How many simulations do we need to train machine learning methods to extract information available from summary statistics of the cosmological density field? Neural methods have shown the potential to extract non-linear information available from cosmological data. Success depends critically on having sufficient simulations for training the networks and appropriate network architectures. In the first detailed convergence study of neural network training for cosmological inference, we show that currently available simulation suites, such as the Quijote Latin Hypercube(LH) with 2000 simulations, do not provide sufficient training data for a generic neural network to reach the optimal regime, even for the dark matter power spectrum, and in an idealized case. We discover an empirical neural scaling law that predicts how much information a neural network can extract from a highly informative summary statistic, the dark matter power spectrum, as a function of the number of simulations used to train the network, for a wide range of architectures and hyperparameters. We combine this result with the Cramer-Rao information bound to forecast the number of training simulations needed for near-optimal information extraction. To verify our method we created the largest publicly released simulation data set in cosmology, the Big Sobol Sequence(BSQ), consisting of 32,768 $\Lambda$CDM n-body simulations uniformly covering the $\Lambda$CDM parameter space. Our method enables efficient planning of simulation campaigns for machine learning applications in cosmology, while the BSQ dataset provides an unprecedented resource for studying the convergence behavior of neural networks in cosmological parameter inference. Our results suggest that new large simulation suites or new training approaches will be necessary to achieve information-optimal parameter inference from non-linear simulations.

This paper presents the Learning the Universe Implicit Likelihood Inference (LtU-ILI) pipeline, a codebase for rapid, user-friendly, and cutting-edge machine learning (ML) inference in astrophysics and cosmology. The pipeline includes software for implementing various neural architectures, training schema, priors, and density estimators in a manner easily adaptable to any research workflow. It includes comprehensive validation metrics to assess posterior estimate coverage, enhancing the reliability of inferred results. Additionally, the pipeline is easily parallelizable, designed for efficient exploration of modeling hyperparameters. To demonstrate its capabilities, we present real applications across a range of astrophysics and cosmology problems, such as: estimating galaxy cluster masses from X-ray photometry; inferring cosmology from matter power spectra and halo point clouds; characterising progenitors in gravitational wave signals; capturing physical dust parameters from galaxy colors and luminosities; and establishing properties of semi-analytic models of galaxy formation. We also include exhaustive benchmarking and comparisons of all implemented methods as well as discussions about the challenges and pitfalls of ML inference in astronomical sciences. All code and examples are made publicly available at https://github.com/maho3/ltu-ili.

Set-based learning is an essential component of modern deep learning and network science. Graph Neural Networks (GNNs) and their edge-free counterparts Deepsets have proven remarkably useful on ragged and topologically challenging datasets. The key to learning informative embeddings for set members is a specified aggregation function, usually a sum, max, or mean. We propose Fishnets, an aggregation strategy for learning information-optimal embeddings for sets of data for both Bayesian inference and graph aggregation. We demonstrate that i) Fishnets neural summaries can be scaled optimally to an arbitrary number of data objects, ii) Fishnets aggregations are robust to changes in data distribution, unlike standard deepsets, iii) Fishnets saturate Bayesian information content and extend to regimes where MCMC techniques fail and iv) Fishnets can be used as a drop-in aggregation scheme within GNNs. We show that by adopting a Fishnets aggregation scheme for message passing, GNNs can achieve state-of-the-art performance versus architecture size on ogbn-protein data over existing benchmarks with a fraction of learnable parameters and faster training time.

Convolutional neural networks (CNNs) have been shown to both extract more information than the traditional two-point statistics from cosmological fields, and marginalise over astrophysical effects extremely well. However, CNNs require large amounts of training data, which is potentially problematic in the domain of expensive cosmological simulations, and it is difficult to interpret the network. In this work we apply the learnable scattering transform, a kind of convolutional neural network that uses trainable wavelets as filters, to the problem of cosmological inference and marginalisation over astrophysical effects. We present two models based on the scattering transform, one constructed for performance, and one constructed for interpretability, and perform a comparison with a CNN. We find that scattering architectures are able to outperform a CNN, significantly in the case of small training data samples. Additionally we present a lightweight scattering network that is highly interpretable.

