted physical fields; the target is not defined as a function of the inputs, so the found equations are not self-definitional. The synthetic FG13 experiment (Appendix D) is an external-style recovery test showing the search can return a known relation from noisy data, which supports that the reported equations are not just relabeled inputs. The model-selection rule requiring training loss below the best analytic model is a filter, not a construction of the test-set result; the reported R2 values are evaluated on a held-out 20% pixel test set and train/test loss ratios are ~1. Self-citations (Orr et al. 2018, 2020; Salim et al. 2015, 2020) are used as baselines or for map-generation conventions and for omitting redshift; none of these carries the load of the main claim, and the analytic baselines are independently stated equations. The paper's own caveats (limited stochasticity capture at 10 Myr, low-redshift bias, 750 pc resolution) are correctness limitations, not circularities. The main residual concern is the random pixel-level split with strong spatial autocorrelation, which is an out-of-sample generalization issue for the test-set comparison, not a circular derivation.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on four main assumptions: the FIRE-2 SF prescription is trustworthy, the 750 pc mapping captures the physics, the eight selected variables are sufficient, and redshift evolution can be ignored. The free parameters are the fitted constants and exponents in the discovered equations, plus the optimized baselines used for comparison.

free parameters (3)
  • Constant offsets in found equations = 4.98, 5.33, 5.19, 5.21 (10 Myr); 6.12, 6.08, 6.06, 6.06 (100 Myr)
    These constants are fitted by PySR during training to match the FIRE-2 data, and they appear in every top equation in Table 4.
  • Exponents in found equations = e.g., 1.52, 0.4, 0.49, 0.64, 1.26, 0.23, 1.29, 1.25, 0.12, 0.86, 0.85, 0.9, 0.91, 0.96
    The exponents on Σgas, Σ*, σgas,z, and the composite terms are optimized by PySR, as shown in Table 4.
  • Analytic baseline efficiencies and normalizations = K98 normalization; ϵff for KDM12, SFK15, SAF20
    The comparison analytic models are fit to the FIRE-2 test dataset (Section 3.1), giving them optimized offsets that affect the baseline loss and the selection of top PySR equations.
assumptions (4)
  • domain assumption FIRE-2 subgrid star formation prescription is a valid physical model
    The training data are pixels from FIRE-2 simulations whose star formation follows a density threshold, H2 fraction, virial parameter, and free-fall rate (Section 2.2). The discovered equations inherit any biases in this prescription.
  • domain assumption The 750 pc pixel scale preserves the relevant physics for the discovered relations
    The paper maps galaxies onto 750 pc pixels (Section 2.2), a scale larger than GMCs. The authors acknowledge this in Section 4.1.1 as a limitation for subgrid use.
  • domain assumption The eight selected variables are sufficient to predict ΣSFR
    The feature selection via XGBoost SHAP plus physical arguments (Section 2.1.1) assumes no important predictor is left out. The paper does not test whether omitted variables such as metallicity or magnetic fields would change the equations.
  • domain assumption Redshift independence of the resolved KS relation
    The paper pools snapshots from z=0 to z=2 without a redshift feature, citing Orr et al. (2018) for no significant redshift evolution (Section 2.2). If this assumption fails, the pooled fit may mix different physical regimes.

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Pith. "Pith review of A data-driven approach for star formation parameterization using symbolic regression." pith.science (2026). https://pith.science/paper/XTJVMXRY

@misc{pith2026250504681,
  author       = {Pith},
  title        = {Pith review of: A data-driven approach for star formation parameterization using symbolic regression},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XTJVMXRY}},
  note         = {Machine review of arXiv:2505.04681}
}
abstract

