Interpretable machine learning of halo gas density profiles: a sensitivity analysis of cosmological hydrodynamical simulations

Problem Statement
Understanding the distribution of gas within galaxies and the intergalactic medium is fundamental to galaxy formation theory, as star formation efficiency depends critically on the amount, configuration, and thermodynamic state of this gas. These properties are regulated by stellar and Active Galactic Nucleus (AGN) feedback processes. However, the specific mechanisms by which feedback shapes the radial gas density profile in host haloes remain poorly constrained. While cosmological hydrodynamical simulations (e.g., EAGLE, IllustrisTNG, Simba) provide detailed descriptions of these systems, they employ distinct sub-grid prescriptions for feedback, leading to significant model-dependent differences in predicted gas distributions. Analytic and semi-analytic models often rely on simplified assumptions (e.g., spherically symmetric power-law profiles) that fail to capture the complex interplay of feedback mechanisms. Conversely, while machine learning (ML) offers data-driven capabilities, it is frequently criticized for lacking physical interpretability ("black box" nature). This work addresses the need for a framework that is both accurate in predicting gas density profiles and physically interpretable regarding how specific galaxy and halo properties influence these profiles across different feedback scenarios.

Methodology
The authors utilize suites of cosmological hydrodynamical simulations: EAGLE, IllustrisTNG, and multiple variants of Simba (including runs with deactivated feedback modes). They extract data for well-resolved haloes ($M_{200c} > 10^{9.5} M_\odot$) across redshifts $0 < z < 4$.

1. Data Construction:

   • Target Variable: Spherically averaged radial profiles of total gas density ($\rho$) in 20 logarithmic bins from $0.01 R_{200c}$ to $5 R_{200c}$. Predictions are restricted to the numerically robust range $0.05 R_{200c} \leq r \leq R_{200c}$.
   • Input Features: Six global properties: total halo mass ($M_{200c}$), central galaxy gas mass ($M_{gas}$), stellar mass ($M_\star$), star formation rate (SFR), central black hole (BH) mass ($M_{BH}$), and BH accretion rate ($\dot{M}_{BH}$).
   • Preprocessing: To handle the wide dynamic range and zero values (e.g., for SFR or BH accretion in quenched systems), input features and target variables are rescaled. Masses and densities use decimal logarithms. Features with potential zero values (SFR, $M_{BH}$, $\dot{M}_{BH}$) undergo a modified inverse hyperbolic sine transformation (Eq. 1) to ensure a smooth transition from linear behavior at low values to logarithmic behavior at high values.

2. Machine Learning Model:

   • A Random Forest (RF) regression algorithm is trained to predict the gas density profile at each radial bin based on the six input features.
   • The dataset is split into training (80%) and test (20%) sets. Hyperparameters (minimum node size, tree depth, number of estimators) are optimized using a validation set.
   • The model is trained separately for each simulation snapshot.

3. Sensitivity Analysis:

   • To interpret the "black box" RF model, the authors apply Sobol statistical sensitivity analysis. This technique quantifies the contribution of each input feature (and their interactions) to the variance of the model's output (predicted gas density).
   • They compute first-order Sobol indices ($S_1$), representing the main effect of varying a single feature, and total-order Sobol indices ($S_T$), representing the total effect including all interactions with other features.
   • A significance threshold is established based on the intrinsic accuracy of the model to distinguish physically meaningful feature importance from noise.

Key Contributions

• Development of an Interpretable ML Framework: The paper presents a Random Forest algorithm capable of predicting halo gas density profiles with high accuracy using only six global properties.
• First Application of Sobol Analysis to Full Hydrodynamical Simulations: The authors apply Sobol sensitivity analysis to full cosmological hydrodynamical simulations for the first time, moving beyond semi-analytic models or emulators. This allows for a rigorous quantification of feature importance and degeneracies in a complex, non-linear physical system.
• Cross-Simulation Comparison: The framework is tested across three major simulation suites (EAGLE, IllustrisTNG, Simba) with distinct feedback implementations, as well as specific Simba variants where feedback modes (stellar, AGN-wind, AGN-jet, X-ray) are selectively deactivated.

Results

• Predictive Accuracy: The RF algorithm reproduces simulated gas density profiles with an average accuracy ($\langle \alpha \rangle$) of $\sim 83-90\%$ over the mass range $10^{9.5} < M_{200c} < 10^{15} M_\odot$ and redshift $0 < z < 4$.
  • EAGLE and IllustrisTNG achieve $\sim 90\%$ accuracy at $z > 1$ and $\sim 86\%$ at $z=0$.
  • Simba simulations show slightly lower accuracy ($\sim 83-87\%$), with a notable systematic underprediction in high-mass haloes ($M_{200c} > 10^{12.2} M_\odot$) at $z=0$.
• Feature Importance (Sensitivity Analysis):
  • Dominant Drivers: Across all simulations and redshifts, the total halo mass ($M_{200c}$) and the central galaxy's gas mass ($M_{gas}$) are the most influential features. $M_{gas}$ is the single most impactful predictor, particularly in the inner halo ($< 0.2 R_{200c}$), while $M_{200c}$ acts as a global predictor.
  • Redshift Evolution: At high redshift ($z=4$), stellar mass ($M_\star$) becomes a significant predictor in the inner regions, reflecting the dominance of stellar feedback. At lower redshifts ($z \le 2$), the importance of BH mass ($M_{BH}$) increases, particularly in Simba, correlating with the onset of significant AGN feedback.
  • Negligible Features: The instantaneous BH accretion rate ($\dot{M}_{BH}$) and Star Formation Rate (SFR) generally show low impact on the global gas density profile. The authors suggest the accretion rate's low impact may be due to timescale mismatches (galactic vs. halo timescales) or the anisotropic nature of AGN feedback which is diluted in spherical averages.
• Feedback Variants: In Simba variants, the activation of AGN feedback modes (specifically AGN-jets) increases the importance of $M_{BH}$ and $M_\star$ in shaping the gas distribution. Conversely, in the "No-feedback" run, $M_{gas}$ remains a strong predictor even at large radii due to the efficient conversion of gas into stars across the halo.

Significance and Claims
The authors position this work as a "proof of concept" demonstrating that data-driven models can be made physically interpretable through sensitivity analysis.

• Interpretability: The study successfully identifies that while feedback physics is complex, the resulting gas density profiles are strongly tied to a few global properties ($M_{gas}$ and $M_{200c}$), with secondary dependencies on stellar and BH properties that evolve with redshift and feedback implementation.
• Utility for Theory: The framework can be embedded into semi-analytic models of galaxy formation to incorporate gas density profiles consistent with various hydrodynamical simulations without running full simulations.
• Future Application: The authors claim the model's accuracy (particularly at $z \gtrsim 1$) is sufficient to serve as a benchmark for future comparisons with observational data (e.g., X-ray surface brightness, absorption/emission lines). They note that while current accuracy at $z=0$ for Simba is slightly below the $\sim 15\%$ observational benchmark, the model is promising for constraining feedback models once extended to predict specific gaseous phases (ISM, CGM) rather than just total density.

The paper concludes that this approach offers a computationally cheap and physically intuitive method to understand the interplay between feedback, galaxy properties, and gas distribution, serving as a foundation for future work linking simulation parameters to observational constraints.