Measuring the Dark Matter Self-Interaction Cross-Section with Deep Compact Clustering for Robust Machine Learning Inference

TL;DR

This work tackles the challenge of constraining the dark matter self-interaction cross-section $ rac{\sigma_{\rm DM}}{m} $ from two-dimensional galaxy cluster mass maps in a trustworthy way. It introduces a semi-supervised, compact deep clustering framework that learns a 7D latent space where simulations with different $ rac{\sigma_{\rm DM}}{m} $ cluster together and where out-of-domain data can be identified via latent-space proximity, enabling both parameter estimation and confidence assessments. By training on two simulation suites (BAHAMAS-SIDM and DARKSKIES) and using an ensemble of networks, the method yields posterior estimates for $ rac{\sigma_{\rm DM}}{m} $ with quantified uncertainty and demonstrates robust OOD detection with random-noise inputs and progressively incorporating more simulations to adapt to new domains. The approach advances transparent, robust inference in cosmology, offering a blueprint for applying domain-aware ML to real observations while highlighting the need for domain adaptation and broader simulation coverage to ensure reliable application to data from next-generation surveys.

Abstract

We have developed a machine learning algorithm capable of detecting ``out-of-domain data'' for trustworthy cosmological inference. By using data from two separate suites of cosmological simulations, we show that our algorithm is able to determine whether ``observed'' data is consistent with its training domain, returning confidence estimates as well as accurate parameter estimations. We apply our algorithm to two-dimensional images of galaxy clusters from the BAHAMAS-SIDM and DARKSKIES simulations with the aim to measure the self-interaction cross-section of dark matter. Through deep compact clustering we construct an informative latent space where galaxy clusters are mapped to the latent space forming ``latent-clusters'' for each simulation, with the location of the latent-cluster corresponding to the macroscopic parameters, such as the cross-section, $σ_{\rm DM}/m$. We then pass through mock observations, where the location of the observed latent-cluster informs us of which properties are shared with the training data. If the observed latent-cluster shares no similarities with latent-clusters from the known simulations, we can conclude that our simulations do not represent the observations and discard any parameter estimations, thus providing us with a method to measure machine learning confidence. This method serves as a blueprint for transparent and robust inference that is in demand in scientific machine learning.

Figures (11)

• Figure 1: The architecture and loss functions used in this paper. The input is the total mass and optionally X-ray maps, which is then compressed using a convolutional NN (blue encoder) into a 7D latent space. From this latent space, we can get the similarity cluster and distance losses, equations \ref{['eq:cclp_loss']} and \ref{['eq:dist_loss']}, or further transform it using a fully connected NN (red classifier) to obtain the classification loss, Equation \ref{['eq:class_loss']}. All losses are then weighted summed using the weights $\lambda_{\rm CCLP}$, $\lambda_{\rm dist}$, and $\lambda_{\rm class}$, Equation \ref{['eq:total-loss']} See Figure \ref{['fig:network-architecture']} for the full encoder and classifier architecture.
• Figure 2: Average classification accuracy from five NNs, normalised to the asymptotic accuracy, against the number of latent dimensions. Two sets of NNs are trained, one with X-rays included (reds/triangles) and the other excluded (blues/non-triangles). Each line shows an increase in the number of simulations included in the training, starting with BAHAMAS-0 and BAHAMAS-SIDM (solid), then BAHAMAS-AGN (dashed), and finally DARKSKIES (dotted). We fit each set of data with an $y=a+\arctan\left( \left(\left|\mathcal{Z}\right|-b\right)/c\right)$ fit (lines).
• Figure 3: The first two components of the PCA of the 7D latent space. Each point corresponds to a galaxy cluster from its colour corresponding simulation with the contours representing the 68% region for that simulation. The unknown datasets, BAHAMAS-0w and BAHAMAS-0s, are represented by the hatched contours and asterisk next to the legend label. We interpret that the first component corresponds to a transformation of $\log{\left(\sigma_{\rm DM}/m\right)}$ and the second component corresponds to the level of AGN feedback.
• Figure 4: Method for building confidence in our ML estimator for data outside the training domain. Left: We train an ensemble of 10 regression NNs on the fiducial hydro BAHAMAS simulations. We show the combined probability distribution functions (PDF) of $\log{\left(\sigma_{\rm DM}/m\right)}$ for the known BAHAMAS simulations (blue shades) and blind uniform random noise (black hatched). We find that the regressor consistently estimates a significant, positive cross-section of $\sim0.6\ {\rm cm^2 g^{-1}}$ with no regard to its confidence, presenting the issue with direct regression estimators. Right: We show the first and third latent dimensions from our compact clustering algorithm trained with the same fiducial BAHAMAS as known (blue shades) and the random noise dataset as unknown (black hatched). The first latent dimension corresponds to $\log{\sigma_{\rm DM}/m}$, which we would naively assume that the noise dataset has a cross-section of $\sim0.1\ {\rm cm^2g^{-1}}$; however, from the third dimension we see that the noise dataset shares no similarities with the known simulations and therefore, cannot be trusted.
• Figure 5: A consistency check on in-distribution testing. Left: We train an ensemble of 3 clustering NNs and show the combined PDF of $\log{\left(\sigma_{\rm DM}/m\right)}$ for the known BAHAMAS-0, BAHAMAS-0.3, and BAHAMAS-1 (solid blue shades) and unknown BAHAMAS-0.1 (black dashed). We find a cross-section of $\sim0.05\ {\rm cm^2 g^{-1}}$ and within $1\sigma$ of $0.1\ {\rm cm^2 g^{-1}}$ for BAHAMAS-0.1. Right: The projected 7D latent space from our clustering algorithm into a 1D distance PDF, where each known simulation (blue shades) is projected in the direction from the centre of their distribution to the centre of BAHAMAS-0.1. BAHAMAS-0.1 (black hatched) is projected in the direction of BAHAMAS-0.3 to show the greatest overlap. The distance has arbitrary units as it depends on the scale of the latent space. BAHAMAS-0.1 shares a large overlap of 49% and 55% with BAHAMAS-0 and BAHAMAS-0.3, respectively, leading to the conclusion that it lies within the training domain.