We examine the use of a novel variant of Physics-Informed Neural Networks to predict cosmological parameters from recent supernovae and baryon acoustic oscillations (BAO) datasets. Our machine learning framework generates uncertainty estimates for target variables and the inferred unknown parameters of the underlying PDE descriptions. Built upon a hybrid of the principles of Evidential Deep Learning, Physics-Informed Neural Networks, Bayesian Neural Networks and Gaussian Processes, our model enables learning of the posterior distribution of the unknown PDE parameters through standard gradient-descent based training. We apply our model to an up-to-date BAO dataset (Bousis et al. 2024) calibrated with the CMB-inferred sound horizon, and the Pantheon$+$ Sne Ia distances (Scolnic et al. 2018), examining the relative effectiveness and mutual consistency among the standard $Λ$CDM, $w$CDM and $Λ_s$CDM models. Unlike previous results arising from the standard approach of minimizing an appropriate $χ^2$ function, the posterior distributions for parameters in various models trained purely on Pantheon$+$ data were found to be largely contained within the $2σ$ contours of their counterparts trained on BAO data. Their posterior medians for $h_0$ were within about $2σ$ of one another, indicating that our machine learning-guided approach provides a different measure of the Hubble tension.

In cosmology, emulators play a crucial role by providing fast and accurate predictions of complex physical models, enabling efficient exploration of high-dimensional parameter spaces that would be computationally prohibitive with direct numerical simulations. Symbolic emulators have emerged as promising alternatives to numerical approaches, delivering comparable accuracy with significantly faster evaluation times. While previous symbolic emulators were limited to relatively narrow prior ranges, we expand these to cover the parameter space relevant for current cosmological analyses. We introduce approximations to hypergeometric functions used for the $Λ$CDM comoving distance and linear growth factor which are accurate to better than 0.001% and 0.05%, respectively, for all redshifts and for $Ω_{\rm m} \in [0.1, 0.5]$. We show that integrating symbolic emulators into a Dark Energy Survey-like $3\times2$pt analysis produces cosmological constraints consistent with those obtained using standard numerical methods. Our symbolic emulators offer substantial improvements in speed and memory usage, demonstrating their practical potential for scalable, likelihood-based inference.

We propose a new field-level emulator that bridges two simulators using unpaired simulation datasets. Our method leverages a flow-based approach to learn the likelihood transport from one simulator to the other. Since multiple transport maps exist, we employ Conditional Optimal Transport Flow Matching (COT-FM) to ensure that the transformation minimally distorts the underlying structure of the data. We demonstrate the effectiveness of this approach by bridging weak lensing simulators: a Lagrangian Perturbation Theory (LPT) to a N-body Particle-Mesh (PM). We demonstrate that our emulator captures the full correction between the simulators by showing that it enables full-field inference to accurately recover the true posterior, validating its accuracy beyond traditional summary statistics.

Machine learning enables powerful cosmological inference but typically requires many high-fidelity simulations covering many cosmological models. Transfer learning offers a way to reduce the simulation cost by reusing knowledge across models. We show that pre-training on the standard model of cosmology, $Λ$CDM, and fine-tuning on various beyond-$Λ$CDM scenarios -- including massive neutrinos, modified gravity, and primordial non-Gaussianities -- can enable inference with significantly fewer beyond-$Λ$CDM simulations. However, we also show that negative transfer can occur when strong physical degeneracies exist between $Λ$CDM and beyond-$Λ$CDM parameters. We consider various transfer architectures, finding that including bottleneck structures provides the best performance. Our findings illustrate the opportunities and pitfalls of foundation-model approaches in physics: pre-training can accelerate inference, but may also hinder learning new physics.

Generative models are a promising tool to produce cosmological simulations but face significant challenges in scalability, physical consistency, and adherence to domain symmetries, limiting their utility as alternatives to $N$-body simulations. To address these limitations, we introduce a score-based generative model with an equivariant graph neural network that simulates gravitational clustering of galaxies across cosmologies starting from an informed prior, respects periodic boundaries, and scales to full galaxy counts in simulations. A novel topology-aware noise schedule, crucial for large geometric graphs, is introduced. The proposed equivariant score-based model successfully generates full-scale cosmological point clouds of up to 600,000 halos, respects periodicity and a uniform prior, and outperforms existing diffusion models in capturing clustering statistics while offering significant computational advantages. This work advances cosmology by introducing a generative model designed to closely resemble the underlying gravitational clustering of structure formation, moving closer to physically realistic and efficient simulators for the evolution of large-scale structures in the universe.

