# Bayesian inference and non-linear extensions of the CIRCE method for quantifying the uncertainty of closure relationships integrated into thermal-hydraulic system codes

## TL;DR

This paper extends the CIRCE method for uncertainty quantification in thermal-hydraulic codes by incorporating Bayesian inference and non-linear modeling using Gaussian process emulators, enabling more accurate and efficient analysis.

## Contribution

It introduces a Bayesian framework and Gaussian process emulators to improve the CIRCE method's handling of non-linearities and statistical uncertainties.

## Key findings

- Bayesian approach provides posterior distributions of parameters.
- GP emulators significantly reduce computational time.
- Method applied to condensation closure relationships with successful results.

## Abstract

Uncertainty Quantification of closure relationships integrated into thermal-hydraulic system codes is a critical prerequisite in applying the Best-Estimate Plus Uncertainty (BEPU) methodology for nuclear safety and licensing processes.The purpose of the CIRCE method is to estimate the (log)-Gaussian probability distribution of a multiplicative factor applied to a reference closure relationship in order to assess its uncertainty. Even though this method has been implemented with success in numerous physical scenarios, it can still suffer from substantial limitations such as the linearity assumption and the difficulty of properly taking into account the inherent statistical uncertainty. In the paper, we will extend the CIRCE method in two aspects. On the one hand, we adopt the Bayesian setting putting prior probability distributions on the parameters of the (log)-Gaussian distribution. The posterior distribution of the parameters is then computed with respect to an experimental database by means of Markov Chain Monte Carlo (MCMC) algorithms. On the other hand, we tackle the more general setting where the simulations do not move linearly against the multiplicative factor(s). MCMC algorithms then become time-prohibitive when the thermal-hydraulic simulations exceed a few minutes. This handicap is overcome by using Gaussian process (GP) emulators which can yield both reliable and fast predictions of the simulations. The GP-based MCMC algorithms will be applied to quantify the uncertainty of two condensation closure relationships at a safety injection with respect to a database of experimental tests. The thermal-hydraulic simulations will be run with the CATHARE 2 computer code.