Finite Volume-Informed Neural Network Framework for 2D Shallow Water Equations: Rugged Loss Landscapes and the Importance of Data Guidance
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866909034062282752 |
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| author | Liu, Xiaofeng |
| author_facet | Liu, Xiaofeng |
| contents | 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 $7\times$ larger than at the trained solution, a shallow basin that an ordinary optimizer falls into; adding even sparse data turns this into a $310\times$ separation, breaking the degeneracy. On a 2D block-in-channel benchmark, just $200$ random velocity measurements drop the velocity-field $L_2$ error by $22\times$ versus physics-only; $50$ measurements still deliver a $7\times$ reduction. A controlled ablation isolates the contribution of the FVM-PINN loss: it reduces velocity-field $L_2$ by $\sim$$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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_11001 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Finite Volume-Informed Neural Network Framework for 2D Shallow Water Equations: Rugged Loss Landscapes and the Importance of Data Guidance Liu, Xiaofeng Machine Learning 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\times$ larger than at the trained solution, a shallow basin that an ordinary optimizer falls into; adding even sparse data turns this into a $310\times$ separation, breaking the degeneracy. On a 2D block-in-channel benchmark, just $200$ random velocity measurements drop the velocity-field $L_2$ error by $22\times$ versus physics-only; $50$ measurements still deliver a $7\times$ reduction. A controlled ablation isolates the contribution of the FVM-PINN loss: it reduces velocity-field $L_2$ by $\sim$$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. |
| title | Finite Volume-Informed Neural Network Framework for 2D Shallow Water Equations: Rugged Loss Landscapes and the Importance of Data Guidance |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2605.11001 |