Estimating bottom topography in shallow water flows

Lucas Pancotto; Patricio Clark Di Leoni

arxiv: 2604.07075 · v1 · submitted 2026-04-08 · ⚛️ physics.flu-dyn · physics.geo-ph

Estimating bottom topography in shallow water flows

Lucas Pancotto , Patricio Clark Di Leoni This is my paper

Pith reviewed 2026-05-10 18:06 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn physics.geo-ph

keywords shallow water flowsbottom topographyinverse problemsphysics-informed neural networksadjoint state methodsurface measurementsvelocity reconstruction

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The pith

Two numerical methods recover bottom topography and surface velocity from surface deformation measurements in shallow water flows.

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

The paper develops and tests two approaches to infer the underwater bottom shape in shallow water flows when only surface height changes are observed. One approach trains physics-informed neural networks to satisfy the shallow-water equations while matching the surface data. The other uses the adjoint state method to compute gradients for an optimization problem that adjusts the bottom to fit the same data. Both recover the topography and the velocity field on synthetic one- and two-dimensional cases, and both tolerate moderate noise and reduced numbers of measurement points.

Core claim

Both the PINN-based inversion and the adjoint-state inversion successfully reconstruct the bottom topography and the surface velocity field from surface deformation data generated by the shallow-water equations, and both remain accurate when the data contain added noise or are spatially or temporally sparse.

What carries the argument

Inversion frameworks that minimize the mismatch between observed surface deformations and predictions from the shallow-water equations, implemented once via physics-informed neural networks and once via the adjoint state method.

If this is right

The methods apply to both one-dimensional and two-dimensional shallow-water problems.
Surface velocity is recovered together with bottom topography without additional measurements.
Moderate levels of measurement noise and data sparsity do not prevent successful reconstruction.
The same surface data suffice for both neural-network and adjoint-based reconstructions.

Where Pith is reading between the lines

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

If the shallow-water approximation holds for a real flow, these techniques could enable mapping of river or coastal bottoms using only drone or satellite surface observations.
The adjoint method may scale more favorably than PINNs when the domain size or resolution increases substantially.
Hybrid use of the two methods on the same dataset could provide cross-validation of the recovered topography.

Load-bearing premise

The shallow-water equations must accurately describe the real flow, and any test data must be generated from the identical forward model used inside the inversion.

What would settle it

An independent field survey that measures the true bottom topography while simultaneously recording surface deformations; large systematic differences between the surveyed bottom and the bottom recovered by either method would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.07075 by Lucas Pancotto, Patricio Clark Di Leoni.

**Figure 2.** Figure 2: Bottom profile hb(x) used in the 1D experiments. The Adjoint State Method (ASM) is a variational approach to tackle inverse problems (Dimet & Talagrand, 1986; Zaki, 2025). A cost function for the available field measurements is minimized subject to the physical equations of the problem. A Lagrangian is formulated that includes both the cost function from data to be extremized and the differential operato… view at source ↗

**Figure 3.** Figure 3: Diagram of 2D flow studied at t/T = 0.5. The blue surface depicts ground truth of field h0 + η/η0, and the diverging red and blue colored surface depicts ground truth of hb/h0. The arrows indicated the direction in which the wave is propagating. rest measured from the lowest point, one can derive the Shallow Water (SW) equations (Pedlosky, 1987) that describe the dynamics of the free surface ∂u ∂t + (u · ∇… view at source ↗

**Figure 4.** Figure 4: Reconstructed bottom topographies hb for the cases with distance between measurements of (a) δx = 3/1024 and (b) δx = 3/8. The true hb is marked with black solid line, the green dashed line denotes the ASM results h˜ b and the red solid line denote PINN results hˆ b. 3.2 PINN parameters The networks were trained using Adam with a mini-batch size of mb = 2000, and a learning rate of µ = 1×10−6 for the 1D fl… view at source ↗

