Multifidelity Surrogate Modeling of Depressurized Loss of Forced Cooling in High-temperature Gas Reactors

Meredith Eaheart; Majdi I. Radaideh

cs.LGMar 14, 2026

Multifidelity Surrogate Modeling of Depressurized Loss of Forced Cooling in High-temperature Gas Reactors

Authors: Meredith Eaheart, Majdi I. Radaideh

Organizations: Department of Nuclear Engineering and Radiological Science, University of Michigan, Ann Arbor, Michigan 48109 · Department of Computer Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109

Abstract

High-fidelity computational fluid dynamics (CFD) simulations are widely used to analyze nuclear reactor transients, but are computationally expensive when exploring large parameter spaces. Multifidelity surrogate models offer an approach to reduce cost by combining information from simulations of varying resolution. In this work, several multifidelity machine learning methods were evaluated for predicting the time to onset of natural circulation (ONC) and the temperature after ONC for a high-temperature gas reactor (HTGR) depressurized loss of forced cooling transient. A CFD model was developed in Ansys Fluent to generate 1000 simulation samples at each fidelity level, with low and medium-fidelity datasets produced by systematically coarsening the high-fidelity mesh. Multiple surrogate approaches were investigated, including multifidelity Gaussian processes and several neural network architectures, and validated on analytical benchmark functions before application to the ONC dataset. The results show that performance depends strongly on the informativeness of the input variables and the relationship between fidelity levels. Models trained using dominant inputs identified through prior sensitivity analysis consistently outperformed models trained on the full input set. The low- and high-fidelity pairing produced stronger performance than configurations involving medium-fidelity data, and two-fidelity configurations generally matched or exceeded three-fidelity counterparts at equivalent computational cost. Among the methods evaluated, multifidelity GP provided the most robust performance across input configurations, achieving excellent metrics for both time to ONC and temperature after ONC, while neural network approaches achieved comparable accuracy with substantially lower training times.

Figures & tables

Figure 1: Computational domains with boundary conditions of the Ansys Fluent simulation

Parameter	Lower Bound	Upper Bound
Initial temperature of heated section (K)	873.15	1498.2
Heat transfer coefficient of unheated section (W/m 2 K)	0.1	10
Air viscosity (kg/m·s) [ 19 , 58 ]	1.85 $\times$ 10 -5	5.16 $\times$ 10 -5
Air thermal conductivity (W/m·K) [ 19 , 58 ]	0.02551	0.08452
Helium viscosity (kg/m·s) [ 24 ]	1.98 $\times$ 10 -5	6.15 $\times$ 10 -5
Helium thermal conductivity (W/m·K) [ 24 ]	0.15525	0.47859

Table 1: Parameter bounds used in the sensitivity and uncertainty analysis. All parameters were sampled using a uniform distribution [ 8 ] .

Figure 2: Global and local uncertainty quantification reproduced from a previous work [ 8 ]

Figure 3: Comparison of (left) low-fidelity (17.5k elements), (center) medium-fidelity (35k elements), and (right) high-fidelity (70k elements) mesh resolutions defining the fidelity hierarchy.

Target Output	Input Set	Included Parameters
Time to ONC	All Inputs	All 8 input parameters
	Dominant	Heated Section Temperature
	Non-Dominant	All parameters excluding Heated Section Temperature
Temperature after ONC	All Inputs	All 8 input parameters
	Dominant	Heated Section Temperature and Unheated Section Heat Transfer Coefficient
	Non-Dominant	All parameters excluding Heated Section Temperature and Unheated Section Heat Transfer Coefficient

Table 2: Definition of input parameter sets for the dominant and non-dominant sensitivity studies. The specific parameters included in the ’Dominant’ and ’Non-Dominant’ sets vary depending on the target output quantity, based on the sensitivity analysis results.

Hyperparameter	Search Values
Number of Layers	[2, 3, 4]
Nodes per Layer	[16, 32, 64, 128]
Learning Rate	[1e-4, 5e-4, 1e-3]
Number of Epochs	500

Table 3: Hyperparameter tuning grid for Multi-fidelity Neural Networks.

