Probabilistic Digital Twin for Misspecified Structural Dynamical Systems via Latent Force Modeling and Bayesian Neural Networks

TL;DR

This work tackles state prediction for dynamical systems with misspecified physics by proposing an end-to-end probabilistic digital twin. It combines Gaussian Process Latent Force Models for diagnosis with a Bayesian Neural Network to learn a probabilistic map from states to latent forces, and uses GPLFM in prognosis with BNN-generated pseudo-measurements to propagate uncertainty under new inputs. The approach yields stable, uncertainty-aware predictions across multiple nonlinear benchmarks, including Duffing-type, local nonlinearities, Silverbox, and Bouc-Wen hysteresis, while highlighting limitations in history-dependent cases and the need for representative training data. The framework offers a principled fusion of physics-based modeling and data-driven learning, with potential extensions to online operation and history-aware mappings for robust real-time digital twins.

Abstract

This work presents a probabilistic digital twin framework for response prediction in dynamical systems governed by misspecified physics. The approach integrates Gaussian Process Latent Force Models (GPLFM) and Bayesian Neural Networks (BNNs) to enable end-to-end uncertainty-aware inference and prediction. In the diagnosis phase, model-form errors (MFEs) are treated as latent input forces to a nominal linear dynamical system and jointly estimated with system states using GPLFM from sensor measurements. A BNN is then trained on posterior samples to learn a probabilistic nonlinear mapping from system states to MFEs, while capturing diagnostic uncertainty. For prognosis, this mapping is used to generate pseudo-measurements, enabling state prediction via Kalman filtering. The framework allows for systematic propagation of uncertainty from diagnosis to prediction, a key capability for trustworthy digital twins. The framework is demonstrated using four nonlinear examples: a single degree of freedom (DOF) oscillator, a multi-DOF system, and two established benchmarks -- the Bouc-Wen hysteretic system and the Silverbox experimental dataset -- highlighting its predictive accuracy and robustness to model misspecification.

Figures (23)

• Figure 1: Schematic of the proposed framework showing the flow of data and information between stages.
• Figure 2: Posterior samples of displacement $q$ versus MFE $\eta$, drawn from $p(\bm{{x}}_k, \eta_k \mid \bm{{y}}_{1:N_t})$ at each time step. In this case, $\eta$ represents an unmodelled cubic displacement nonlinearity $K_{\text{nl}}q^3$, where $K_{\text{nl}}$ is a constant. Vertical dashed lines indicate selected values of $q$ (e.g., $q = 0.01$, $0.1$), along which the corresponding conditional distributions $p\left(\eta \mid q\right)$ are shown. The associated $\eta$ samples are highlighted in matching colors. The nonlinear trend and variation in spread of $\eta$ across $q$ illustrate the heteroskedastic and state-dependent nature of $p(\eta \mid \bm{{x}})$, motivating the use of BNNs for mapping.
• Figure 3: Schematic of the BNN architecture mapping system states $\bm{{x}}$ to model-form errors $\bm{{\eta}}$. The output layer predicts the mean $\bm{{\mu}}^\eta$ and the Cholesky factor $\mathbf{{L}}$ to reconstruct the full covariance $\mathbf{{\Sigma}}^\eta = \mathbf{{L}} \mathbf{{L}}^T$.
• Figure 4: External force input signals used during the prognosis phase to evaluate model generalization. (a) Periodic sinusoidal input representing structured forcing. (b) Filtered broadband noise representing stochastic excitation.
• Figure 5: Schematic of the SDOF oscillator with Duffing-type nonlinear stiffness, excited by ground acceleration $\ddot{u}_g$ during diagnosis.