**Figure 5.** Figure 5: Fourier spectra Ehb of the ground truth (black solid line), the ASM reconstruction (green dashed line), and the PINN reconstruction (red solid line). Each panel corresponds to cases with different separation between measurements: (a) δx = 3/1024, (b) δx = 3/64, (c) δx = 3/16 and (d) δx = 3/8. The blue arrow indicates the wavelength corresponding to separation δx in each case. eters A, B, x0, σ in B + A e−… view at source ↗

**Figure 6.** Figure 6: Dense measurement case, δ = 3/1024. (a-d): True and reconstructed velocity fields u/c at different times. (e-h): True and reconstructed surface fluctuations η/η0. Same legends as figure 4. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 x/L 0 2.00×10−5 4.00×10−5 6.00×10−5 8.00×10−5 |∂tu ˆ+u ˆ∂xu ˆ+g∂xh ˆ| [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

**Figure 7.** Figure 7: Absolute value of the SW momentum equation evaluated on PINN predicted [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗

**Figure 8.** Figure 8: Sparse measurement case, δ = 3/8. (a-d): true and reconstructed velocity fields u/c at different times. (e-h): true and reconstructed surface fluctuations η/η0. Same legends as figure 4. ted against the true hb field (black solid line) for (a) δx = 3/1024 and (b) δx = 3/8. Both methods can reconstruct the overall shape of the bottom topography with good accuracy, even in the sparser case. The PINN produce… view at source ↗

**Figure 9.** Figure 9: Errors as function of the measurement distance [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗

**Figure 10.** Figure 10: Reconstructed bottom topographies hb for the cases with noise amplitudes (a) ϵ/η0 = 2 × 10−3 and (b) ϵ/η0 = 1 × 10−1 . The true hb is marked with black solid line, the green dashed line denotes the ASM results h˜ b and the red solid line denote PINN results hˆ b. The blue shade represents the standard deviation between different PINN realizations, for the same data points. achieves better results than PIN… view at source ↗

**Figure 11.** Figure 11: Errors as function of the added noise ϵ/η0. (a) Global errors in the topography reconstruction Ehb , and (b) global errors in the velocity reconstruction Eu. The errors for the ASM are marked with the green dashed line, while the errors for the PINN are marked with the red solid line. −1.00×10−4 0 1.00×10−4 2.00×10−4 u / c (a) t =0.0T Ground truth ASM PINN (b) t =0.25T (c) t =0.5T (d) t =0.75T 0 1 2 3 x/… view at source ↗

**Figure 12.** Figure 12: Case with noise amplitude ϵ/η0 = 0.02 (a-d): True and reconstructed velocity fields u/c at different times. (e-h): True and reconstructed surface fluctuations η/η0. Same legends as figure 4. –18– [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗

**Figure 13.** Figure 13: Reconstructed bottom topographies hb using velocity measurements instead of height measurements. The true hb is marked with black solid line, the green dashed line denotes the ASM results h˜ b and the red solid line denote PINN results hˆ b. −1.00×10−4 0 1.00×10−4 2.00×10−4 u / c (a) t =0.0T Ground truth ASM PINN (b) t =0.25T (c) t =0.5T (d) t =0.75T 0 1 2 3 x/L 0.0 0.2 0.4 0.6 0.8 1.0 η / η 0 (e) 0 1 2 3… view at source ↗

**Figure 14.** Figure 14: Case with velocity information. (a-d): True and reconstructed velocity fields [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗

**Figure 15.** Figure 15: Surface plot of field hb/h0 for (a) ground truth and (b) prediction using PINN method. The global error produced by predictions from both methods for each amplitude of noise added to data is plotted in figure 11, for (a) Ehb and (b) Eu [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗

**Figure 16.** Figure 16: (a) Field hb/h0 for prediction using PINN method and (b) the error of the PINN prediction. 0 1 2 3 4 x/L 0 1 2 3 4 y / L (a) 0 1 2 3 4 x/L (b) 0.00 0.01 0.02 0.03 0.04 u/c 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 √ (u−uˆ)2 max x,y,t (√ u 2) [PITH_FULL_IMAGE:figures/full_fig_p021_16.png] view at source ↗

