Machine learning assisted state prediction of misspecified linear dynamical system via modal reduction

TL;DR

This work introduces a comprehensive framework for MFE estimation and correction in high-dimensional finite element (FE) based structural dynamical systems and offers a potential means to uphold digital twin accuracy amid inherent modeling uncertainties.

Abstract

Accurate prediction of structural dynamics is imperative for preserving digital twin fidelity throughout operational lifetimes. Parametric models with fixed nominal parameters often omit critical physical effects due to simplifications in geometry, material behavior, damping, or boundary conditions, resulting in model form errors (MFEs) that impair predictive accuracy. This work introduces a comprehensive framework for MFE estimation and correction in high-dimensional finite element (FE) based structural dynamical systems. The Gaussian Process Latent Force Model (GPLFM) represents discrepancies non-parametrically in the reduced modal domain, allowing a flexible data-driven characterization of unmodeled dynamics. A linear Bayesian filtering approach jointly estimates system states and discrepancies, incorporating epistemic and aleatoric uncertainties. To ensure computational tractability, the FE system is projected onto a reduced modal basis, and a mesh-invariant neural network maps modal states to discrepancy estimates, permitting model rectification across different FE discretizations without retraining. Validation is undertaken across five MFE scenarios-including incorrect beam theory, damping misspecification, misspecified boundary condition, unmodeled material nonlinearity, and local damage demonstrating the surrogate model's substantial reduction of displacement and rotation prediction errors under unseen excitations. The proposed methodology offers a potential means to uphold digital twin accuracy amid inherent modeling uncertainties.

Figures (23)

• Figure 1: A two-layer neural network maps mean modal states $[\bm{{q}}(t),\,\dot{\bm{{q}}}(t)]$ to the mean latent discrepancy force $\bm{{\eta}}(t)$ in the reduced space. Hidden units use sigmoid activations, and the output is linear. The neural network is parameterized by $\bm{{\psi}} = [\bm{{\psi}}^{(1)^\top}_{w},\psi^{(1)^\top}_{b},\bm{{\psi}}^{(2)^\top}_{w},\psi^{(2)^\top}_{b}]$.
• Figure 2: Rectification of the model discrepancy is done mainly in three stages as follows: (A) Estimate state $\bm{{q}}(t),\bm{{\dot{q}}}(t)$ and model form error $\bm{{\eta}}(t)$ (B) Train neural network using the estimated $\bm{{q}}(t),\bm{{\dot{q}}}(t),\bm{{\eta}}(t)$ (C) Test the rectified model against unseen external forcing function ($\bm{{p}}^*(t) = \Phi^\top \bm{{f}}^*(t)$)
• Figure 3: Experimental setup: simply supported beam with point input force given at three locations 2 m, 5 m and 8 m from left side of the beam. Sensors are placed at equidistance starting from location 1 m up to 9 m of the beam and are represented by a red dot
• Figure 4: Averaged log–magnitude FFT of displacement (and rotation) from the nominal linear beam at the nine sensor locations. Peaks at the first four structural natural frequencies are shown in dotted vertical lines; mode 4 is weak but discernible, while no clear peaks are observed beyond mode 4.
• Figure 5: Spatio–temporal displacement estimation on the full beam for a representative window (3.5–3.6 s). Grid of panels: columns show (i) true displacement (Timoshenko), (ii) nominal displacement (model as specified in each example), (iii) GPLFM estimate (0% noise), (iv) GPLFM estimate (5% noise). Rows correspond to Examples 1–4: (1) beam theory mismatch, (2) damping misspecification, (3) boundary condition misspecification, (4) nonlinear constitutive law. Color scale is consistent along rows.