Inferring Cosmological Parameters with Evidential Physics-Informed Neural Networks

TL;DR

The paper develops an evidential physics-informed neural network (E-PINN) framework, augmented with Gaussian Process supervision, to infer cosmological parameters from Pantheon+ SN Ia and BAO data while yielding calibrated posterior uncertainties. By modeling the luminosity-distance relation as a surrogate with a t-distribution likelihood and enforcing a PDE constraint through the Hubble function H(z;Ω), the approach yields posterior distributions for Ω and H0 under ΛCDM, wCDM, and ΛsCDM. Empirical coverage and model-evidence analyses reveal dataset-dependent tensions between Pantheon+ and BAO when trained separately, while combining data reduces discrepancies and yields more coherent posteriors, albeit with larger uncertainties than traditional χ2-based analyses. The work introduces data-informed priors and GP-guided uncertainty learning, offering a flexible, uncertainty-calibrated alternative for cosmological parameter inference. It also points to future improvements, such as model-independent r_d treatments and incorporating data correlations, to further refine the Hubble tension landscape.

Abstract

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.

Figures (11)

• Figure 1: Diagrams showing both the Pantheon$+$ and BAO datasets together with the fitted Gaussian Process Regression curves. The $1 \sigma$ confidence bands were used to supervise epistemic uncertainty whereas the empirical error bars were used to guide learning of aleatoric uncertainty in our model.
• Figure 2: Plots of empirical coverage probabilities vs their nominal values for a couple of models. The perfectly calibrated uncertainty-aware model would exhibit a straight line joining the origin to $(1,1)$. Models trained on BAO data exhibited less ideal ECP plots compared to those trained on Pantheon data, most likely attributable to the much smaller size of the dataset.
• Figure 3: Evolution of loss function and log $M$ for a couple of models (Top: $\Lambda$CDM trained on Pantheon data; Bottom: $w$CDM model trained on BAO data). All models have been checked to display convergence with a relative tolerance $<10^{-4}$ in both the loss and log $M$ term.
• Figure 4: Left diagram shows residuals between BAO data and Pantheon$+$ data-trained models, normalized by the combined model uncertainties, highlighting systematic differences induced by dataset choice. Right diagram collects all model predictions.
• Figure 5: Joint marginal distributions of $h_0$ and $\Omega_m$ for the three models, each trained separately on the Pantheon$+$ (light brown) and BAO data (dark brown). The 68% and 95% credible contours are shown for each panel. For the $w$CDM and $\Lambda_s$CDM models, the distributions shown were obtained after marginalizing over $w$ and $\Lambda_s$ parameters respectively.