Towards a digital twin of a full-scale hydro turbine, we propose a data-driven physics-informed twin for a lab-scale cantilever beam system featuring two springs at its free end which introduces geometric nonlinearities. To achieve accurate predictions, estimate latent parameters, and quantify uncertainty, we leverage a physics-informed neural network framework as a trial solution for the finite element formulation to reduce the physical residual loss of the neural network. By doing so, we aim to bridge the gap between machine learning techniques and computational mechanics. Our approach uses the weak form of physical equations to reduce the order of derivatives; while strategically incorporating boundary terms and forcing terms into the weak formulation to minimize competing loss functions. Two critical challenges we address are the limited knowledge of the system’s underlying physics and the presence of noisy experimental data—both of which introduce uncertainties during training and impact the reliability and robustness of predictions. To mitigate these issues, we combine known physics with an additional unknown function, employing Bayesian inference methods to quantify both modeling and measurement uncertainties. This allows the neural network to effectively handle incomplete or imprecise physical models. The effectiveness of our methodology will be validated through various case studies, spanning both linear and nonlinear systems, including multi-degree-of-freedom setups, fluid-solid interaction models, beam-moving mass systems, and finally, the lab-scale cantilever beam configuration. The ultimate goal is to estimate the spring stiffness values and predict beam responses at any point along its length, even in the presence of uncertain model parameters and noisy measurements. By achieving these objectives, we aim to apply our insights and methodology to real-world hydro turbine operations, enhancing both operational efficiency and predictive maintenance strategies.