Machine learning assisted state prediction of misspecified linear dynamical system via modal reduction Rohan Vittal Thorat a, Rajdip Nayek a a Department of Applied Mechanics, Indian Institute of Technology Delhi, New Delhi, 110016, IndiaAbstract 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 dis-cretizations 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. Keywords: Model bias, Gaussian Process, Latent Force Model, Bayesian filtering, Modal reduction, Digital twin 1. Introduction The reliable simulation of structural dynamical systems is central to engineering analysis, design, and decision-making. In practice, high-fidelity models are often impractical due to limited information, computational constraints, or simplifying assumptions in geometry, boundary conditions, damping mechanisms, and material constitutive laws. These idealizations lead to model form errors (MFEs)--systematic discrepancies between the predicted and actual system responses--which, if unaccounted for, can significantly degrade predictive accuracy. This challenge is especially critical in the context of digital twins, where model predictions directly inform monitoring and decision-making. Digital twins of structural systems integrate computational models with real-time or historical measurement data to enable continuous prediction, monitoring, and decision making [1, 2].

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.

Ferry quays experience rapid deterioration due to their exposure to harsh maritime environments and ferry impacts. Vibration-based structural health monitoring offers a valuable approach to assessing structural integrity and understanding the structural implications of these impacts. However, practical limitations often restrict sensor placement at critical locations. Consequently, virtual sensing techniques become essential for establishing a Digital Twin and estimating the structural response. This study investigates the application of the Gaussian Process Latent Force Model (GPLFM) for virtual sensing on the Magerholm ferry quay, combining in-operation experimental data collected during a ferry impact with a detailed physics-based model. The proposed Physics-Encoded Machine Learning model integrates a reduced-order structural model with a data-driven GPLFM representing the unknown impact forces via their modal contributions. Significant challenges are addressed for the development of the Digital Twin of the ferry quay, including unknown impact characteristics (location, direction, intensity), time-varying boundary conditions, and sparse sensor configurations. Results show that the GPLFM provides accurate acceleration response estimates at most locations, even under simplifying modeling assumptions such as linear time-invariant behavior during the impact phase. Lower accuracy was observed at locations in the impact zone. A numerical study was conducted to explore an optimal real-world sensor placement strategy using a Backward Sequential Sensor Placement approach. Sensitivity analyses were conducted to examine the influence of sensor types, sampling frequencies, and incorrectly assumed damping ratios. The results suggest that the GP latent forces can help accommodate modeling and measurement uncertainties, maintaining acceptable estimation accuracy across scenarios.

Gaussian Processes (GPs) provide a principled Bayesian framework for function approximation, making them particularly useful in many applications requiring uncertainty calibration [Rasmussen and Williams, 2006], such as Bayesian optimisation [Snoek et al., 2012] and time-series analysis [Roberts et al., 2013]. Despite offering reasonable uncertainty estimation, shallow GPs often struggle to model complex, non-stationary processes present in practical applications. To overcome this limitation, Deep Gaussian Processes (DGPs) employ a compositional architecture by stacking multiple GP layers, thereby enhancing representational power while preserving the model's intrinsic capability to quantify uncertainty [Damianou and Lawrence, 2013]. However, the conventional variational formulation of DGPs heavily depends on local inducing point approximations across intermediate GP layers [Titsias, 2009, Salimbeni and Deisenroth, 2017], which hinder the model from capturing the global structures commonly found in real-world scenarios. Incorporating Fourier features into GP models has shown promise in addressing this challenge in GP inference due to the periodic nature of these features. A line of research uses Random Fourier Features (RFFs, [Rahimi and Recht, 2007]) of stationary kernels to convert the original (deep) GPs into Bayesian networks in weight space [Lázaro-Gredilla et al., 2010, Gal and Turner, 2015, Cutajar et al., 2017]. Building on this concept within a sparse variational GP framework, recent advancements in inter-domain GPs [Lázaro-Gredilla and Figueiras-Vidal, 2009a, Van der Wilk et al., 2020] directly approximate the posterior of the original GPs by introducing fixed Variational Fourier Features (VFFs) through process projection onto a Reproducing Kernel Hilbert Space (RKHS)[Hensman et al., 2018, Rudner et al., 2020]. VFFs are derived by projecting GPs onto a different domain.

Effectively modeling phenomena present in highly nonlinear dynamical systems whilst also accurately quantifying uncertainty is a challenging task, which often requires problem-specific techniques. We present a novel, domain-agnostic approach to tackling this problem, using compositions of physics-informed random features, derived from ordinary differential equations. The architecture of our model leverages recent advances in approximate inference for deep Gaussian processes, such as layer-wise weight-space approximations which allow us to incorporate random Fourier features, and stochastic variational inference for approximate Bayesian inference. We provide evidence that our model is capable of capturing highly nonlinear behaviour in real-world multivariate time series data. In addition, we find that our approach achieves comparable performance to a number of other probabilistic models on benchmark regression tasks.