Probabilistic mass-mapping with neural score estimation

“To overcome these limitations, Remy et al. (2022) introduced an approach allowing to sample from the full posterior distribution of the mass-mapping problem.”

“At the time of writing there are only three examples of score-based (or diffusion) modelling in the astronomy literature [184,217,218].”

“[20] to produce samples from the posterior of convergence maps from weak lensing data.”

“Additionally, machine learning methods have demonstrated the ability to outperform traditional estimators in mass mapping—reconstructing the weak lensing convergence maps from observed galaxy ellipticities [37–40].”

“…Feng et al. 2023; Dia et al. 2023; Drozdova et al. 2024; Feng et al. 2024), deconvolution (e.g. Xue et al. 2023; Adam et al. 2023), mass modelling (Remy et al. 2023), cosmological fields or cosmological parameters recovery (Mudur & Finkbeiner 2022; Heurtel-Depeiges et al. 2023; Fl¨oss et al.…”

“…2023; Kawar et al., 2022; Dou & Song, 2024; Chung et al., 2023; Feng et al., 2023; Dia et al., 2023; Drozdova et al., 2024; Xue et al., 2023; Remy et al., 2023; Flöss et al., 2024; Feng et al., 2024), noise models for explicit likelihood inference ( e.g. , Legin et al., 2023; Adam et al.,…”

“…shear and convergence has also permitted to build Bayesian reconstruction schemes of the convergence based on an observed shear field with the help of informed priors [see for example 34–36] as well as machine learning reconstruction algorithms [37] and more recently clever mix of the two [38].”

“Following Remy et al. (2023), the relative probability between two points x ∗ and x n drawn from a distribution p ( x ) can be obtained from the score by solving the line integral We compute the integral in Eq.”

“…transformation [74], informative priors for example to regularize inverse problems [ e.g. 75], data-driven priors potentially specified by a generative model [ e.g. 42, 76, 77], or data-informed priors, where the posterior of an a priori analysis is used as the prior for an analysis with new data…”

“Diffusion models have been successfully applied to solve inverse problems (Kawar et al., 2022; Remy et al., 2023; Adam et al., 2022; Song et al., 2022).”

“…shear and convergence has also permitted to build Bayesian reconstruction schemes of the convergence based on an observed shear field with the help of informed priors [see for example 34–36] as well as machine learning reconstruction algorithms [37] and more recently clever mix of the two [38].”

“First, a prior encapsulating the space of all possible galaxy morphologies must be defined; diffusion models are well known to effectively represent such complex priors [19–21].”

“…such as the reconstruction of source galaxies from gravitational lensing observations [ 106 ] , dark matter mass mapping from weak lensing data [ 107 ] , gravitational lensing potential from distorted Cosmic Microwave Background observations [ 108 ] , and the separation of astrophysical sources…”

“However, since the likelihoods considered in this work are very informative, we can employ the convolved likelihood approximation [31, 32], which allows us to write where A ≡ S F P beam , Σ jk = δ jk + δ j 0 δ k 0 , where δ jk denotes the Kronecker delta, and Γ ∈ 2 m × 2 m is the covariance matrix…”

“The closest existing works which we are aware of are [6] and [7], which also aim to improve the posterior of a noisy observation of a Gaussian field, by using a learned prior.”

“In addition, DM has also been applied in mass reconstruction (B. Remy et al. 2023; S. S. Boruah et al. 2025).”

“…in high-energy particle physics experiments, e.g., Bellagente et al., 2020; Diefenbacher et al., 2024; Pazos et al., 2025, or in astrophysics, e.g., Remy et al., 2023; Adam et al., 2022; Feng et al., 2023; Legin et al., 2024) or geosciences (e.g Andry et al., 2025; Savary et al., 2025; Rozet &…”

“The seminal work of Kaiser and Squires [24] introduced a direct inversion method, which has since been improved via sparsity-based regularization [30,31], inverse-variance filtering [47], or supervised deep learning [22,43] to better handle the severe noise.”

“The weak lensing community is served by publicly available mapping tools like lenspack (Peel, 2020) and jax-lensing (Remy et al., 2023), each with their own strengths. jax-lensing excels at neural-network-based approaches and deep learning methods, but applying it to survey-specific data products…”

“Related decompositions are known: Remy et al. (2023) combine an analytic Gaussian score with a learned residual trained by residual denoising score matching, while Wang & Vastola (2024) show that learned diffusion scores are well approximated by a Gaussian linear score at moderate to high noise…”

“More recent developments include deep-learning-based reconstructions employing U-Net convolutional architectures [19] and diffusion models [5, 6, 37], which learn nonlinear priors directly from simulated data.”

“Diffusion and score-based methods have also been applied to weak-lensing mass map denoising and reconstruction [18–20].”

“Existing UQ approaches are often tied to specific sets of cosmological parameters by the training of data-driven prior models on simulated data [17, 18] or by performing conformal prediction with a set of simulated calibration data [10, 11].”

“Data-driven specifications include those potentially specified by a generative model (e.g. [23, 24, 25, 26]), and data-informed priors, where the posterior of an a priori analysis is used as the prior for an analysis with new data (e.g. [27]).”

“In addition to GAN, several previous works employed DM [Remy et al., 2023, Boruah et al., 2025] for denoising mass maps motivated by the fact that DM outperforms GAN in various tasks.”

“This is commonly formulated as a Bayesian inference problem [42, 53]: where p ( κ ) is a prior on the convergence field, and p ( γ obs | κ ) is the likelihood.”

“…Wiener filtering using architectures such as DeepMass Jeffrey et al. (2020).. Probabilistic extensions have enabled uncertainty quantification using Bayesian sampling techniques Remy et al. (2023) or conformalized regression to estimate credible intervals on recovered structures Leterme et al. (2025).”

“This also leads to leakage of the E and B modes of the reconstructed convergence field (Remy et al. 2022).”