In online structural health monitoring (SHM) frameworks, digital twinning plays a valuable role by allowing to continually update a virtual model on the basis of - even real-time - monitoring data from the physical system. In this context, surrogate modeling based on machine learning (ML) techniques aims to replicate the dynamics of high-fidelity computational models, thereby enabling faster simulations. However, traditional ML approaches, such as neural networks (NNs), often act as black-box models, generating predictions that may lack physical interpretability and consistency. To address these limitations, we propose a novel surrogate model that leverages physics-informed neural networks (PINNs) to identify ballistic impact damage on a helicopter transmission shaft. By incorporating dynamic equations derived from a physics-based model of the operating structure into the network's loss function, the proposed approach ensures predictions that adhere to physical laws. This physics-based model, referred to as a "white-box", simulates damage-induced vibration loads and system responses. The physics-informed surrogate model is continuously updated using real-time monitoring data from the operation, coupled with the physical laws derived from the white-box model. The novelty of the proposed PINN framework lies in its dual updating scheme, which facilitates both solution discovery (e.g., excitation force estimation) and parameter identification (e.g., stiffness estimation), thus allowing for advanced damage identification. The efficacy and cost-effectiveness of this framework are demonstrated through numerical studies, highlighting its robustness in delivering accurate predictions across a wide range of impact damage scenarios.