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arxiv: 2511.20429 · v2 · submitted 2025-11-25 · 🌌 astro-ph.CO · astro-ph.GA

Estimating the triaxiality of massive clusters from 2D observables in MillenniumTNG with machine learning

Ana Maria Delgado , Michelle Ntampaka , Sownak Bose , Fulvio Ferlito , Boryana Hadzhiyska , Lars Hernquist , John Soltis , John F. Wu
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Mikaeel Yunus John ZuHone
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Pith reviewed 2026-05-17 04:25 UTC · model grok-4.3

classification 🌌 astro-ph.CO astro-ph.GA
keywords galaxy clusterstriaxialitymachine learningcosmologyhydrodynamical simulationsneural networkscluster observables2D projections
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The pith

A fusion neural network estimates the triaxial shapes and orientations of massive galaxy clusters from 2D images and member data.

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

Massive galaxy clusters deviate from perfect spherical symmetry, which biases estimates of their mass, concentration, and other properties that are sensitive to cosmology. This work trains a deep learning model on hydrodynamical simulations to predict the three-dimensional geometry directly from two-dimensional multi-wavelength images and graphs of galaxy member observables. The model combines convolutional networks to analyze X-ray and thermal Sunyaev-Zel'dovich images with graph networks to process positions, velocities, and luminosities of cluster members. It achieves a thirty percent improvement over the common assumption of spherical symmetry, with strong performance on predicting the major axis length and identifying prolate clusters aligned with the line of sight. If the method holds, cluster-based cosmological tests can achieve higher precision by accounting for realistic shapes.

Core claim

The paper demonstrates that a multi-modal network integrating convolutional and graph neural networks can extract three-dimensional triaxiality and orientation information for massive clusters from idealized two-dimensional observables in the MillenniumTNG simulations, yielding an R-squared score of 0.85 for major axis length regression and correctly classifying 71 percent of prolate clusters with line-of-sight elongations, which is a 30 percent improvement compared to spherical models.

What carries the argument

The multi-modal fusion network that uses a convolutional neural network to process multi-wavelength 2D cluster images and a graph neural network to handle mathematical graph representations of cluster member observables.

Load-bearing premise

The idealized 2D images and graph data generated from the simulations accurately represent the measurements obtainable from real galaxy clusters with current instruments.

What would settle it

Running the model on a sample of real observed clusters and checking if the predicted shapes match independent determinations from strong gravitational lensing or multi-wavelength tomography.

Figures

Figures reproduced from arXiv: 2511.20429 by Ana Maria Delgado, Boryana Hadzhiyska, Fulvio Ferlito, John F. Wu, John Soltis, John ZuHone, Lars Hernquist, Michelle Ntampaka, Mikaeel Yunus, Sownak Bose.

