Information fusion and machine learning for sensitivity analysis using physics knowledge and experimental data

Berkcan Kapusuzoglu; Sankaran Mahadevan

cs.CEAug 18, 2026

Information fusion and machine learning for sensitivity analysis using physics knowledge and experimental data

Authors: Berkcan Kapusuzoglu, Sankaran Mahadevan

Organizations: Department of Civil and Environmental Engineering Vanderbilt University, Nashville, TN 37235, USA

Abstract

When computational models (either physics-based or data-driven) are used for the sensitivity analysis of engineering systems, the sensitivity estimate is affected by the accuracy and uncertainty of the model. This paper considers global sensitivity analysis (GSA) for situations where both a physics-based model and experimental observations are available, and investigates physics-informed machine learning strategies to effectively combine the two sources of information in order to maximize the accuracy of the sensitivity estimate. Two representative machine learning (ML) techniques are considered, namely, deep neural networks (DNN) and Gaussian process (GP) modeling, and two strategies for incorporating physics knowledge within these techniques are investigated, namely: (i) incorporating loss functions in the ML models to enforce physics constraints, and (ii) pre-training and updating the ML model using simulation and experimental data respectively. Four different models are built for each type (DNN and GP), and the uncertainties in these models are included in the Sobol indices computation. The DNN-based models, with many degrees of freedom in terms of model parameters and training options, are found to result in smaller bounds on the sensitivity estimates when compared to the GP-based models. The proposed methods are illustrated for additive manufacturing and lake temperature modeling examples.

Figures & tables

Figure 1: A deep neural network with two hidden layers.

Figure 2: PIML strategies: (a) incorporating physics-based loss functions in the ML models to enforce physics constraints, (b) pre-training an ML model with physics model input-output (

\mathbf{X_{\mathrm{phy}}}

\mathbf{Y_{\mathrm{phy}}}

) and updating it with experimental training data (

\mathbf{X_{\mathrm{obs}}}

\mathbf{Y_{\mathrm{obs}}}

), (c) the trained ML model prediction

(\mathbf{\hat{Y}})

, (d) updated ML model prediction

(\mathbf{\hat{Y}})

Figure 3: Cross-sectional geometry of an FFF specimen printed with nozzle temperature 240 ∘ C and speed 42 mm/s.

Models	Training [in minutes]	Sobol’ indices calculation [in minutes]
1. $\mathbf{\rm GP}$	1	3
2. $\mathbf{\rm GP^{\mathcal{L}_{\rm phy}}}$	2	5
3. $\mathbf{\rm GP^{\rm upd}}$	3	7
4. $\mathbf{\rm GP^{\rm upd,\mathcal{L}_{\rm phy}}}$	3	8
5. $\mathbf{\rm DNN}$	.3	1
6. $\mathbf{\rm DNN^{\mathcal{L}_{\rm phy}}}$	.7	2

Table 1: Computational effort of eight models for training and estimation of first-order and total effect Sobol’ indices using 5000 MC samples with

n=39

number of observations.

Model	$n=5$	$n=10$	$n=15$	$n=20$	$n=30$
1. $\mathbf{\rm GP}$	0.020( $\pm$ 0.004)	0.026( $\pm$ 0.005)	0.021( $\pm$ 0.001)	0.021( $\pm$ 0.002)	0.020( $\pm$ 0.001)
2. $\mathbf{\rm GP^{\mathcal{L}_{\rm phy}}}$	0.019( $\pm$ 0.004)	0.026( $\pm$ 0.004)	0.024( $\pm$ 0.004)	0.021( $\pm$ 0.002)	0.019( $\pm$ 0.001)
3. $\mathbf{\rm GP^{\rm upd}}$	0.021( $\pm$ 0.003)	0.028( $\pm$ 0.003)	0.020( $\pm$ 0.001)	0.020( $\pm$ 0.001)	0.019( $\pm$ 0.001)
4. $\mathbf{\rm GP^{\rm upd,\mathcal{L}_{\rm phy}}}$	0.021( $\pm$ 0.005)	0.023( $\pm$ 0.003)	0.022( $\pm$ 0.003)	0.020( $\pm$ 0.002)	0.019( $\pm$ 0.001)

Table 2: Effect of different amounts of training data on the RMSE of the GP models.

