Simulation-trained generative models are emerging as a powerful framework for field-level inference in cosmology, particularly in strongly nonlinear regimes where analytical approaches become intractable. However, the finite size of training datasets often leads to variance shrinkage, indicating that part of the realization-level variability of the underlying physical process is not captured by the learned generative dynamics. In this paper, we interpret variance shrinkage as a diagnostic of missing effective degrees of freedom and introduce physics-informed latent modifications to the generative dynamics of a continuous normalizing flow. We show that these latent corrections recover a substantial fraction of the missing variability without requiring additional training simulations. Using weak-lensing field-level inference as a representative cosmological application, we further demonstrate through posterior analysis that the learned latent directions are actively exploited during inference and behave as effective nuisance parameters that absorb part of the mismatch between the training and target distributions. As a result, the model exhibits improved robustness to out-of-distribution shifts in field-level inference. Our work establishes variance shrinkage as a practical diagnostic for identifying missing degrees of freedom in simulation-trained generative models and provides a framework for recovering them through physics-informed modifications of generative dynamics.
Weak gravitational lensing is a powerful cosmological probe that enables direct mapping of the matter distribution and precise constraints on the fundamental parameters governing structure formation in the universe. In recent years, machine learning has shown strong potential for extracting cosmological information from weak-lensing data beyond traditional summary statistics. However, progress has been limited by the lack of standardized benchmarks that capture the complexity of the dataset, simulator imperfections, and various systematic biases encountered in realistic analyses. To address this gap, we present the FAIR Universe Weak Lensing Machine Learning Uncertainty Challenge, one of the NeurIPS 2025 competitions, as a benchmark dataset and evaluation study. Our dataset reflects realistic weak-lensing observations, incorporating multiple systematic biases, and thereby providing a controlled testbed for robust cosmological inference under uncertainties. We summarize the main methodological advances developed by the winners of our challenge, including data augmentation with conditional optimal transport, an AI-agent-driven approach for parameter inference, ensemble learning, Vision Transformer-based architectures, and physics-informed feature integration. By unifying the benchmark dataset and the winning solutions, this work provides a community resource for developing and evaluating machine learning methods for weak lensing cosmology and, more broadly, for scientific inference under systematic uncertainties.
We introduce a spectral hierarchy of cosmic-web classifications obtained by applying simple scale-weighting kernels to the density field before performing a standard eigenvalue-based web classification. This unifies and extends several widely used web definitions within a single framework: the familiar potential/tidal web (large-scale, nonlocal), a curvature-based web (more local, peak- and ridge-sensitive), and additional higher-derivative levels that progressively emphasize smaller-scale structure. Because the classification is built from second derivatives of the filtered field, successive hierarchy levels align naturally with operator families that appear in renormalised bias and effective descriptions of large-scale structure, providing an explicit bridge between cosmic-web environments and long- and short-range nonlocal bias ingredients. We quantify the information content of the hierarchy with a compact statistic: we map each cell to one of four ordered web types (void, sheet, filament, knot), construct a corresponding ``web contrast'' field, and measure its cross-correlation with halos from the AbacusSummit simulation suite on a coarse mesh with $\Delta L\simeq 5.5\,h^{-1}\mathrm{Mpc}$. We find that the hierarchy retains significant tracer-relevant information from very large scales down to the mesh Nyquist limit, with the more local (curvature/higher-derivative) levels dominating toward nonlinear scales. This makes the spectral hierarchy a practical, interpretable conditioning basis for fast mock-galaxy production and field-level modelling, and a flexible tool for studying environment-dependent clustering and assembly bias.