Star formation (SF) in the interstellar medium (ISM) is fundamental to understanding galaxy evolution and planet formation. However, efforts to develop closed-form analytic expressions that link SF with key influencing physical variables, such as gas density and turbulence, remain challenging. In this work, we leverage recent advancements in machine learning (ML) and use symbolic regression (SR) techniques to produce the first data-driven, ML-discovered analytic expressions for SF using the publicly available FIRE-2 simulation suites. Employing a pipeline based on training the genetic algorithm of SR from an open software package called PySR, in tandem with a custom loss function and a model selection technique which compares candidate equations to analytic approaches to describing SF, we produce symbolic representations of a predictive model for the star formation rate surface density ($\Sigma_\mathrm{SFR}$) averaged over both 10 Myr and 100 Myr based on eight extracted variables from FIRE-2 galaxies. The resulting model that PySR finds best describes SF, on both averaging timescales, features equations that incorporates the surface density of gas, $\Sigma_\mathrm{gas}$, the velocity dispersion of gas $\sigma_{\mathrm{gas,~z}}$ and the surface density of stars $\Sigma_\mathrm{*}$. Furthermore, we find that the equations found for the longer SFR timescale all converge to a scaling-relation-like equation, all of which also closely capture the intrinsic physical scatter of the data within the Kennicutt-Schmidt (KS) plane. This observed convergence to physically interpretable scaling relations at longer SFR timescales demonstrates that our method successfully identifies robust physical relationships rather than fitting to stochastic fluctuations.

Figures

Figures reproduced from arXiv: 2505.04681 by the authors.

Figure 1
Figure 1. Chart showing how data is manipulated and flows through our procedure. In all panels, filled green boxes represent a machine learning-based method or algorithm that is utilized, and uncolored green outlined boxes represent a predictive model for our target variable, star formation rate surface density. The top panel shows how our initial raw data is prepared and filtered to determine the variables with which to run … view at source ↗
Figure 2
Figure 2. Star formation rate surface density (ΣSFR) maps of galaxy ”m12b” at redshift z = 0 with 750 pc pixel size, averaged over 10 Myr (left panel) and 100 Myr (right panel). and/or the strength of the variables that are found to describe the two target variables respectively. From this point onward, the averaging timescale of the experiment will be denoted as ⟨tSFR⟩ [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Distribution of points across the KS plane of the FIRE data (grey contours) predicted by the respective models labeled in the top-left orange boxes for the ⟨tSFR⟩=10 Myr (left subfigure) and ⟨tSFR⟩=100 Myr (right subfigure) experiments. For all contours (grey and green), the regions represent the 95%, 80%, 50% and 20% data inclusion region in the plane (from lightest to darkest colored, outside-in). The dashed, dark… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Evolution of the total loss values as defined by Equation 6 (top rows) and the quantile loss values as defined by Equation 2.1.2 (bottom rows) by training epoch for the 10 Myr averaged SFR target variable experiment (left subfigure) and the 100 Myr averaged SFR target …
Figure 5
Figure 5. Figure 5: Points indicating the equation ”score” as originally defined by M. Cranmer (2023) (top row) and the R 2 (bottom row) of all equations found during the 4000 epoch training run for the two experiments when SF is averaged over ⟨tSFR⟩ = 10 Myr (left) and when averaged over…
Figure 6
Figure 6. Figure 6: The predicted ΣSFR of the top four models found by PySR in comparison to the native ΣSFR of the FIRE-2 test sample, with the colour indicating the value of Σgas at that point for the ⟨tSFR⟩ = 10 Myr (left column) and 100 Myr (right column) experiments. The overlaid con…
Figure 7
Figure 7. Figure 7: Distribution of points across the KS (logarithmic Σgas-ΣSFR) plane, the logarithmic σgas,z − ΣSFR plane and the Σ∗ − ΣSFR planes of the native FIRE data (grey contours) and ΣSFR attained for the respective equations found by PySR (colored contours; green, purple and bl…
Figure 8
Figure 8. Figure 8: Same as in [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: Distribution of points across the KS plane of the native FIRE data (grey contours), with the found models at fiducial values for features other than Σgas overlaid, colored by complexity of the models. Style as [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]
Figure 10
Figure 10. Figure 10: SHAP values for the input features used to train an XGBoost model for when SF is averaged over ⟨tSFR⟩ = 10 Myr (left) and when averaged over ⟨tSFR⟩ = 100 Myr (right). Each point in this figure indicates a the SHAP value of that corresponding data point, colored by the…
Figure 11
Figure 11. Figure 11: SHAP values for the four input features used to create the synthetic FG13 dataset and train an XGBoost model. Each point in this figure indicates a the SHAP value of that corresponding data point, colored by the relative strength of that particular feature, with red i…

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