Fidelity Setup	$n_{\text{LF}}$	$n_{\text{MF}}$	$n_{\text{HF}}$	Total Cost
Total Cost = 300
2F (LF + HF)	200	0	25	300
2F (LF + MF)	200	50	0	300
2F (MF + HF)	0	100	25	300
3F (LF + MF + HF)	150	50	12	298
Total Cost = 600

Table 4: Cost-matched simulation budgets for two-fidelity (2F) and three-fidelity (3F) experiments. Costs are normalized relative to low-fidelity simulations (LF = 1, MF = 2, HF = 4).

Fidelity	Method	Hidden Layers	Learning Rate	Mean RMSE
2	GPmimic	(128,128,128,128)	0.001	38.5905
2	MF-NN-Delta	(64, 64, 64, 64)	0.001	8.4437
2	MFNN-Intermediate	(128, 128, 128, 128)	0.001	9.8412
2	MFNN-TwoStep	(64, 64, 64, 64)	0.001	120.2910
2	MFNN-ThreeStep	(128,128,128,128)	0.001	93.8388
2	MFNN-Flag	(128,128,128,128)	0.001	10.0238

Table 5: Best hyperparameter configurations across two- and three-fidelity methods. All models were trained for 500 epochs. The RMSE is averaged across all benchmarks for each method.

Fidelity	Method	$\alpha$	$w_{h}$	$w_{m}$	$w_{l}$	$\lambda$	Mean RMSE
2	GPmimic	0.05	–	–	–	1e-05	38.4414
2	MFNN-Intermediate	0.05	–	–	–	0.1	8.3376
3	GPmimic3f	–	0.5	0.2	0.3	0.0001	160.2200
3	MFNN-Intermediate3f	–	0.7	0.2	0.1	0.001	75.4586

Table 6: Optimal fidelity-weighting and penalty parameters selected on the benchmark problems for two- and three-fidelity methods. For three-fidelity models, weights satisfy

w_{l}+w_{m}+w_{h}=1

Benchmark	GPmimic	MF-GP	MF-NN-Delta	MFNN-Flag	MFNN-Interm.
Forrester (1D)	0.0411	0.0000	0.0370	0.1320	0.0827
Booth (2D)	32.7409	0.0067	13.1901	25.3527	25.9151
Branin (2D)	14.7299	0.0154	8.9195	8.3862	9.9333
Park91A (4D)	0.1610	0.0116	0.1629	0.2407	0.1358
Hartmann6 (6D)	0.1529	0.1028	0.1125	0.1512	0.1550
Borehole (8D)	3.2645	0.0820	5.3271	4.6495	4.9388

Table 7: RMSE comparison across benchmarks for two-fidelity methods, ordered by input dimension.

Figure 4: Parity plot comparing predicted and true high-fidelity outputs (top) and residual distribution (bottom) for the Booth (2D) benchmark. Labels: y_pred = predicted value, y_true=ground truth.

Figure 5: Parity plot comparing predicted and true high-fidelity outputs (top) and residual distribution (bottom) for the Borehole (8D) benchmark. Labels: y_pred = predicted value, y_true=ground truth

Benchmark	GPmimic	MFNN-Intermediate	MFNN-Flag
Forrester 1D	0.0417	0.0340	0.0606
Rastrigin 2D	0.1509	0.0718	0.0855
Rastrigin 5D	0.5669	0.5302	0.7285
Rosenbrock 2D	10.6742	9.6140	24.7987
Rosenbrock 5D	281.1450	317.0519	410.2674

Table 8: RMSE comparison across benchmarks for three-fidelity methods.

Figure 6: Parity plot comparing predicted and true high-fidelity outputs (top) and residual distribution (bottom) for the Rosenbrock (2D) benchmark on three-fidelity methods. Labels: y_pred = predicted value, y_true=ground truth

Figure 7: Parity plot comparing predicted and true high-fidelity outputs (top) and residual distribution (bottom) for the Rastrigin (5D) benchmark on three-fidelity methods.

