Learning Complex Physical Regimes via Coverage-oriented Uncertainty Quantification: An application to the Critical Heat Flux

Michele Cazzola; Alberto Ghione; Lucia Sargentini; Julien Nespoulous; Riccardo Finotello

cs.LGFeb 25, 2026

Learning Complex Physical Regimes via Coverage-oriented Uncertainty Quantification: An application to the Critical Heat Flux

Authors: Michele Cazzola, Alberto Ghione, Lucia Sargentini, Julien Nespoulous, Riccardo Finotello

Organizations: Université Paris Saclay, Cea, Service Thermo-hydraulique et Mécanique des Fluides (Stmf), Gif-sur-Yvette, F-91191, France · Université Paris Saclay, Cea, Service de Génie Logiciel pour la Simulation (Sgls), Gif-sur-Yvette, F-91191, France

Abstract

A central challenge in scientific machine learning (ML) is the correct representation of physical systems governed by multi-regime behaviours. In these scenarios, standard data analysis techniques often fail to capture the nature of the data, as the system's response varies significantly across the state space due to its stochasticity and the different physical regimes. Uncertainty quantification (UQ) should thus not be viewed merely as a safety assessment, but as a support to the learning task itself, guiding the model to internalise the behaviour of the data. We address this by focusing on the Critical Heat Flux (CHF) benchmark and dataset presented by the OECD/NEA Expert Group on Reactor Systems Multi-Physics. This case study represents a test for scientific ML due to the non-linear dependence of CHF on the inputs and the existence of distinct microscopic physical regimes. These regimes exhibit diverse statistical profiles, a complexity that requires UQ techniques to internalise the data behaviour and ensure reliable predictions. In this work, we conduct a comparative analysis of UQ methodologies to determine their impact on physical representation. We contrast post-hoc methods, specifically conformal prediction, against end-to-end coverage-oriented pipelines, including (Bayesian) heteroscedastic regression and quality-driven losses. These approaches treat uncertainty not as a final metric, but as an active component of the optimisation process, modelling the prediction and its behaviour simultaneously. We show that while post-hoc methods ensure statistical calibration, coverage-oriented learning effectively reshapes the model's representation to match the complex physical regimes. The result is a model that delivers not only high predictive accuracy but also a physically consistent uncertainty estimation that adapts dynamically to the intrinsic variability of the CHF.

Figures & tables

Panel of Figure 3 — Figure 3.1 : Experimental setting of CHF measurements in the NRC dataset.

Figure 4.1 : Beta probability density functions, as functions of the variables

\upalpha

and

m

Panel of Figure 4 — Figure 4.2 : Architecture of the double-head network for HR for predicting the CHF value $\upmu(\mathbf{x};\uptheta)$ and its uncertainty $\upsigma(\mathbf{x};\uptheta)$ .

Panel of Figure 5 — Figure 5.1 : Triangular distribution used for the uncertainty calibration.

	Previous work [ 21 ]	Current work
Depth	$30$	$8$
Width	$64$	$64$
Learning rate	$1\text{\times}{10}^{-3}$	$1\text{\times}{10}^{-3}$
smoothness ( $\upbeta$ )	—	$5$
Weight decay	$1\text{\times}{10}^{-7}$	$1\text{\times}{10}^{-7}$
RMSPE (%)	$11.0$	10.1

Table 6.1 : Hyperparameters and results of the ResNet architecture.

Figure 6.1 : Measured vs predicted CHF of previous and current work.

Variant	Bounds (%)		PICP (%)
Variant	Mean	Std	Test set	Full dataset
Vanilla	31.4	30.5	93.1	95.1
Adaptive	22.8	16.9	96.9	98.6

Table 6.2 : Comparison between vanilla and adaptive CP.

Figure 6.2 : Uncertainty bounds of adaptive CP.

Figure 6.3 : Uncertainty bounds vs outlet quality for adaptive CP.

Variant	INF (%)	CLB (%)	UQF (%)
Vanilla	70.1	74.9	64.4
Adaptive	77.3	74.6	72.5

Table 6.3 : UQ metrics for CP.

