Finite Volume-Informed Neural Network Framework for 2D Shallow Water Equations: Rugged Loss Landscapes and the Importance of Data Guidance

Physics-informed neural networks (PINNs) are a simple surrogate-modelling paradigm for partial differential equations, but their standard strong-form residual formulation is ill suited to the shallow water equations (SWE). It cannot enforce local conservation, handle discontinuities, or leverage the boundary-conforming unstructured meshes used in real-world applications. We introduce ``Data-Guided FVM-PINN'', a framework that replaces the strong-form residual with a differentiable, well-balanced Roe Riemann-solver finite-volume (FVM) loss evaluated on unstructured meshes. The major finding is that physics-only FVM-PINN training often fails on realistic 2D problems: the network collapses to a trivial low-momentum state that nearly satisfies the FVM-PINN residual but bears no resemblance to the true flow. A loss-landscape diagnostic shows that the FVM-PINN loss at zero momentum is only about 7× larger than at the trained solution, a shallow basin that an ordinary optimizer falls into; adding even sparse data turns this into a 310× separation, breaking the degeneracy. On a 2D block-in-channel benchmark, just 200 random velocity measurements drop the velocity-field L2 error by 22× versus physics-only; 50 measurements still deliver a 7× reduction. A controlled ablation isolates the contribution of the FVM-PINN loss: it reduces velocity-field L2 by ~23% in the sparse-data regime and is essentially neutral when dense reference data is available. On a real-world Savannah River reach (1306 cells, 3600~s simulation, five Manning zones), the framework constructs an accurate surrogate from SRH-2D anchor data, with time-window decomposition reducing error monotonically via progressive initial-condition handoff.