Adaptive surrogate modeling for high-dimensional spatio-temporal output

Berkcan Kapusuzoglu; Shunsaku Matsumoto; Yoshitomo Miyagi; Daigo Watanabe; Sankaran Mahadevan

cs.CEAug 18, 2026

Adaptive surrogate modeling for high-dimensional spatio-temporal output

Authors: Berkcan Kapusuzoglu, Shunsaku Matsumoto, Yoshitomo Miyagi, Daigo Watanabe, Sankaran Mahadevan

Organizations: Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN 37235, USA · Strength Research Department, Research and Innovation Center, Mitsubishi Heavy Industries, Ltd., Nagasaki, 851-0392, Japan · Strength Research Department, Research and Innovation Center, Mitsubishi Heavy Industries, Ltd., Takasago, 676-8686, Japan

Abstract

This paper develops an adaptive surrogate modeling method for problems with very high-dimensional spatio-temporal outputs. The analysis of spatio-temporal multi-physics systems is computationally expensive and consists of a large number of inputs and outputs. Surrogate models are often constructed to replace the physics-based model to achieve computational efficiency in analyses such as uncertainty quantification and optimization that require many function calls. In order to address the challenge introduced by the high dimensionality of spatio-temporal output, a dimension reduction method is first employed to map the high-dimensional output to a low-dimensional latent space. This is followed by the construction of the surrogate model in the low-dimensional space. The prediction error in the original space, which includes both the reconstruction error and surrogate model error, is evaluated using different error metrics. Based on the prediction accuracy of the surrogate model, new training points are identified for adaptive improvement of the surrogate model. We present a novel adaptive sampling technique that combines exploration and exploitation to improve the surrogate model accuracy with the fewest possible runs of the expensive physics-based model. Thermo-mechanical analysis of a gas turbine engine blade is used to analyze the effectiveness of the proposed method.

Figures & tables

Figure 1: Schematic describing the proposed procedure for surrogate model construction with high-dimensional field data using rSVD

Figure 2: Evolution of the NRMSE from the initial surrogate model (20 samples) to the final surrogate model (45 samples) using different adaptive sampling techniques, for the seven benchmark functions in Appendix A

Figure 3: Adaptively selected training point locations for the Branin’s function and the final surrogate model accuracy in terms of NRMSE are plotted in contours (Black dots: initial 20 samples, Red dots: additional 25 samples)

Figure 4: Adaptively selected training point locations for the Schwefel function and the final surrogate model accuracy in terms of NRMSE are plotted in contours (Black dots: initial 20 samples, Red dots: additional 25 samples)

Input	Values
TBC thickness	$\frac{2}{3}\times$ nominal to $\frac{4}{3}\times$ nominal
T1T rate	80 $\times$ nominal to 100 $\times$ nominal
Transient creep rate	-3 $\sigma$ to 3 $\sigma$
Constant creep rate	-3 $\sigma$ to 3 $\sigma$
Output	Nomenclature
Creep equivalent strain	CEEQ

Table 1: Physics model input and output parameters

Figure 5: FEM model of engine rotor blade

	Model
Fold #	Extra-Trees	LightGBM	XGBoost	CatBoost	DNN	GP
1	0.1722	0.2029	0.1823	0.1780	0.2102	0.2205
2	0.1885	0.2146	0.2113	0.1843	0.2243	0.2315
3	0.1329	0.1829	0.1732	0.1362	0.1856	0.1795
4	0.1090	0.1257	0.1219	0.1155	0.1351	0.1583
5	0.1123	0.1286	0.1199	0.1136	0.1209	0.1348

Table 2: 7-fold cross-validation results in terms of RMSE

Figure 6: Predicted vs. observed values of important features for FEM run 23

Figure 7: Actual Feature 1 values vs. TBC thickness and T1T rate

Batch id	$\beta_{1}$	$\beta_{2}$	$\alpha$	$\gamma$
First batch: FEM 47-56	0.30	0.03	1	1
Second batch: FEM 57-60	0.56	0.23	1	1
Third batch: FEM 61-62	1	1	1	1
Fourth batch: FEM 63-66	0.53	0.04	1	1

Table 3: Optimal hyperparameter values of each batch

Figure 8: Initial FEM training data and adaptively selected additional samples

Figure 9: Final surrogate model prediction accuracy in the original space for leave-one-out FEM runs

Figure 10: Final surrogate model prediction compared against FEM run 36 at node 5057

Training data	Avg. PNMAE	Avg. PNrMAE	Avg. RMSE
FEM 1-46	27.9	19.3	0.00254
FEM 1-56	26.5	17.6	0.00222
FEM 1-60	26.3	17.3	0.00219
FEM 1-62	24.9	16.1	0.00209
FEM 1-66	24.6	15.8	0.00193

Table 4: Average LOOCV results of the first 46 FEM runs in terms of PNMAE, PNrMAE, and RMSE

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Panel of Table 5 — Table 5: Test problems

Surrogate Modeling

Surrogate modeling builds faster or more interpretable approximations of expensive simulations and black-box models.

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In many engineering applications, a single high-fidelity model produces multiple quantities of interest (QoIs) under the same input parameters, e.g. finite element models of complex physical systems. To alleviate the high computational cost of direct model evaluations, surrogate models are widely used to construct efficient approximations of model responses. Naturally, the accuracy of surrogates strongly depends on the quality of the experimental design (ED). However, a single ED may not provide an adequate representation for all outputs simultaneously, especially when different outputs exhibit varying sensitivities to the input variables. A straightforward solution is to perform separate sampling for each output, but this results in increased sampling complexity and computational cost. From a statistical perspective, such an approach also ignores potential correlations among all outputs and may compromise data consistency. To address this issue, an adaptive sequential sampling method for constructing polynomial chaos expansion surrogate models is generalized for vector valued QoIs. The method sequentially selects new samples from a candidate pool based on their local contribution to the output variance, while balancing distance-based exploration of the input space and exploitation of aggregated variance information across all outputs. Its performance is compared with non-sequential Latin Hypercube Sampling through several numerical examples from engineering problems. Numerical results demonstrate that the proposed strategy improves both surrogate accuracy and stability, and provides a more reliable estimation of second-order statistics.

Abstract

Surrogate Modeling

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