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

arXiv: https://arxiv.org/abs/2605.11001 · HTML: https://mlsanity.com/paper/2605.11001/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 35 references that are in the map.
- First posted on arXiv 9 May 2026.

## Builds on (most cited first, 10 of 10)

- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Training Deep Nets with Sublinear Memory Cost](https://mlsanity.com/paper/1604.06174/) (2016)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Physics-informed neural networks for the shallow-water equations on the sphere](https://mlsanity.com/paper/2104.00615/) (2021)
- [wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws](https://mlsanity.com/paper/2207.08483/) (2022)
- [GoRINNs: Godunov-Riemann Informed Neural Networks for Learning Hyperbolic Conservation Laws](https://mlsanity.com/paper/2410.22193/) (2024)
- [Bathymetry Inversion using a Deep-Learning-Based Surrogate for Shallow Water Equations Solvers](https://mlsanity.com/paper/2203.02821/) (2022)
- [Scientific Machine Learning of Flow Resistance Using Universal Shallow Water Equations with Differentiable Programming](https://mlsanity.com/paper/2502.12396/) (2025)
- [FFV-PINN: A Fast Physics-Informed Neural Network with Simplified Finite Volume Discretization and Residual Correction](https://mlsanity.com/paper/2603.24114/) (2026)

## Abstract (arXiv)

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.