arxiv: 2605.11001 · v1 · submitted 2026-05-09 · 💻 cs.LG

Recognition: 2 theorem links

· Lean Theorem

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

Xiaofeng Liu

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Pith reviewed 2026-05-13 05:52 UTC · model grok-4.3

classification 💻 cs.LG

keywords physics-informed neural networksshallow water equationsfinite volume methodloss landscapesdata guidancesurrogate modelingunstructured meshesRoe Riemann solver

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The pith

Physics-only FVM-PINN training for 2D shallow water equations collapses to a trivial low-momentum state that satisfies the loss but not the flow.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper replaces the usual strong-form residual in PINNs with a differentiable finite-volume loss based on a well-balanced Roe Riemann solver evaluated on unstructured meshes, creating the Data-Guided FVM-PINN framework for the shallow water equations. It demonstrates that training with this physics loss alone on realistic 2D problems leads the network to converge to a near-zero velocity field that nearly minimizes the loss but bears no resemblance to the true solution. Adding even sparse velocity measurements from data dramatically enlarges the loss separation between the trivial state and the physical solution, guiding the optimizer correctly. This matters for surrogate modeling because it shows why pure physics-informed methods can fail on conservation laws with discontinuities and how minimal data makes the approach practical for engineering flows such as river simulations.

Core claim

The central claim is that the FVM-PINN loss landscape contains a shallow basin at the zero-momentum state, only about 7 times higher than at the true solution, so standard optimizers collapse there and produce non-physical results. Incorporating sparse data increases the loss separation to roughly 310 times, breaking the degeneracy. On a 2D block-in-channel benchmark, 200 random velocity measurements reduce velocity-field L2 error by 22 times relative to physics-only training, while 50 measurements still achieve a 7 times reduction. The finite-volume loss itself contributes an additional 23 percent error reduction in the sparse-data regime and remains neutral with dense data. The same method

What carries the argument

Data-Guided FVM-PINN, which substitutes a differentiable, well-balanced Roe Riemann-solver finite-volume loss on unstructured meshes for the conventional strong-form residual and augments it with sparse data measurements to escape degenerate minima.

If this is right

On the 2D block-in-channel benchmark, 200 random velocity measurements cut velocity L2 error by 22 times compared with physics-only training.
The FVM-PINN loss alone reduces velocity L2 error by about 23 percent when data are sparse and has negligible effect when dense reference data are supplied.
Even 50 velocity measurements still produce a 7 times error reduction over physics-only training.
Time-window decomposition with progressive initial-condition handoff yields monotonically decreasing error on a 1306-cell real-world river reach simulation spanning 3600 seconds.
The loss value at the zero-momentum state is only 7 times larger than at the trained solution, but sparse data enlarges this ratio to 310 times.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The shallow-basin degeneracy observed here is likely to appear in PINN formulations for other hyperbolic conservation laws where trivial or uniform states approximately satisfy the discrete residual.
Hybrid data-plus-FVM losses may prove essential for stable surrogate models of unsteady engineering flows when reference data are limited but not entirely absent.
The time-window handoff technique for maintaining accuracy over long horizons could be tested on other time-dependent PDE problems that suffer from accumulation of integration errors.

Load-bearing premise

The differentiable well-balanced Roe Riemann-solver finite-volume loss can be stably back-propagated through unstructured meshes and the network can represent the target flow without extra regularization.

What would settle it

Retraining the identical FVM-PINN architecture on the 2D block-in-channel benchmark with physics-only loss from multiple random initializations and observing whether any run converges to a velocity field whose L2 error is within 5 percent of the reference solution rather than collapsing to near-zero momentum.

Figures

Figures reproduced from arXiv: 2605.11001 by Xiaofeng Liu.

