The author develops a physics-informed neural network (PINN) for the shallow water equations (used in flood modeling) by replacing the usual point-wise PDE residual with a finite-volume (FVM) loss computed on an unstructured mesh. The main finding is that training on physics alone often collapses to a trivial "no flow" solution because the FVM loss landscape has a shallow basin near zero momentum. Adding even a small amount of real data (50–200 scattered measurements) breaks the degeneracy: the gap between the trivial solution's loss and the true solution's loss jumps from 7× to 310×, and prediction error drops by 7–22×. The framework is demonstrated on a 2D benchmark and a real Savannah River reach.
Main takeaways:
- Standard strong-form PINNs can't handle discontinuities or enforce conservation for shallow water equations; the author uses a differentiable finite-volume Riemann solver instead.
- Physics-only training frequently gets stuck in a low-momentum state that nearly satisfies the FVM loss but doesn't match real flow.
- Loss-landscape analysis shows the trivial solution sits in a shallow basin; sparse data (e.g., 200 velocity measurements) deepens the basin around the correct solution by 44×.
- On a 2D channel benchmark, just 50 data points cut velocity error by 7×; 200 points yield a 22× improvement over physics-only.
- The FVM-PINN loss contributes most in the sparse-data regime (≈23% error reduction) and becomes neutral when dense reference data is available.