We present an implicit likelihood approach to quantifying cosmological information over discrete catalogue data, assembled as graphs. To do so, we explore cosmological parameter constraints using mock dark matter halo catalogues. We employ Information Maximising Neural Networks (IMNNs) to quantify Fisher information extraction as a function of graph representation. We a) demonstrate the high sensitivity of modular graph structure to the underlying cosmology in the noise-free limit, b) show that graph neural network summaries automatically combine mass and clustering information through comparisons to traditional statistics, c) demonstrate that networks can still extract information when catalogues are subject to noisy survey cuts, and d) illustrate how nonlinear IMNN summaries can be used as asymptotically optimal compressed statistics for Bayesian simulation-based inference. We reduce the area of joint $\Omega_m, \sigma_8$ parameter constraints with small ($\sim$100 object) halo catalogues by a factor of 42 over the two-point correlation function, and demonstrate that the networks automatically combine mass and clustering information. This work utilises a new IMNN implementation over graph data in Jax, which can take advantage of either numerical or auto-differentiability. We also show that graph IMNNs successfully compress simulations away from the fiducial model at which the network is fitted, indicating a promising alternative to n-point statistics in catalogue simulation-based analyses.

When you finish reading this sentence, look away from the screen for a moment and repeat it back in your head. Do you know exactly where in your brain this inner "voice" is speaking from? Researchers have tried to map out the regions of the brain responsible for this "inner monologue" for years. One promising candidate is an area called the supramarginal gyrus, which sits a little north of your eyeballs and slightly behind your ears. What's new -- According to new research presented at the recent Society for Neuroscience conference, the supramarginal gyrus could help scientists translate people's inner thoughts.

The interplay between models and observations is a cornerstone of the scientific method, aiming to inform which theoretical models are reflected in the observed data. Within cosmology, as both models and observations have substantially increased in complexity over time, the tools needed to enable a rigorous comparison have required updating as well. With an eye towards the next decade in cosmology, the vast data volumes to be delivered by ongoing and upcoming surveys, as well as the ever-expanding theoretical search-space, motivate a re-thinking of the statistical machinery used. In particular, we are now at a crucial juncture where we may be limited by the statistical and data-driven tools themselves rather than the quality or volume of the available data. Methods based on artificial intelligence (AI) and machine learning (ML) have recently emerged as promising tools for cosmological applications, demonstrating the ability to overcome some of the computational bottlenecks associated with traditional statistical techniques. Machine learning is starting to see increased adoption across different subfields of and for various applications within cosmology. At the same time, the nascent and emergent nature of practical artificial intelligence motivates careful continued development and significant care when it comes to their application in the sciences, as well as cognizance of their potential for broader societal impact. In this white paper, we provide an overview of some of the ways machine learning methods are becoming increasingly central to the way cosmological data is collected, analyzed, and interpreted. Along the way, we highlight our vision for necessary developments, framing these as recommendations--both technological as well as sociological--for the widespread safe and equitable adoption of machine learning methods within cosmology in the coming decade.

High-dimensional probability density estimation for inference suffers from the "curse of dimensionality". For many physical inference problems, the full posterior distribution is unwieldy and seldom used in practice. Instead, we propose direct estimation of lower-dimensional marginal distributions, bypassing high-dimensional density estimation or high-dimensional Markov chain Monte Carlo (MCMC) sampling. By evaluating the two-dimensional marginal posteriors we can unveil the full-dimensional parameter covariance structure. We additionally propose constructing a simple hierarchy of fast neural regression models, called Moment Networks, that compute increasing moments of any desired lower-dimensional marginal posterior density; these reproduce exact results from analytic posteriors and those obtained from Masked Autoregressive Flows. We demonstrate marginal posterior density estimation using high-dimensional LIGO-like gravitational wave time series and describe applications for problems of fundamental cosmology.