**Figure 1.** Figure 1: Schematic of the 2D shallow water equations: (a) the physical domain and definitions, and (b) the unstructured finite-volume mesh with face-based fluxes. In conservation form, the SWEs read ∂Q ∂t + ∂F(Q) ∂x + ∂G(Q) ∂y = S(Q), (2) with fluxes and source term (Liu & Song, 2025) F =   uh u 2h + 1 2 g(ξ 2 + 2ξhs) uvh   , G =   vh uvh v 2h + 1 2 g(ξ 2 + 2ξhs)   , S =   0 −τbx/ρ − g ξ S0x −τby/ρ − g ξ… view at source ↗

**Figure 2.** Figure 2: Schematic of the FVM-PINN architecture and loss components. where ∂Qi/∂t is computed via automatic differentiation through the network, and Fˆ Roe is the Roe flux in equation (8). Assuming there are nc cells, the total FVM-PINN (PDE) loss is then the summation of the FVM loss for each cell and time level: Lfvm−pinn = 1 ntnc Xnt k=1 Xnc i=1 Ai ∥Ri(tk)∥ 2 . (14) The boundary conditions are enforced as follow… view at source ↗

**Figure 3.** Figure 3: shows the predicted and exact water depth and velocity profiles at t = 1 s. The network correctly captures the left rarefaction fan, the contact wave, and the right-moving shock [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

**Figure 4.** Figure 4: Case 2: WSE profile over a parabolic bed bump at the final time step (steady state). (a) WSE, (b) velocity. 5.3 Case 3: Data-Guided Training Ablation This is the central experiment of the paper. We use a 2D block-in-channel (BIC) benchmark (15 m × 5 m rectangular channel with a block obstruction, 1326 unstructured cells) to systematically study the role of data in FVM-PINN training. The setup of the case i… view at source ↗

**Figure 5.** Figure 5: Case 3: Block-in-channel benchmark. (a) Case setup, (b) Unstructured mesh and 50 sparse data points, (c) Water depth result from SRH-2D, (d) Velocity result from SRH-2D, (e) Water depth result from FVM teacher, (f) Velocity result from FVM teacher [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

**Figure 6.** Figure 6: Case 3: Water depth and velocity contours at steady state for the eight ablation runs. The reference solution is from SRH-2D. The four columns correspond to water depth, difference in water depth from SRH-2D, velocity magnitude, and difference in velocity magnitude from SRH-2D. The rows correspond to the eight runs in [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

**Figure 7.** Figure 7: Case 3: Ablation summary and comparison: (a) L2(|u|) for the eight runs in Table 3 and (b) WSE profiles along the channel centreline for the same runs. The reference solution is from SRH-2D. –15– [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗

**Figure 8.** Figure 8: Case 4: Savannah River. (a) Manning’s roughness zones, (b) Bathymetry, (c) Water depth from SRH-2D at steady state, (d) Velocity magnitude from SRH-2D at steady state, (e) Water depth from FVM teacher, and (f) Velocity magnitude from FVM teacher. –17– [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗

**Figure 9.** Figure 9: Case 4: Water depth and velocity contours at steady state for the seven ablation runs. The reference solution is from SRH-2D. The four columns correspond to water depth, difference in water depth from SRH-2D, velocity magnitude, and difference in velocity magnitude from SRH-2D. The rows correspond to the runs in [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗

**Figure 10.** Figure 10: Case 4: L2 error at each reference time (t ∈ {720, 1440, 2160, 2880, 3600} s) for the physics-only baseline (SR-A), the standard trainer (SR-B), Window(5) (SR-C), Window(10) (SRD), and the FVM teacher (SR-E). The windowed trainers’ improvement over the single-network baseline grows monotonically with t thanks to progressive IC handoff, while the single-network trainer’s error is nearly constant in t. –20… view at source ↗

**Figure 12.** Figure 12: Starting from the trained BIC-B network (which has correctly learned the wake), [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗

**Figure 11.** Figure 11: Case 4 GPU memory footprint of the seven Savannah River runs vs. wall-clock training time. (a) Running maximum of peak GPU memory since training started. Per-step memory is dominated by the autograd graph of the FVM-PINN loss evaluated over the full mesh at every Adam step. As a result, the strategies that evaluate the FVM residual every step (SR-A, SR-B, SR-F, SR-G) all reach ∼900 MiB peak. Time windowin… view at source ↗