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.

Next-generation galaxy surveys promise unprecedented precision in testing gravity at cosmological scales. However, realising this potential requires accurately modelling the non-linear cosmic web. We address this challenge by exploring conditional generative modelling to create 3D dark matter density fields via score-based (diffusion) and flow-based methods. Our results demonstrate the power of diffusion models to accurately reproduce the matter power spectra and bispectra, even for unseen configurations. They also offer a significant speed-up with slightly reduced accuracy, when flow-based reconstructing the probability distribution function, but they struggle with higher-order statistics. To improve conditional generation, we introduce a novel multi-output model to develop feature representations of the cosmological parameters. Our findings offer a powerful tool for exploring deviations from standard gravity, combining high precision with reduced computational cost, thus paving the way for more comprehensive and efficient cosmological analyses null .

Finding the best parametrization for cosmological models in the absence of first-principle theories is an open question. We propose a data-driven parametrization of cosmological models given by the disentangled 'latent' representation of a variational autoencoder (VAE) trained to compress cosmic microwave background (CMB) temperature power spectra. We consider a broad range of $\Lambda$CDM and beyond-$\Lambda$CDM cosmologies with an additional early dark energy (EDE) component. We show that these spectra can be compressed into 5 ($\Lambda$CDM) or 8 (EDE) independent latent parameters, as expected when using temperature power spectra alone, and which reconstruct spectra at an accuracy well within the Planck errors. These latent parameters have a physical interpretation in terms of well-known features of the CMB temperature spectrum: these include the position, height and even-odd modulation of the acoustic peaks, as well as the gravitational lensing effect. The VAE also discovers one latent parameter which entirely isolates the EDE effects from those related to $\Lambda$CDM parameters, thus revealing a previously unknown degree of freedom in the CMB temperature power spectrum. We further showcase how to place constraints on the latent parameters using Planck data as typically done for cosmological parameters, obtaining latent values consistent with previous $\Lambda$CDM and EDE cosmological constraints. Our work demonstrates the potential of a data-driven reformulation of current beyond-$\Lambda$CDM phenomenological models into the independent degrees of freedom to which the data observables are sensitive.

Conventional calibration of Baryon Acoustic Oscillations (BAO) data relies on estimation of the sound horizon at drag epoch $r_d$ from early universe observations by assuming a cosmological model. We present a recalibration of two independent BAO datasets, SDSS and DESI, by employing deep learning techniques for model-independent estimation of $r_d$, and explore the impacts on $\Lambda$CDM cosmological parameters. Significant reductions in both Hubble ($H_0$) and clustering ($S_8$) tensions are observed for both the recalibrated datasets. Moderate shifts in some other parameters hint towards further exploration of such data-driven approaches.

Building upon [2308.02636], this article investigates the potential constraining power of persistent homology for cosmological parameters and primordial non-Gaussianity amplitudes in a likelihood-free inference pipeline. We evaluate the ability of persistence images (PIs) to infer parameters, compared to the combined Power Spectrum and Bispectrum (PS/BS), and we compare two types of models: neural-based, and tree-based. PIs consistently lead to better predictions compared to the combined PS/BS when the parameters can be constrained (i.e., for $\{\Omega_{\rm m}, \sigma_8, n_{\rm s}, f_{\rm NL}^{\rm loc}\}$). PIs perform particularly well for $f_{\rm NL}^{\rm loc}$, showing the promise of persistent homology in constraining primordial non-Gaussianity. Our results show that combining PIs with PS/BS provides only marginal gains, indicating that the PS/BS contains little extra or complementary information to the PIs. Finally, we provide a visualization of the most important topological features for $f_{\rm NL}^{\rm loc}$ and for $\Omega_{\rm m}$. This reveals that clusters and voids (0-cycles and 2-cycles) are most informative for $\Omega_{\rm m}$, while $f_{\rm NL}^{\rm loc}$ uses the filaments (1-cycles) in addition to the other two types of topological features.