Figure 1
Figure 1. Figure 1: Histogram of MTNG cluster mass. We define massive clusters as those having of M200 > 1014 M⊙, yield￾ing 4,117 massive clusters in MTNG. Most cluster masses fall between log10(14.0 − 14.5) M⊙. For machine learning applications, we must ensure that low-frequency clusters are properly distributed across train/validation/test splits. example, along the Cartesian z-axis of the simulation box, which we adopt as … view at source ↗
Figure 2
Figure 2. Figure 2: Visualization of massive MTNG clusters. In each panel, dark matter particles (sampled at 1-in-100) are shown in blue, gas particles (sampled at 1-in-100) are shown in magenta, and subhalos are shown in orange. Corresponding ellipses are drawn for the ϵ+ of the positions. Blue ellipses correspond to ϵ+ measured from the dark matter positions weighted by particle mass, magenta ellipses correspond to that of … view at source ↗
Figure 3
Figure 3. Figure 3: Views of different cluster morphologies. Each column illustrates views of four different clusters belonging to the same morphology. The clusters are grouped based on their ∆ϵ parameter value and scatter plots are made of their subhalo positions. Each row shows a different line-of-sight. We draw the reader’s attention to the middle column containing the clusters of great interest to our study, which we refe… view at source ↗
Figure 4
Figure 4. Figure 4: A histogram of the distribution of the princi￾ple axis orientations, |e1z |, from different morphologies of massive MTNG clusters as defined in Section 2.2.1. While spheroids (blue) and ellipsoids (orange) are abundant, false spheres (green) are much more rare and have a limited |e1z | range and low frequency. Correctly identifying these false spheres in observations could reduce systematic errors in weak … view at source ↗
Figure 5
Figure 5. Figure 5: Visualization of our 4-channel cluster images. Each image is normalized as per Eq. 9 with dimensions 128 x 128 pixels. We show images for the most massive cluster in our catalog (top), the median mass cluster (middle) and the least massive cluster (bottom). Differences across the four wavelengths are noticeable by eye. images and complex spatial patterns, making it well￾suited for extracting triaxiality in… view at source ↗
Figure 6
Figure 6. Figure 6: Visualization of the hNN pipeline. Our hNN consists of two main branches for feature extraction: a GNN branch designed for graph-level predictions, which takes a graph representation for each cluster as input, and a CNN branch that takes 4-channel 128x128 multi-wavelength images of each cluster as inputs. The latent representations from each branch are then combined in a fusion layer where further feature … view at source ↗
Figure 7
Figure 7. Figure 7: Distribution of galaxy members within MTNG clusters. We define galaxies as subhalos resolved with at least 50 stellar particles corresponding to M∗ ≳ 1.5 × 109 M⊙. Most clusters in our catalog have ∼ 50 galaxies. ing node xj ) with their corresponding edge attributes (εij ). This concatenated representation is processed through a three-layer MLP with layer normalization, GELU activations, 8 hidden channels… view at source ↗
Figure 8
Figure 8. Figure 8: A toy illustration of a graph with edge features angle α. Each node represents a galaxy, and angle α is seen as the angle that subtends the central galaxy by a pair of two other galaxies [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: A toy illustration of edge feature angle φ shown with two nodes representing a pair of galaxies. φ is the align￾ment angle between the major-axis of a galaxy and the vector connecting its pair. galaxy as a node feature, but found that this did not improve our model’s performance. CNN image features: The CNN branch is trained on 4-channel 128x128 ide￾alized images: • three channels are of Xray photon intens… view at source ↗
Figure 10
Figure 10. Figure 10: hNN regression results from our fiducial model. R 2 scores for each parameter are shown in the bottom right corner of the panels. The black solid line depicts where a perfect prediction would fall. Error bars (grey) are the upper and lower bounds calculated from three separate training iterations of the hNN. Our fiducial model is able to account for most of the variation in the data across all six variabl… view at source ↗
Figure 12
Figure 12. Figure 12: The percent of clusters in the test set for which orientation was correctly classified and triaxiality (all three semi-axes) was estimated within an absolute error threshold of MTNG True. We only show the absolute error within 50% of MTNG (along the x-axis). The dashed black line is the result from our fiducial hNN model. We compare our model against two hypothetical models: one predicting cor￾rect orient… view at source ↗
Figure 11
Figure 11. Figure 11: hNN classification results from our fiducial model. (Top) Confusion matrix results from the full test set for the classification of the principal axis orientation, |e1z |, of all clusters. The accuracy averaged for all classes is reported above the confusion matrix. The highest frequency corre￾sponds to the correct classification, 68%, of clusters elon￾gated along our LoS. Errors are the upper and lower b… view at source ↗
Figure 13
Figure 13. Figure 13: An approach to feature importance analysis. We performed an ablation study by systematically omitting features or channels and retraining models with identical architecture, hyperparameters and duration as our fiducial model. The above results show the percentage of test set clusters within a given threshold of MTNG (similar to Fig￾ure 12) for triaxiality regression only (without conditioning on classific… view at source ↗
Figure 14
Figure 14. Figure 14: Feature importance analysis via ablation study for the classification task. Confusion matrices are shown for each of the ablated variants. Accuracy scores are also reported for each model. Results from our fiducial model (top panel of [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Mean population saliency maps from the CNN branch of the network show by wavelength (rows) as a function of binned mass (columns). We note that the number of clusters in each column decreases as mass increases, with the lowest mass bin containing 238 clusters while the highest mass bin contains 17 clusters. Each map shows the mean of activated pixels from the stacked test set. The centers of the galaxies … view at source ↗
Figure 16
Figure 16. Figure 16: Similar to [PITH_FULL_IMAGE:figures/full_fig_p024_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Saliency maps showing consensus activation across wavelengths for individual clusters for the prediction of principal axis orientation. Each row in this figure is a different ellipsoidal cluster whose correct principal axis orientation label is shown in the leftmost column. The rightmost column shows consensus: mean activation for pixels where there is ≥ 0.10 activation in every wavelength. The consensus … view at source ↗
Figure 18
Figure 18. Figure 18: Similar to [PITH_FULL_IMAGE:figures/full_fig_p026_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Similar to [PITH_FULL_IMAGE:figures/full_fig_p026_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Similar to [PITH_FULL_IMAGE:figures/full_fig_p027_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Similar to [PITH_FULL_IMAGE:figures/full_fig_p027_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Similar to [PITH_FULL_IMAGE:figures/full_fig_p028_22.png] view at source ↗
read the original abstract