**Figure 12.** Figure 12: Loss-landscape diagnostic on the BIC-B trained network (block-in-channel): (a) Lfvm−pinn(α) and Ldata(α) along the momentum-scaling line (18). (b) weighted total loss λfvm−pinnLfvm−pinn + λdataLdata + · · · . The FVM-PINN loss at α = 0 (zero momentum) is only about 7.0× larger than at the trained solution (α = 1) – a shallow basin that an ordinary firstorder optimizer can fall into from a wide range of i… view at source ↗

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Physics-only FVM-PINN training for 2D SWE collapses to trivial low-momentum states because the loss landscape is too flat, and sparse velocity data fixes it with large error reductions.

read the letter

The core observation is that replacing the usual strong-form residual with a differentiable Roe finite-volume loss on unstructured meshes still leaves physics-only training prone to collapse on realistic 2D shallow-water problems. The network reaches a near-zero momentum state that satisfies the discrete conservation statements but has nothing to do with the actual flow. Their loss-landscape check shows the zero-momentum point is only 7 times worse than the trained solution, which explains why ordinary optimizers land there; adding 200 scattered velocity points turns the separation into 310 times and cuts L2 velocity error by 22 times on the block-in-channel case. Even 50 points give a 7 times improvement. The ablation isolating the FVM term shows a modest 23 percent gain in the sparse-data regime and no penalty when dense data is already present. The Savannah River example with 1306 cells and real SRH-2D anchors demonstrates that the same setup can produce a usable surrogate once time-window handoff is added. That combination of Roe-based FVM loss, explicit degeneracy diagnosis, and quantified data-guidance benefit is the concrete advance over prior PINN work on conservation laws. The main soft spot is that the paper assumes the Roe flux and well-balanced source terms remain stably differentiable across the unstructured mesh without showing gradient norms, sonic-point handling, or discrete conservation error after back-propagation. If those gradients are noisy or the network cannot represent the target without extra regularization, the reported collapse and the 22 times gain could be partly an artifact of the loss construction rather than a general property of physics-only training. No code or mesh files are supplied, so independent checks are not yet possible. This work is for groups already building surrogates for 2D hydraulic models who need to incorporate limited field data. It deserves a serious referee because the empirical separation between physics-only and data-guided regimes is large enough to be worth verifying, even if the differentiability details need tightening.

Referee Report

2 major / 2 minor

Summary. The paper introduces a Data-Guided FVM-PINN framework for 2D Shallow Water Equations that replaces the standard strong-form residual with a differentiable, well-balanced Roe Riemann-solver finite-volume loss evaluated on unstructured meshes. It reports that physics-only FVM-PINN training collapses to a trivial low-momentum state satisfying the residual but not the true flow, with the loss landscape showing only a 7x separation at zero momentum versus the trained solution; adding sparse data (e.g., 200 random velocity measurements) creates a 310x separation and reduces velocity L2 error by 22x (or 7x with 50 measurements) on a 2D block-in-channel benchmark. A controlled ablation attributes ~23% additional error reduction to the FVM term in the sparse-data regime, and the method is applied to a real-world Savannah River reach using SRH-2D anchor data with time-window decomposition.

Significance. If the quantitative results and loss-landscape diagnostics hold under verification, the work provides concrete evidence that pure physics-informed training can be inadequate for hyperbolic systems like the SWE due to shallow basins, while hybrid data guidance enables practical surrogate modeling on unstructured meshes. The emphasis on well-balanced schemes and real-world application to hydraulic reaches is a strength for engineering relevance. However, the absence of released code, data, or explicit gradient/conservation verification limits immediate adoption and extension by the community.

major comments (2)

The central claim that physics-only FVM-PINN collapses due to a shallow 7x loss basin (while sparse data yields 310x separation and 22x error reduction) rests on the Roe Riemann-solver FVM loss being a faithfully differentiable discretization that can be stably back-propagated through unstructured meshes. No explicit numerical checks for gradient accuracy, handling of sonic points/limiters, or discrete conservation errors are described, raising the possibility that the observed degeneracy is an artifact of an ill-posed loss rather than an intrinsic feature of physics-only training.
The ablation isolating the FVM-PINN loss contribution (~23% velocity L2 reduction in the sparse-data regime) does not specify whether the network architecture, optimizer, or regularization were held identical between the physics-only and hybrid cases; any unstated differences could confound attribution of the gain to the finite-volume term.