Fidelity	Method	Layers	Nodes per Layer	Learning Rate	RMSE (seconds)
2	GPmimic	3	128	0.0010	386.8805
2	MF-NN-Delta	4	128	0.0010	194.1512
2	MFNN-Flag	4	128	0.0010	194.7106
2	MFNN-Intermediate	4	128	0.0010	199.8040
2	MFNN-ThreeStep	2	128	0.0010	2977.1743
2	MFNN-TwoStep	3	32	0.0010	3065.7572

Table 9: Best hyperparameter configurations on the real dataset (All

\rightarrow

Time to ONC) across two- and three-fidelity methods. All methods were trained for 500 epochs.

		Time to ONC (Seconds)			Temp. after ONC (K)
Inputs	Method	RMSE	$R^{2}$	Time (s)	RMSE	$R^{2}$	Time (s)
All Inputs
	GPmimic (2F)	312.59	0.8100	5.57	110.96	0.6103	5.75
	GPmimic3f (3F)	286.92	0.8399	86.94	95.88	0.7090	73.92
	MF-GP (2F)	102.35	0.9796	392.29	1.08	1.0000	227.39
	MF-NN-Delta (2F)	139.13	0.9624	8.60	25.14	0.9800	8.94

Table 10: Performance comparison for the LF+HF configuration at a fixed total computational budget of 1800. The Time (s) column refers to the computatiuonal time needed by each method

		Time to ONC			Temp. After ONC
Budget	Method	RMSE	$R^{2}$	Time (s)	RMSE	$R^{2}$	Time (s)
Total Budget = 300
	GPmimic	149.23	0.9567	22.42	95.30	0.7125	4.10
	MF-GP	115.79	0.9739	7.61	6.24	0.9988	2.60
	MF-NN-Delta	118.81	0.9726	40.56	21.47	0.9854	6.44
	MFNN-Flag	129.59	0.9673	23.81	38.42	0.9533	3.68

Table 11: Scaling behavior for LF+HF (2F) configuration across increasing total budgets for dominant inputs .

		Time to ONC			Temp After ONC
Budget	Method	RMSE	$R^{2}$	Time (s)	RMSE	$R^{2}$	Time (s)
Total Budget = 300
	GPmimic	670.01	0.1272	3.90	200.77	-0.2757	3.91
	MF-GP	137.84	0.9631	6.15	8.95	0.9975	5.61
	MF-NN-Delta	301.93	0.8228	6.43	80.15	0.7967	6.54
	MFNN-Flag	177.22	0.9389	3.68	86.25	0.7645	3.44

Table 12: Scaling behavior for LF+HF (2F) configuration across increasing total budgets for all inputs .

Figure 8: MF-GP runtime as a function of total computational budget for different input selections and Time to ONC output. Results are shown for all inputs, dominant inputs, and non-dominant inputs.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

		Time to ONC			Temp after ONC
Inputs	Method	RMSE	$R^{2}$	Time (s)	RMSE	$R^{2}$	Time (s)
All Inputs
	GPmimic (2F)	555.05	0.3888	4.52	103.57	0.6490	5.90
	MF-GP (2F)	527.92	0.4471	495.54	0.89	1.0000	295.25
	MF-NN-Delta (2F)	684.92	0.0693	6.76	14.09	0.9935	9.34
	MFNN-Flag (2F)	513.92	0.4760	4.68	36.75	0.9558	5.90

Appendix

Table 13: Performance comparison for the LF+MF (2F) configuration at a total computational budget of 1800.

		Time to ONC			Temp after ONC
Inputs	Method	RMSE	$R^{2}$	Time (s)	RMSE	$R^{2}$	Time (s)
All Inputs
	GPmimic (2F)	401.26	0.6869	5.03	84.21	0.7756	5.00
	MF-GP (2F)	416.79	0.6622	80.26	0.97	1.0000	69.38
	MF-NN-Delta (2F)	135.73	0.9642	8.01	23.39	0.9827	7.40
	MFNN-Flag (2F)	189.38	0.9303	4.52	39.91	0.9496	4.79

Appendix

Table 14: Performance comparison for the MF+HF (2F) configuration at a total computational budget of 1800.

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