Deep Learning-Based Metamodeling of Nonlinear Stochastic Dynamic Systems under Parametric and Predictive Uncertainty

Haimiti Atila; Seymour M. J. Spence

cs.LGMar 12, 2026

Deep Learning-Based Metamodeling of Nonlinear Stochastic Dynamic Systems under Parametric and Predictive Uncertainty

Authors: Haimiti Atila, Seymour M. J. Spence

Organizations: Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI 48109, USA

Abstract

Modeling high-dimensional, nonlinear dynamic structural systems under natural hazards presents formidable computational challenges, especially when simultaneously accounting for uncertainties in external loads and structural parameters. Studies have successfully incorporated uncertainties related to external loads from natural hazards, but few have simultaneously addressed loading and parameter uncertainties within structural systems while accounting for prediction uncertainty of neural networks. To address these gaps, three metamodeling frameworks were formulated, each coupling a feature-extraction module implemented through a multi-layer perceptron (MLP), a message-passing neural network (MPNN), or an autoencoder (AE) with a long short-term memory (LSTM) network using Monte Carlo dropout and a negative log-likelihood loss. The resulting architectures (MLP-LSTM, MPNN-LSTM, and AE-LSTM) were validated on two case studies: a multi-degree-of-freedom Bouc-Wen system and a 37-story fiber-discretized nonlinear steel moment-resisting frame, both subjected to stochastic seismic excitation and structural parameter uncertainty. All three approaches achieved low prediction errors: the MLP-LSTM yielded the most accurate results for the lower-dimensional Bouc-Wen system, whereas the MPNN-LSTM and AE-LSTM provided superior performance on the more complex steel-frame model. Moreover, a consistent correlation between predictive variance and actual error confirms the suitability of these frameworks for active-learning strategies and for assessing model confidence in structural response predictions.

Figures & tables

Figure 1: 2D steel frame with Bouc–Wen model of hysteresis : (a) structural layout; and (b) shear building model.

Floors	Beams	Interior columns [cm]	Exterior columns [cm]
1	W36 $\times$ 282	66 $\times$ 7.6	66 $\times$ 6.4
2-10	W36 $\times$ 282	56 $\times$ 7.6	51 $\times$ 6.4
11-20	W36 $\times$ 194	51 $\times$ 5.0	51 $\times$ 5.0
21-30	W33 $\times$ 169	46 $\times$ 2.5	46 $\times$ 2.5
31-40	W27 $\times$ 84	46 $\times$ 1.9	46 $\times$ 1.9

Table 1: Section sizes used in the Bouc-Wen shear building.

Parameter	Mean	COV	Distribution
Young’s Modulus	200 GPa	0.04	Lognormal
Building Density	150 $kg/m^{3}$	0.1	Normal
Damping Ratio	0.005	0.40	Lognormal

Table 2: Statistical properties of uncertain structural parameters

Proposed Schemes	MAE	MSE	RMSE
MLP–LSTM	0.0616	0.0090	0.0680
MPNN–LSTM	0.0642	0.0115	0.0700
AE–LSTM	0.0674	0.0114	0.0741

Table 3: Summary of the performance of the various proposed schemes for the Bouc-Wen case study.

Figure 2: Box plot of the sample-wise X-direction displacement response MSE of different proposed schemes for the Bouc-Wen case study.

Figure 3: Sample displacement time histories for Bouc-Wen case study: (a) horizontal displacement of node 20; (b) horizontal displacement of node 40.

Figure 4: Correlation between maximum MAE with maximum variance over test sample: (a) MLP-LSTM scheme; (b) MPNN-LSTM scheme; (c) AE-LSTM. Note: the figures are in log-log scale for easier visualization.

Figure 5: Layout of the 2D 37-story six-bay steel frame: (a) Plan view; and (b) Elevation view.

Floors	Beams	Box columns [cm]
1 - 10	W24 $\times$ 192	51.25 $\times$ 2.5
11 - 20	W24 $\times$ 192	51.25 $\times$ 2.5
21- 30	W24 $\times$ 103	41 $\times$ 2.0
31 - 37	W24 $\times$ 103	35.9 $\times$ 1.8

Table 4: Section sizes used in the steel frame.