minor comments (2)

The abstract and results mention time-window decomposition with progressive initial-condition handoff, but the precise mechanism for transferring states across windows and its effect on long-time stability is not detailed enough for reproduction.
Notation distinguishing the FVM residual term from the data-guidance term in the composite loss should be introduced explicitly to avoid ambiguity when comparing physics-only versus hybrid training.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and for recognizing the potential significance of the data-guided FVM-PINN approach for hyperbolic systems. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and verifications.

read point-by-point responses

Referee: The central claim that physics-only FVM-PINN collapses due to a shallow 7x loss basin (while sparse data yields 310x separation and 22x error reduction) rests on the Roe Riemann-solver FVM loss being a faithfully differentiable discretization that can be stably back-propagated through unstructured meshes. No explicit numerical checks for gradient accuracy, handling of sonic points/limiters, or discrete conservation errors are described, raising the possibility that the observed degeneracy is an artifact of an ill-posed loss rather than an intrinsic feature of physics-only training.

Authors: We agree that explicit verification of the FVM loss differentiability and numerical properties would strengthen the central claim. In the revised manuscript we will add a dedicated subsection with: (i) finite-difference gradient checks on a 1D Riemann problem to quantify back-propagation accuracy, (ii) monitoring of discrete mass and momentum conservation errors during training (shown to remain at machine precision), and (iii) explicit confirmation that the Roe solver includes the standard entropy fix for sonic points together with the minmod limiter. These additions will demonstrate that the loss is well-posed and that the observed collapse arises from the shallow basin rather than discretization artifacts. revision: yes
Referee: The ablation isolating the FVM-PINN loss contribution (~23% velocity L2 reduction in the sparse-data regime) does not specify whether the network architecture, optimizer, or regularization were held identical between the physics-only and hybrid cases; any unstated differences could confound attribution of the gain to the finite-volume term.

Authors: The ablation experiments used identical network architectures (same depth and width), optimizer (Adam with the same learning-rate schedule), and regularization (identical weight decay) in all cases; the only controlled difference was the presence or absence of the FVM loss term. To remove any ambiguity we will revise the ablation description in Section 4.2 to state this explicitly and add a supplementary table listing the shared hyperparameters. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical results rest on training outcomes and ablations

full rationale

The paper presents an empirical framework replacing strong-form PINN residuals with a differentiable well-balanced Roe FVM loss on unstructured meshes, then reports training collapses under physics-only conditions and quantitative error reductions from sparse data (22x velocity L2 drop with 200 measurements, 23% FVM contribution in sparse regime). No derivation chain exists that reduces by construction to fitted inputs, self-citations, or ansatzes; claims are supported by loss-landscape diagnostics, controlled ablations on the block-in-channel benchmark, and real-world Savannah River results. The differentiability assumption on the Roe solver is stated as a prerequisite rather than derived from the target result, leaving the central findings externally falsifiable via the reported experiments.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard domain assumptions from finite-volume methods for hyperbolic conservation laws; no free parameters or invented entities are introduced beyond typical neural-network hyperparameters.

axioms (1)

domain assumption A well-balanced Roe Riemann solver provides a differentiable and accurate discretization of the shallow water equations on unstructured meshes
Invoked as the basis for the replacement loss function.

pith-pipeline@v0.9.0 · 5637 in / 1317 out tokens · 71314 ms · 2026-05-13T05:52:33.592649+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The major finding is that physics-only FVM-PINN training often fails... the FVM-PINN loss at zero momentum is only about 7× larger than at the trained solution... adding even sparse data turns this into a 310× separation
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

a differentiable, well-balanced Roe Riemann-solver finite-volume (FVM) loss evaluated on unstructured meshes

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Reference graph

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