Physics-informed Neural Network Model for Transient Wave Propagation in A Pressurized Pipeline

This paper explores the use of neural networks (NNs) to model water-hammer waves propagation in a bounded pipe system. The training dataset is obtained from a numerical solution using the Method of Characteristics (MOC). It is found that conventional NNs fail to model the water-hammer waves propagation when trained using only the boundary conditions and require large datasets for training. However, the use of physics-informed NN (PINN), wherein the one-dimensional wave equation, along with the boundary conditions are imposed as part of the loss function, provides accurate results. The PINN model parameters (i.e., weights and biases) are optimized using (i) the Adaptive Moment Estimation (ADAM), and (ii) the limited-memory Broyden-Fletcher- Goldfarb-Shanno (L-BFGS) algorithms. The results show that L-BFGS based solution is in good agreement with the exact MOC solution, whereas the ADAM-based solution has slow convergence. Moreover, it is found that the PINN solution is sensitive to changes in boundary and initial conditions as well as to the time length of the wave field solution. In particular, our results show that as the time length (or bandwidth) of the signals increases, a denser NN model architecture (in terms of number of neurons) is required to accurately predict the response. It is also found that the PINN model is sensitive to the location of additional measurement points used as a constraining dataset. Detailed sensitivity analysis concerning model architecture and the constraining dataset is yet to be investigated. The advantages and implications of using PINN in hydraulic transient modeling are discussed.