Response Estimation and System Identification of Dynamical Systems via Physics-Informed Neural Networks

TL;DR

Three key applications of Physics-Informed Neural Networks are investigated: state estimation in systems with sparse sensing, joint state-parameter estimation, when both system response and parameters are unknown, and parameter estimation from full-field observation, within a Bayesian framework to quantify uncertainties.

Abstract

The accurate modelling of structural dynamics is crucial across numerous engineering applications, such as Structural Health Monitoring (SHM), seismic analysis, and vibration control. Often, these models originate from physics-based principles and can be derived from corresponding governing equations, often of differential equation form. However, complex system characteristics, such as nonlinearities and energy dissipation mechanisms, often imply that such models are approximative and often imprecise. This challenge is further compounded in SHM, where sensor data is often sparse, making it difficult to fully observe the system's states. To address these issues, this paper explores the use of Physics-Informed Neural Networks (PINNs), a class of physics-enhanced machine learning (PEML) techniques, for the identification and estimation of dynamical systems. PINNs offer a unique advantage by embedding known physical laws directly into the neural network's loss function, allowing for simple embedding of complex phenomena, even in the presence of uncertainties. This study specifically investigates three key applications of PINNs: state estimation in systems with sparse sensing, joint state-parameter estimation, when both system response and parameters are unknown, and parameter estimation within a Bayesian framework to quantify uncertainties. The results demonstrate that PINNs deliver an efficient tool across all aforementioned tasks, even in presence of modelling errors. However, these errors tend to have a more significant impact on parameter estimation, as the optimization process must reconcile discrepancies between the prescribed model and the true system behavior. Despite these challenges, PINNs show promise in dynamical system modeling, offering a robust approach to handling uncertainties.

Figures (14)

• Figure 1: MDOF Nonlinear Oscillator
• Figure 2: Framework of a general PINN, highlighting where the data-driven and physics-knowledge are embedded within the process
• Figure 3: Visualisation of domain definitions for different schemes and motivations that employ PINNs. The blue areas represent the continuous collocation domain, and the red dots represent the coverage and sparsity of the discrete observation domain.
• Figure 4: Estimated input and response for 6DOF Duffing system with no model error. The DOF is indicated on the left, with an asterisk indicating unobserved DOFs.
• Figure 5: Estimated input and response for 20DOF Duffing system with no model error. The DOF is indicated on the left, with an asterisk indicating unobserved DOFs.