Physics-informed neural networks involving unsteady friction for transient pipe flow

A robust physics-informed neural network (PINN) approach is developed to accurately predict pressure and flow velocity during the water hammer event, while an experimental system is designed to validate the proposed approach further. Compared to forward numerical methods, PINN can retrieve hydraulic information from sensor data at any location in a real complex pipe system or network. However, the nonlinear nature of hydraulic transients may lead to unstable optimization for PINN, and the available labeled data are sparse. Therefore, it is necessary to explore a more feasible solution for this analysis. In this paper, a locally adaptive activation function (LAAF) is adopted to improve PINN performance. To account for real-world uncertainties in sensing data (such as pipe friction, viscoelasticity, and noise), an unsteady friction model with a self-adaptive coefficient is incorporated. This allows the partial differential equations to better reflect actual conditions and be seamlessly integrated into the PINN without additional training costs. Based on these two optimizations, four PINN schemes with different components are used to conduct a series of ablation studies in numerical tests. LAAF for PINN exhibits enhanced robustness for high-frequency data. Integrated with the Brunone model, it effectively avoids local minima and achieves excellent agreement with the reference solutions in predicting hydraulic parameters across the global flow field. In experimental studies, the proposed approach successfully extrapolates hydraulic information even with noisy data from only two sensors, attaining a relative error below 7.00e-2. In addition, it is observed that training points closer to the hydraulic transient yield richer physical insights conducive to PINN training.