Parameter	Mean	COV	Distribution
Young’s Modulus	200 GPa	0.04	Lognormal
Yield Strength	0.38 GPa	0.06	Lognormal
Damping Ratio	0.015	0.40	Lognormal

Table 5: Statistical properties of uncertainty structural parameters

Proposed Schemes	MAE	MSE	RMSE
MLP–LSTM	0.0356	0.0034	0.0444
MPNN–LSTM	0.0314	0.0025	0.0391
AE–LSTM	0.0324	0.0027	0.0404

Table 6: Summary of the performance of the various proposed schemes for the fiber-discretized nonlinear frame case study.

Figure 6: Box plot of the sample-wise direction-wise displacement response MSE of different proposed schemes for the fiber-discretized nonlinear frame case study: (a) horizontal direction; (b) vertical direction; (c) rotational direction.

Figure 7: Sample displacement time histories for fiber-discretized nonlinear frame case study: (a) horizontal displacement of node 147; (b) horizontal displacement of node 266.

Figure 8: Sample displacement time histories for fiber-discretized nonlinear frame case study: (a) vertical displacement of node 147; (b) vertical displacement of node 266.

Figure 9: Sample displacement time histories for fiber-discretized nonlinear frame case study: (a) rotational displacement of node 147; (b) rotational displacement of node 266.

Figure 10: Correlation between maximum MAE with maximum variance over test sample: (a) MLP-LSTM scheme; (b) MPNN-LSTM scheme; (c) AE-LSTM. Note: the figures are in log-log scale for easier visualization.

Neural Surrogate Modeling

Neural surrogate modeling uses learned models to approximate costly simulations, enabling faster prediction, design, control, and inference.

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Apr 2, 2026J/K move · Enter open · S save

Variational LSTM with Augmented Inputs: Nonlinear Response History Metamodeling with Aleatoric and Epistemic Uncertainty

Manisha Sapkota, Min Li, Bowei Li

Department of Civil, Environmental, and Construction Engineering, Texas Tech University, 2500 Broadway St, Lubbock 79409, TX, USA · Department of Civil and Environmental Engineering, Rensselaer Polytechnic Institute, 110 8th Street, Troy 12180, NY, USA · National Wind Institute, Texas Tech University, 1009 Canton Ave, Lubbock 79409, TX, USA

Surrogate Modeling Uncertainty Propagation Long Short-Term Memory Networks Epistemic Uncertainty Structural Mechanics Aleatoric Uncertainty

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Uncertainty propagation in high-dimensional nonlinear dynamic structural systems is pivotal in state-of-the-art performance-based design and risk assessment, where uncertainties from both excitations and structures, i.e., the aleatoric uncertainty, must be considered. This poses a significant challenge due to heavy computational demands. Machine learning techniques are thus introduced as metamodels to alleviate this burden. However, the "black box" nature of Machine learning models underscores the necessity of avoiding overly confident predictions, particularly when data and training efforts are insufficient. This creates a need, in addition to considering the aleatoric uncertainty, of estimating the uncertainty related to the prediction confidence, i.e., epistemic uncertainty, for machine learning-based metamodels. We developed a probabilistic metamodeling technique based on a variational long short-term memory (LSTM) with augmented inputs to simultaneously capture aleatoric and epistemic uncertainties. Key random system parameters are treated as augmented inputs alongside excitation series carrying record-to-record variability to capture the full range of aleatoric uncertainty. Meanwhile, epistemic uncertainty is effectively approximated via the Monte Carlo dropout scheme. Unlike computationally expensive full Bayesian approaches, this method incurs negligible additional training costs while enabling nearly cost-free uncertainty simulation. The proposed technique is demonstrated through multiple case studies involving stochastic seismic or wind excitations. Results show that the calibrated metamodels accurately reproduce nonlinear response time histories and provide confidence bounds indicating the associated epistemic uncertainty.

Abstract

Neural Surrogate Modeling

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