Figure 6.4 : Beta distribution (in red) and relative frequencies of the coverage values (in blue).

Variant	Depth	$\bm{\upgamma}$	RMSPE (%)	Bounds (%)		PICP (%)
Variant	Depth	$\bm{\upgamma}$	RMSPE (%)	Mean	Std	Test set	Dataset
Base	12	0.4	10.8	20.8	12.2	97.2	99.1
MTL	11 + 0	0.4	10.6	19.7	12.6	96.7	98.9
TL	8	0.4	10.3	39.6	1.5	99.2	99.4

Table 6.4 : Hyperparameters and results of different variants of HR.

Figure 6.5 : Uncertainty bounds of HR with TL.

Figure 6.6 : Uncertainty bounds vs outlet quality for HR trained from scratch.

Variant	RMSPE (%)	INF (%)	CLB (%)	UQF (%)
Base	10.8	79.2	73.7	74.1
MTL	10.6	80.3	72.5	73.8
TL	10.3	60.4	85.5	69.9

Table 6.5 : UQ metrics for HR.

Variant	Depth	$\uplambda$	$s$	$1-\upgamma$
Base	8	$1\text{\times}{10}^{2}$	$1\text{\times}{10}^{-1}$	$1\text{\times}{10}^{-6}$
MTL	6 + 1	$1\text{\times}{10}^{3}$	$1\text{\times}{10}^{-1}$	$1\text{\times}{10}^{-6}$
TL	8	$1\text{\times}{10}^{3}$	$1\text{\times}{10}^{-1}$	$1\text{\times}{10}^{-6}$

Table 6.6 : Hyperparameters for QD uncertainty estimation.

Variant	RMSPE	Bounds (%)				Coverage (%)
	(%)	Lower		Upper		Test set	Dataset
	(%)	Mean	Std	Mean	Std	Test set	Dataset
Base	10.1	19.2	6.3	28.7	12.8	96.0	97.8
MTL	10.5	28.7	11.8	38.2	21.9	96.2	96.9
TL	10.2	21.6	5.1	35.0	16.5	97.6	98.9

Table 6.7 : Results of QD uncertainty estimation.

Figure 6.7 : Uncertainty bounds of QD uncertainty estimation with MTL.

Figure 6.8 : Uncertainty bounds vs outlet quality for QD uncertainty estimation trained from scratch.

Variant	RMSPE (%)	INF (%)	CLB (%)	UQF (%)
Base	10.1	76.0	77.1	73.8
MTL	10.5	67.4	81.2	70.7
TL	10.2	71.7	78.8	72.9

Table 6.8 : UQ metrics for QD techniques.

Variant	Depth	$\bm{\upbeta}_{\text{KL }}$	RMSPE (%)	Bounds (%)		Coverage (%)
Variant	Depth	$\bm{\upbeta}_{\text{KL }}$	RMSPE (%)	Mean	Std	Test set	Dataset
Base	12	1	11.1	17.5	9.4	95.6	97.9
MTL	11 + 0	$1\text{\times}{10}^{-3}$	11.3	18.7	9.7	95.4	97.8

Table 6.9 : Hyperparameters and results for BHR.

Variant	Test set (%)			Dataset (%)
Variant	Prediction	Model	Aleatoric	Prediction	Model	Aleatoric
Base	8.9	2.2	8.6	8.9	2.2	8.6
MTL	9.6	2.4	9.2	9.7	2.4	9.2

Table 6.10 : Contribution of uncertainty components in BHR.

Figure 6.9 : Uncertainty bounds of BHR, with single-head architecture.

Figure 6.10 : Uncertainty bounds vs outlet quality for BHR with single-head architecture.

Variant	RMSPE (%)	INF (%)	CLB (%)	UQF (%)
Base	11.1	82.5	69.4	72.5
MTL	11.3	81.3	67.4	70.6

Table 6.11 : UQ metrics of BHR.

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