Many scientific simulations are computationally expensive, limiting their use in simulation-based inference, uncertainty quantification, and decision making. We introduce macrocanonical generator networks, a framework for learning fast, data-efficient neural surrogates for amortized simulation of stationary physical processes by matching multiscale statistics of a target process. Inspired by microcanonical maximum-entropy synthesis, which matches feature statistics per sample by inner-loop optimization, our approach instead trains a neural generator to satisfy the same constraints in expectation, preserving physically realistic sample-to-sample fluctuations. We prove that gradient-descent training induces preconditioned gradient-descent dynamics in sample space that inherit the symmetry-preservation properties of the microcanonical gradient descent of Bruna and Mallat, amortizing its per-sample inner-loop optimization into a single training run after which every sample is generated in one forward pass. We further show that residual generator parameterizations initialized near the identity, trained with a small learning rate, preserve entropy locally during early optimization and mitigate mode collapse. Experiments on cosmology and fluid simulations show that macrocanonical generators recover realistic samples from extremely small training sets, in some cases a single training example, while generating new realizations more than $10^6$ times faster per sample than the reference simulator and more than $10^3$ times faster than microcanonical gradient descent. The trained generators can be reused for downstream pretraining, including simulation-based inference, providing a data-efficient route to multi-fidelity scientific machine learning workflows.
Mapping the three-dimensional distribution of dark matter is key to tracing the growth of cosmic structure. Two-dimensional weak lensing mass maps recover its projected distribution, but they blend structures at different distances along the line of sight. We reconstruct the three-dimensional matter distribution in the 0.54 deg$^2$ COSMOS-Web field using deep James Webb Space Telescope imaging, with an effective source density of 121 galaxies arcmin$^{-2}$ We combine a physical weak lensing forward model with a diffusion-based Bayesian generative prior learned from cosmological simulations, yielding a posterior distribution of the matter field across 20 lens planes to $z \simeq 2$, doubling the redshift reach of previous Hubble Space Telescope-based 3D reconstructions and approaching cosmic noon, an epoch of rapid galaxy growth. The reconstruction separates structures that overlap in projection and recovers non-Gaussian matter structure across a range of spatial scales, including over-densities associated with independently identified galaxy clusters that were not detected in earlier weak lensing maps. Posterior sampling quantifies the range of three-dimensional matter distributions allowed by the data, enabling a probabilistic visualization of the cosmic web while propagating uncertainties in its inferred structure. Null and simulation-based tests show that the recovered structures are supported by the lensing signal and are consistent with expectations from the fiducial $\Lambda$CDM cosmology. These results extend three-dimensional weak lensing mapping into the high-redshift, non-Gaussian regime and establish a route towards probabilistic mapping of the evolving cosmic web with current wide-field lensing surveys.
How many latent variables are needed to fit the cosmic microwave background (CMB) temperature and polarisation power spectra when a neural network is given no cosmological model, and to what extent do they map onto the standard cosmological parameters? We use multi-task learning with variational autoencoders trained on CLASS-generated spectra (TT, TE, EE) over $2\le\ell\le2500$. No cosmological model is built into the network, and the number of latent variables, $L$, is varied over $L=5$, $6$ and $7$. Their fits are compared to the Planck 2018 data using $\hatχ^2$, AIC, BIC and Bayesian evidence. Their correspondence with cosmological parameters is measured with normalised mutual information post-training. The loss weights are set within a single training run by uncertainty weighting, with the weight of the disentanglement term either fixed (semi-automated, SA) or learned (completely automated, CA). The reconstruction loss is split over multipole ranges in three configurations that separate the roles of temperature and polarisation and of the low- and high-$\ell$ ranges. In four of the six combinations of configuration and strategy, the fit improves from $L=5$ to $L=6$ and not beyond, and $L=6$ is the best, or close to the best, in five of the six cases. So, six latent variables are needed, as many as the free parameters of the standard, model-dependent approach. SA gives clearer disentanglement, while CA needs no a priori choice of the disentanglement weight and has the higher evidence at $L=5$ and $6$: the two strategies are points on a trade-off between interpretability and fit to the data. Notably, the latent variables for $τ$ and $A_s$ share almost no information in any model, consistent with large-scale polarisation separating the two, as in standard analyses. A deep-learning approach can thus give a compact, data-driven parametrisation of the CMB spectra.