Andrew Pontzen

Davide Piras, Hiranya V. Peiris, Andrew Pontzen et al.

We develop the use of mutual information (MI), a well-established metric in information theory, to interpret the inner workings of deep learning models. To accurately estimate MI from a finite number of samples, we present GMM-MI (pronounced $``$Jimmie$"$), an algorithm based on Gaussian mixture models that can be applied to both discrete and continuous settings. GMM-MI is computationally efficient, robust to the choice of hyperparameters and provides the uncertainty on the MI estimate due to the finite sample size. We extensively validate GMM-MI on toy data for which the ground truth MI is known, comparing its performance against established mutual information estimators. We then demonstrate the use of our MI estimator in the context of representation learning, working with synthetic data and physical datasets describing highly non-linear processes. We train deep learning models to encode high-dimensional data within a meaningful compressed (latent) representation, and use GMM-MI to quantify both the level of disentanglement between the latent variables, and their association with relevant physical quantities, thus unlocking the interpretability of the latent representation. We make GMM-MI publicly available.

Luisa Lucie-Smith, Hiranya V. Peiris, Andrew Pontzen et al.

The density profiles of dark matter halos are typically modeled using empirical formulae fitted to the density profiles of relaxed halo populations. We present a neural network model that is trained to learn the mapping from the raw density field containing each halo to the dark matter density profile. We show that the model recovers the widely-used Navarro-Frenk-White (NFW) profile out to the virial radius, and can additionally describe the variability in the outer profile of the halos. The neural network architecture consists of a supervised encoder-decoder framework, which first compresses the density inputs into a low-dimensional latent representation, and then outputs $ρ(r)$ for any desired value of radius $r$. The latent representation contains all the information used by the model to predict the density profiles. This allows us to interpret the latent representation by quantifying the mutual information between the representation and the halos' ground-truth density profiles. A two-dimensional representation is sufficient to accurately model the density profiles up to the virial radius; however, a three-dimensional representation is required to describe the outer profiles beyond the virial radius. The additional dimension in the representation contains information about the infalling material in the outer profiles of dark matter halos, thus discovering the splashback boundary of halos without prior knowledge of the halos' dynamical history.

Luisa Lucie-Smith, Hiranya V. Peiris, Andrew Pontzen

We use explainable neural networks to connect the evolutionary history of dark matter halos with their density profiles. The network captures independent factors of variation in the density profiles within a low-dimensional representation, which we physically interpret using mutual information. Without any prior knowledge of the halos' evolution, the network recovers the known relation between the early time assembly and the inner profile, and discovers that the profile beyond the virial radius is described by a single parameter capturing the most recent mass accretion rate. The results illustrate the potential for machine-assisted scientific discovery in complicated astrophysical datasets.

Luisa Lucie-Smith, Hiranya V. Peiris, Andrew Pontzen et al.

The evolution of linear initial conditions present in the early universe into extended halos of dark matter at late times can be computed using cosmological simulations. However, a theoretical understanding of this complex process remains elusive; in particular, the role of anisotropic information in the initial conditions in establishing the final mass of dark matter halos remains a long-standing puzzle. Here, we build a deep learning framework to investigate this question. We train a three-dimensional convolutional neural network (CNN) to predict the mass of dark matter halos from the initial conditions, and quantify in full generality the amounts of information in the isotropic and anisotropic aspects of the initial density field about final halo masses. We find that anisotropies add a small, albeit statistically significant amount of information over that contained within spherical averages of the density field about final halo mass. However, the overall scatter in the final mass predictions does not change qualitatively with this additional information, only decreasing from 0.9 dex to 0.7 dex. Given such a small improvement, our results demonstrate that isotropic aspects of the initial density field essentially saturate the relevant information about final halo mass. Therefore, instead of searching for information directly encoded in initial conditions anisotropies, a more promising route to accurate, fast halo mass predictions is to add approximate dynamical information based e.g. on perturbation theory. More broadly, our results indicate that deep learning frameworks can provide a powerful tool for extracting physical insight into cosmological structure formation.