Model	$n=5$	$n=10$	$n=15$	$n=20$	$n=30$
5. $\mathbf{\rm DNN}$	0.027( $\pm$ 0.007)	0.017( $\pm$ 0.009)	0.019( $\pm$ 0.006)	0.013( $\pm$ 0.003)	0.013( $\pm$ 0.003)
6. $\mathbf{\rm DNN^{\mathcal{L}_{\rm phy}}}$	0.017( $\pm$ 0.007)	0.015( $\pm$ 0.008)	0.014( $\pm$ 0.005)	0.014( $\pm$ 0.003)	0.013( $\pm$ 0.002)
7. $\mathbf{\rm DNN^{\rm upd}}$	0.014( $\pm$ 0.002)	0.009( $\pm$ 0.002)	0.014( $\pm$ 0.001)	0.013( $\pm$ 0.001)	0.013( $\pm$ 0.001)
8. $\mathbf{\rm DNN^{\rm upd,\mathcal{L}_{\rm phy}}}$	0.014( $\pm$ 0.002)	0.009( $\pm$ 0.001)	0.014( $\pm$ 0.001)	0.014( $\pm$ 0.001)	0.013( $\pm$ 0.001)

Table 3: Effect of different amounts of training data on the RMSE of the DNN models.

Models
	1. $\mathbf{\rm GP}$	2. $\mathbf{\rm GP^{\mathcal{L}_{\rm phy}}}$	3. $\mathbf{\rm GP^{\rm upd}}$	4. $\mathbf{\rm GP^{\rm upd,\mathcal{L}_{\rm phy}}}$	5. $\mathbf{\rm DNN}$	6. $\mathbf{\rm DNN^{\mathcal{L}_{\rm phy}}}$	7. $\mathbf{\rm DNN^{\rm upd}}$	8. $\mathbf{\rm DNN^{\rm upd,\mathcal{L}_{\rm phy}}}$
$n=39$	0.021( $\pm$ 0.002)	0.020( $\pm$ 0.002)	0.020( $\pm$ 0.001)	0.019( $\pm$ 0.002)	0.015( $\pm$ 0.004)	0.013( $\pm$ 0.004)	0.013( $\pm$ 0.003)	0.013( $\pm$ 0.003)

Table 4: Tenfold cross-validation average RMSE results of GP and DNN models

Figure 4: RMSE values of DNN models with varying dropout rates.

Figure 5: First-order sensitivity index estimators for (a) nozzle temperature, and (b) nozzle speed, using two different pre-training and updating approaches for GP.

Figure 6: Total effect sensitivity index estimates for (a) nozzle temperature, and (b) nozzle speed, using two different pre-training and updating approaches for GP.

Figure 7: First-order sensitivity index estimates for (a) nozzle temperature, and (b) nozzle speed, using the GP models.

Figure 8: Total effect sensitivity index estimates for (a) nozzle temperature, and (b) nozzle speed, using the GP models.

Models	Lower 95% confidence limit					Upper 95% confidence limit
	n=5	n=10	n=20	n=30	n=39	n=5	n=10	n=20	n=30	n=39
1. $\mathbf{\rm GP}$	0.00	0.08	0.13	0.20	0.19	0.37	0.25	0.34	0.35	0.33
2. $\mathbf{\rm GP^{\mathcal{L}_{\rm phy}}}$	0.00	0.00	0.05	0.16	0.18	0.55	0.33	0.43	0.34	0.33
3. $\mathbf{\rm GP^{\rm upd}}$	0.03	0.14	0.10	0.16	0.17	0.77	0.33	0.34	0.31	0.29
4. $\mathbf{\rm GP^{\rm upd,\mathcal{L}_{\rm phy}}}$	0.06	0.11	0.14	0.18	0.20	0.71	0.38	0.34	0.31	0.32

Table 5: First-order sensitivity estimate prediction bounds of nozzle temperature for different amounts of experimental training data, using the GP models.

