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

arXiv: https://arxiv.org/abs/2603.12012 · HTML: https://mlsanity.com/paper/2603.12012/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 57 references that are in the map.
- First posted on arXiv 12 Mar 2026.

## Builds on (most cited first, 17 of 17)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [MLP-Mixer: An all-MLP Architecture for Vision](https://mlsanity.com/paper/2105.01601/) (2021)
- [Graph Convolutional Neural Networks for Web-Scale Recommender Systems](https://mlsanity.com/paper/1806.01973/) (2018)
- [Concrete Dropout](https://mlsanity.com/paper/1705.07832/) (2017)
- [A Theoretically Grounded Application of Dropout in Recurrent Neural Networks](https://mlsanity.com/paper/1512.05287/) (2015)
- [On the Pitfalls of Heteroscedastic Uncertainty Estimation with Probabilistic Neural Networks](https://mlsanity.com/paper/2203.09168/) (2022)
- [Pitfalls of Epistemic Uncertainty Quantification through Loss Minimisation](https://mlsanity.com/paper/2203.06102/) (2022)
- [Physics-Informed Multi-LSTM Networks for Metamodeling of Nonlinear Structures](https://mlsanity.com/paper/2002.10253/) (2020)
- [Machine Learning Approach to Model Order Reduction of Nonlinear Systems via Autoencoder and LSTM Networks](https://mlsanity.com/paper/2109.11213/) (2021)
- [On Information-Theoretic Measures of Predictive Uncertainty](https://mlsanity.com/paper/2410.10786/) (2024)
- [Learning to simulate and design for structural engineering](https://mlsanity.com/paper/2003.09103/) (2020)
- [Neural Operators for Stochastic Modeling of Nonlinear Structural System Response to Natural Hazards](https://mlsanity.com/paper/2502.11279/) (2025)
- [Parameter estimation of structural dynamics with neural operators enabled surrogate modeling](https://mlsanity.com/paper/2410.11712/) (2024)
- [A view on model misspecification in uncertainty quantification](https://mlsanity.com/paper/2210.16938/) (2022)
- [Surrogate Structure-Specific Probabilistic Dynamic Responses of Bridge Portfolios using Deep Learning with Partial Information](https://mlsanity.com/paper/2503.00335/) (2025)
- [Adaptive Machine Learning-Driven Multi-Fidelity Stratified Sampling for Failure Analysis of Nonlinear Stochastic Systems](https://mlsanity.com/paper/2508.00734/) (2025)

## Abstract (arXiv)

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.