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

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.

Deep learning has proven highly effective for nonlinear system identification, but heavily parameterized neural networks are prone to overfitting in low-data regimes and lack reliable uncertainty quantification. The recently developed manifold meta-learning framework addresses the data efficiency problem by restricting the model parameters to a meta-learned low-dimensional manifold. However, that method is purely deterministic. We propose a fully probabilistic extension of the manifold meta-learning framework, based on amortized Variational Inference, where a generative prior over the low-dimensional parameter manifold is learned. During task-specific adaptation, we combine Maximum A Posteriori estimation with the Laplace approximation to yield a mathematically grounded posterior approximation. Evaluated on a static regression task and the Bouc--Wen dynamical system benchmark, the proposed approach achieves predictive accuracy comparable to its deterministic counterpart while successfully providing calibrated uncertainty bounds in severely low-data regimes.

System identification (SysID) is critical for modeling dynamical systems from experimental data, yet traditional approaches often fail to capture nonlinear behaviors. While deep learning offers powerful tools for modeling such dynamics, incorporating uncertainty quantification is essential to ensure reliable predictions. This paper presents a systematic framework for constructing and training interval Neural Networks (INNs) for uncertainty-aware SysID. By extending crisp neural networks into interval counterparts, we develop Interval LSTM and NODE models that propagate uncertainty through interval arithmetic without probabilistic assumptions. This design allows them to represent uncertainty and produce prediction intervals. For training, we propose two strategies: Cascade INN (C-INN), a two-stage approach converting a trained crisp NN into an INN, and Joint INN (J-INN), a one-stage framework jointly optimizing prediction accuracy and interval precision. Both strategies employ uncertainty-aware loss functions and parameterization tricks to ensure reliable learning. Comprehensive experiments on multiple SysID datasets demonstrate the effectiveness of both approaches and benchmark their performance against well-established uncertainty-aware baselines: C-INN achieves superior point prediction accuracy, whereas J-INN yields more accurate and better-calibrated prediction intervals. Furthermore, to reveal how uncertainty is represented across model parameters, the concept of channel-wise elasticity is introduced, which is used to identify distinct patterns across the two training strategies. The results of this study demonstrate that the proposed framework effectively integrates deep learning with uncertainty-aware modeling.