Properties of massive galaxy clusters, such as mass abundance and concentration, are sensitive to cosmology, making cluster statistics a powerful tool for cosmological studies. However, favoring a more simplified, spherically symmetric model for galaxy clusters can lead to biases in the estimates of cluster properties. In this work, we present a deep-learning approach for estimating the triaxiality and orientations of massive galaxy clusters (those with masses $\gtrsim 10^{14}\,M_\odot h^{-1}$) from 2D observables. We utilize the flagship hydrodynamical volume of the suite of cosmological-hydrodynamical MillenniumTNG (MTNG) simulations as our ground truth. Our model combines the feature extracting power of a convolutional neural network (CNN) and the message passing power of a graph neural network (GNN) in a multi-modal, fusion network. Our model is able to extract 3D geometry information from 2D idealized cluster multi-wavelength images (soft X-ray, medium X-ray, hard X-ray and tSZ effect) and mathematical graph representations of 2D cluster member observables (line-of-sight radial velocities, 2D projected positions and V-band luminosities). Our network improves cluster geometry estimation in MTNG by $30\%$ compared to assuming spherical symmetry. We report an $R^2 = 0.85$ regression score for estimating the major axis length of triaxial clusters and correctly classifying $71\%$ of prolate clusters with elongated orientations along our line-of-sight.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a multi-modal deep learning model that fuses a convolutional neural network (CNN) operating on idealized 2D multi-wavelength images (soft/medium/hard X-ray and tSZ) with a graph neural network (GNN) on projected galaxy positions, line-of-sight velocities, and V-band luminosities. The model is trained and evaluated on massive clusters (M ≳ 10^14 M_⊙ h^{-1}) from the MillenniumTNG hydrodynamical simulations to infer 3D triaxial shapes and orientations, claiming a 30% improvement over spherical symmetry, an R² = 0.85 for major-axis length regression, and 71% accuracy in classifying prolate clusters aligned with the line of sight.