Figure 9: First-order sensitivity index estimators for (a) nozzle temperature, and (b) nozzle speed, using DNN models with MC dropout.

Figure 10: Total effect sensitivity index estimators for (a) nozzle temperature, and (b) nozzle speed, using DNN models with MC dropout.

Models	Lower 95% confidence limit					Upper 95% confidence limit
	n=5	n=10	n=20	n=30	n=39	n=5	n=10	n=20	n=30	n=39
5. $\mathbf{\rm DNN}$	0.45	0.22	0.17	0.23	0.30	0.50	0.23	0.18	0.25	0.32
6. $\mathbf{\rm DNN^{\mathcal{L}_{\rm phy}}}$	0.52	0.18	0.15	0.17	0.22	0.59	0.19	0.17	0.18	0.24
7. $\mathbf{\rm DNN^{\rm upd}}$	0.78	0.30	0.12	0.22	0.25	0.81	0.32	0.15	0.24	0.27
8. $\mathbf{\rm DNN^{\rm upd,\mathcal{L}_{\rm phy}}}$	0.78	0.29	0.10	0.18	0.22	0.82	0.33	0.12	0.21	0.25

Table 6: First-order sensitivity estimate prediction bounds of nozzle temperature for different amounts of experimental training data, using the DNN models.

Models	Training [in minutes]	Sobol’ indices calculation [in minutes]
1. $\mathbf{\rm GP}$	37.5	40.1
2. $\mathbf{\rm GP^{\mathcal{L}_{\rm phy}}}$	37.9	41.2
3. $\mathbf{\rm GP^{\rm upd}}$	36.8	39.7
4. $\mathbf{\rm GP^{\rm upd,\mathcal{L}_{\rm phy}}}$	36.9	39.8
5. $\mathbf{\rm DNN}$	1.3	2.8
6. $\mathbf{\rm DNN^{\mathcal{L}_{\rm phy}}}$	2.7	5.7

Table 7: Computational effort of eight models for training and estimation of first-order and total effect Sobol’ indices using 5000 MC samples with

n=1000

number of temperature observations.

Figure 11: First-order sensitivity index estimators for the eleven variables of the lake temperature modeling using the GP models.

Figure 12: Total effect sensitivity index estimates for the eleven variables of the lake temperature modeling using the GP models.

Figure 13: First-order sensitivity index estimators for the eleven variables of the lake temperature modeling using the DNN models.

Figure 14: Total effect sensitivity index estimates for the eleven variables of the lake temperature modeling using the DNN models.

Models	Lower 95% confidence limit					Upper 95% confidence limit
	n=100	n=250	n=500	n=750	n=1000	n=100	n=250	n=500	n=750	n=1000
1. $\mathbf{\rm GP}$	0.00	0.27	0.67	0.61	0.24	0.38	0.71	0.82	0.77	0.53
2. $\mathbf{\rm GP^{\mathcal{L}_{\rm phy}}}$	0.01	0.51	0.60	0.63	0.32	0.36	0.76	0.77	0.77	0.55
3. $\mathbf{\rm GP^{\rm upd}}$	0.54	0.48	0.70	0.65	0.49	0.82	0.78	0.82	0.80	0.68
4. $\mathbf{\rm GP^{\rm upd,\mathcal{L}_{\rm phy}}}$	0.46	0.40	0.59	0.55	0.42	0.69	0.66	0.69	0.68	0.57
5. $\mathbf{\rm DNN}$	0.06	0.14	0.08	0.45	0.16	0.08	0.16	0.12	0.49	0.19

Table 8: First-order sensitivity estimate prediction bounds of

X_{1}

(day of year) for different amounts of experimental training data, using the GP and DNN models.

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Abstract

Sensitivity Analysis

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