Cosmological Analysis with Calibrated Neural Quantile Estimation and Approximate Simulators

TL;DR

NQE is introduced, a new Simulation-Based Inference method that leverages a large number of approximate simulations for training and a small number of high-fidelity simulations for calibration that guarantees an unbiased posterior and achieves near-optimal constraining power when the approximate simulations are reasonably accurate.

Abstract

A major challenge in extracting information from current and upcoming surveys of cosmological Large-Scale Structure (LSS) is the limited availability of computationally expensive high-fidelity simulations. We introduce Neural Quantile Estimation (NQE), a new Simulation-Based Inference (SBI) method that leverages a large number of approximate simulations for training and a small number of high-fidelity simulations for calibration. This approach guarantees an unbiased posterior and achieves near-optimal constraining power when the approximate simulations are reasonably accurate. As a proof of concept, we demonstrate that cosmological parameters can be inferred at field level from projected 2-dim dark matter density maps up to $k_{\rm max}\sim1.5\,h$/Mpc at $z=0$ by training on $\sim10^4$ Particle-Mesh (PM) simulations with transfer function correction and calibrating with $\sim10^2$ Particle-Particle (PP) simulations. The calibrated posteriors closely match those obtained by directly training on $\sim10^4$ expensive PP simulations, but at a fraction of the computational cost. Our method offers a practical and scalable framework for SBI of cosmological LSS, enabling precise inference across vast volumes and down to small scales.

Figures (8)

• Figure 1: Flowchart of the proposed method, which first applies Neural Quantile Estimation (NQE) to infer cosmological parameters from cosmological maps using $\sim\!10^4$ approximate simulations, followed by calibration of the posterior with $\sim\!10^2$ high-fidelity simulations. An example 2-dim projected dark matter density field is shown in the upper right corner.
• Figure 2: Comparison of power spectrum ratios and cross-correlation coefficients between the Particle-Particle (PP) simulation and two Particle-Mesh (PM) variants: the raw PM simulation and PM with transfer function correction (PM+TF). The PM+TF correction achieves sub-percent accuracy in the power spectrum up to $k \sim 1.5\,h$/Mpc but has no effect on the cross-correlation coefficient. Error bands represent the 25%, 50%, and 75% quantiles across 100 realizations.
• Figure 3: Posterior comparison for $k_{\rm max}\!=\!1.5\,h$/Mpc with CNN as the data compressor, trained on different simulations but tested on PP data. The uncalibrated estimator trained on PM+TF simulations is biased, but calibration effectively removes this bias. All calibrated estimators are unbiased; however, the calibrated PM+TF estimator achieves constraining power comparable to that of an estimator trained on $10^4$ PP simulations, and outperforms the estimator trained on 440 PP simulations.
• Figure 4: Empirical coverage of different estimators with CNN as data compressor. Uncalibrated estimators may exhibit bias due to limited training data (e.g., PP 440) or model misspecification (e.g., PM and PM+TF). By calibrating with 100 PP simulations, we achieve a diagonal coverage curve across all methods, indicating unbiased posterior estimates.
• Figure 5: Comparison of the constraining power across different methods, as measured by the inverse volume of the 1-$\sigma$ credible region of the 3-dim posterior. All estimators have been calibrated to have a diagonal coverage curve (see \ref{['fig:cover']}). The proposed method, which utilizes PM+TF simulations to train the estimator and CNN to extract information, achieves comparable constraining power to estimators trained directly on a significantly larger set of PP simulations.