Significance. If the performance metrics generalize beyond the idealized simulation inputs, the work could reduce systematic biases in cluster mass and concentration estimates that arise from spherical assumptions, thereby strengthening cosmological constraints from cluster abundance and clustering statistics. The multi-modal CNN-GNN fusion and use of the large-volume MTNG suite are clear strengths that enable direct comparison to a spherical baseline without additional free parameters in the improvement metric.

major comments (2)
  1. Abstract and Methods: The central performance claims (30% improvement, R² = 0.85, 71% classification accuracy) are reported without any description of train/test splits, cross-validation procedure, or error bars on the metrics. This omission prevents verification that the quoted numbers reflect generalization rather than overfitting to the specific MTNG realization.
  2. Abstract: The idealized 2D images and graph representations contain no modeling of instrumental effects (PSF convolution, noise, foreground subtraction, or selection biases) that would be present in Chandra, XMM-Newton, or eROSITA data. Because the improvement metric is defined relative to spherical symmetry on these clean inputs, it is unclear whether the quoted gains survive realistic observational systematics.
minor comments (2)
  1. The mass threshold is written as ≳ 10^{14} M_⊙ h^{-1}; consistent use of the same notation throughout the text would improve readability.
  2. The abstract refers to 'mathematical graph representations' without specifying the exact node and edge features; a brief enumeration in the main text would clarify the GNN input construction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and insightful comments, which have helped us identify areas where the manuscript can be clarified and strengthened. We provide point-by-point responses to the major comments below and indicate the revisions we will implement.

read point-by-point responses

  1. Referee: Abstract and Methods: The central performance claims (30% improvement, R² = 0.85, 71% classification accuracy) are reported without any description of train/test splits, cross-validation procedure, or error bars on the metrics. This omission prevents verification that the quoted numbers reflect generalization rather than overfitting to the specific MTNG realization.

    Authors: We agree that the absence of explicit details on data partitioning and validation in the Abstract (and insufficient emphasis in the main text) makes it difficult to assess generalization. The manuscript Methods section does describe a random split of the MTNG cluster sample into training and test sets with no overlap, but we did not report cross-validation folds or uncertainty estimates on the metrics. In the revised manuscript we will (i) add a concise statement to the Abstract summarizing the splitting procedure and (ii) include error bars on all quoted metrics obtained via bootstrap resampling of the test set. These changes will directly address the concern about overfitting versus generalization. revision: yes

  2. Referee: Abstract: The idealized 2D images and graph representations contain no modeling of instrumental effects (PSF convolution, noise, foreground subtraction, or selection biases) that would be present in Chandra, XMM-Newton, or eROSITA data. Because the improvement metric is defined relative to spherical symmetry on these clean inputs, it is unclear whether the quoted gains survive realistic observational systematics.

    Authors: The referee is correct that the current results are obtained on idealized, noise-free inputs; this was an intentional first step to quantify the maximum information content extractable from the multi-wavelength and galaxy observables. We do not assert that the precise numerical gains will persist once realistic observational effects are included. In the revised manuscript we will add a dedicated paragraph in the Discussion section that (a) explicitly acknowledges this limitation, (b) discusses the likely directions in which performance may degrade, and (c) outlines planned follow-up work that will incorporate mock observations with PSF, noise, and selection effects. This revision will make the scope and limitations of the present study transparent. revision: partial

Circularity Check

0 steps flagged

No significant circularity; standard supervised ML on simulation ground truth

full rationale

The paper trains a multi-modal CNN-GNN network on idealized 2D multi-wavelength images and projected member graphs extracted from MTNG hydrodynamical simulations to regress 3D triaxiality parameters and classify orientations. Reported performance (30% improvement over spherical symmetry baseline, R^2=0.85 for major-axis length, 71% prolate LOS classification) is measured against an independent external baseline rather than any fitted input or self-defined quantity. No load-bearing self-citations, uniqueness theorems, or ansatzes are invoked; the derivation is a conventional supervised regression task whose outputs are not equivalent to its inputs by construction. The setup is self-contained against the simulation benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the fidelity of the MillenniumTNG hydrodynamical volume as ground truth and on the assumption that the idealized 2D observables match real telescope data; no free parameters or new entities are introduced beyond standard neural-network training.

axioms (1)
  • domain assumption MillenniumTNG hydrodynamical simulations provide accurate 3D ground-truth triaxiality and orientations for massive clusters.
    The paper uses MTNG outputs directly as labels for supervised training and evaluation.

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