CLiENT: A new tool for emulating cosmological likelihoods using deep neural networks

TL;DR

The paper introduces CLiENT, a neural-network framework that directly emulates cosmological likelihoods to yield an auto-differentiable surrogate, avoiding expensive observable calculations. It employs a relative $\chi^2$-based MSRE loss with a Wilson–Hilferty regularization, tempered training densities, and MCMC-driven data generation to focus learning where it matters. Across synthetic Gaussian, Planck $\Lambda$CDM, and sterile-neutrino extensions, CLiENT achieves credible intervals within $\sim 0.1\sigma$ of true posteriors and $\Delta\chi^2$ around $0.3$–$0.5$ near the optimum using under $2 \times 10^4$ evaluations, comparable to observable-emulator pipelines but with full differentiability. The framework is designed to be general, compatible with MontePython or Cobaya, and is released as open-source for broader use in cosmology and beyond.

Abstract

Cosmological emulation of observables such as the Cosmic Microwave Background (CMB) spectra and matter power spectra have become increasingly common in recent years because of the potential for saving computation time in connection with cosmological parameter inference or model comparison. In this paper we present CLiENT (Cosmological Likelihood Emulator using Neural networks with TensorFlow), a new method which circumvents the computation of observables in favour of directly emulating the likelihood function for a data set given a model parameter vector. We find that the method is competitive with observable emulators in terms of the required number of function evaluations, but has the distinct advantage of producing a surrogate likelihood which is completely auto-differentiable. Using less than $2 \times 10^4$ function evaluations CLiENT typically achieves credible intervals within better than $0.1 σ$ of those obtained using the true likelihood and single-point emulator precision better than $Δχ^2 \sim 0.5$ across relevant regions in parameter space.

Figures (10)

• Figure 1: Surrogate error as a function of exact $\chi^2$ value for networks trained on the 27-dimensional analytic Gaussian likelihood. The left panel shows the region closest to the best-fit while the right panel shows the error deep in the tails. A lower value of the parameter $\kappa$ in the loss function clearly reduce the scatter of the emulated likelihood around the best-fit at the expense of increasing the scatter further away.
• Figure 2: Convergence metrics for the synthetic Gaussian likelihood as a function of iterations. Top panel: RMS of the KL divergence as defined in equation \ref{['eq:divKLrms']} and the Gelman–Rubin statistic of samples between iterations. Middle panel: Maximum of the 68% and 95% Credible Metrics defined in equation \ref{['eq:crediblemetric']}. Bottom panel:$\Delta \chi^2$ as evaluated by the network at the maximum likelihood point. Note that the plot is linear from $-1 < \Delta \chi^2 < 1$ and logarithmic otherwise.
• Figure 3: Triangle plot for the synthetic 27-dimensional Gaussian likelihood after the full 10 iterations. 19334 evaluations of the true likelihood with 14076 accepted points beyond the initial latin hypercube of 5000 points. Only the first 6 parameters are shown in the figure. The points show the final point cloud on which the iteration 10 network is trained.
• Figure 4: Convergence metrics for the synthetic likelihood that has two non-Gaussian parameters as a function of iterations. Top panel: RMS of the KL divergence as defined in equation \ref{['eq:divKLrms']} and the Gelman–Rubin statistic of samples between iterations. Middle panel: Maximum of the 68% and 95% Credible Metrics defined in equation \ref{['eq:crediblemetric']}. Bottom panel:$\Delta \chi^2$ as evaluated by the network at the maximum likelihood point. Note that the plot is linear from $-1 < \Delta \chi^2 < 1$ and logarithmic otherwise.
• Figure 5: Triangle plot for the synthetic 29-dimensional non-Gaussian likelihood after the full 10 iterations. 17094 evaluations of the true likelihood with 11806 accepted points beyond the initial latin hypercube of 5000 points. Only parameters 1 , 2 , 3 , 4 , 28 , and 29 are shown in the figure. The points show the final point cloud on which the iteration 10 network is trained.