Physics-Informed Neural Network for Reconstructing Free-Surface Transient Flow Fields in Long-Distance Water-Conveyance Tunnels from Sparse Observations

Long-distance free-surface water-conveyance tunnels require a spatially continuous representation of transient water depth, yet in practice flow is monitored at only a few sections. This study develops a physics-informed neural network (PINN) that reconstructs the transient water-depth field of unsteady free-surface flow in such tunnels from two- or three-point sensors. The one-dimensional Saint-Venant equations, closed with a Darcy–Weisbach steady friction term in hydraulic-radius form, are embedded as a soft constraint in the training loss, so that sparse depth observations are combined with the governing conservation laws to recover the field at unobserved interior and downstream locations. High-resolution finite-volume (FVM) solutions of a 500 m circular tunnel under a flood-rise scenario provide the reference data. The PINN reduces the relative L2 error at unobserved sections to approximately one-quarter of that of an otherwise identical, physics-free ANN (2.50% versus 10.05% at an interior section; 4.77% versus 13.69% at an extrapolation section). Runs repeated with different random seeds confirm statistical stability at zero noise, while revealing that a minority of trainings at 10% noise converge to spurious solutions. Sensor-placement experiments, including layouts anchored at the true domain boundaries (x = 0 and 500 m), show that boundary anchoring—particularly of the upstream boundary—governs both accuracy and noise robustness: boundary-anchored two-sensor layouts remain accurate in most runs under 10–20% observation noise, whereas interior-only layouts degrade sharply. The method is presented as an offline reconstruction tool; its extension to streaming data assimilation is discussed as future work.