**Figure 17.** Figure 17: (a) Field u/c for prediction using PINN method and (b) the error of the PINN prediction. 4.4 Two-dimensional case In this subsection we discuss the results of data assimilation, using PINNs, for the 2D case. The recovery of fields hb, u, and v via assimilation of surface measurements of field h took approximately 3 days running on a single Nvidia T4. In general, hb was recovered satisfactorily, as was fi… view at source ↗

**Figure 18.** Figure 18: (a) Field v/c for prediction using PINN method and (b) the error of the PINN prediction. 0 1 2 3 4 x/L 0 1 2 3 4 y / L (a) 0 1 2 3 4 x/L (b) 0.0 0.2 0.4 0.6 0.8 1.0 η/η0 0.000 0.001 0.002 0.003 0.004 √ (η−ηˆ)2 max x,y,t (√ η2) [PITH_FULL_IMAGE:figures/full_fig_p022_18.png] view at source ↗

**Figure 19.** Figure 19: (a) Field η/η0 for prediction using PINN method and (b) the error of the PINN prediction. u is of one order of magnitude less than that of hb. For field v, we also observe a higher relative error than u [PITH_FULL_IMAGE:figures/full_fig_p022_19.png] view at source ↗

**Figure 20.** Figure 20: (a-d): True and reconstructed velocity fields [PITH_FULL_IMAGE:figures/full_fig_p023_20.png] view at source ↗

read the original abstract

We present two methods to estimate bottom topography in a shallow water flow using only surface deformation measurements. One is based on Physics-Informed Neural Networks (PINNs) and the other on the Adjoint State Method. We test both methods using synthetic data in 1D and 2D cases. Both are able to successfully reconstruct not only the bottom topography but also the surface velocity. Both also show robustness against noise and data sparsity up to reasonable levels.

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

PINNs and adjoint methods recover bottom topography in matched synthetic shallow-water tests, but real-world robustness remains untested.

read the letter

This paper tests two standard techniques—physics-informed neural networks and the adjoint state method—for estimating bottom topography in shallow water flows from surface deformation data alone. In 1D and 2D synthetic cases, both recover the topography and the surface velocity, and they tolerate reasonable noise and data gaps. The concrete pairing of these solvers with the shallow-water inverse problem is new enough for the subfield. The demonstrations show that the inverse problem is tractable when the forward model matches the data generator exactly. That provides a baseline for feasibility. The work is straightforward and focused. It applies established frameworks without claiming new theory, which keeps the scope clear. The robustness checks against noise and sparsity are practical additions. The main limitation is the exclusive use of synthetic data from the same shallow-water model. Success here shows the methods work under perfect model assumptions, but it does not address mismatches that occur in real flows, such as non-hydrostatic pressure, friction, or three-dimensional effects. Without tests against a different forward model or real measurements, the practical utility stays provisional. The abstract also omits specific quantitative error metrics or convergence studies, making it difficult to assess precision. This paper suits readers in computational hydraulics and coastal modeling who need tools for bathymetry recovery. It offers a starting point for implementation rather than a broad advance. I would recommend sending it for peer review. The application is relevant and the synthetic results are a necessary first step, though the referees will likely ask for more validation and quantitative details.

Referee Report

2 major / 2 minor

Summary. The paper presents two methods—Physics-Informed Neural Networks (PINNs) and the Adjoint State Method—to estimate bottom topography in shallow water flows from surface deformation measurements alone. Both methods are tested on synthetic data for 1D and 2D cases, and the authors claim successful reconstruction of the bottom topography as well as the surface velocity, along with robustness to noise and data sparsity.

Significance. If the central claims hold under broader validation, the work offers practical numerical tools for inferring underwater topography from surface observations, with potential applications in coastal engineering and environmental monitoring. The dual-method strategy (data-driven PINN versus optimization-based adjoint) is a positive feature that enables internal cross-checking. The synthetic tests establish that the inverse problem is well-posed when the forward operator matches exactly.

major comments (2)

[Numerical experiments] Numerical experiments section: Synthetic data are generated from the identical shallow-water equations used inside both inversion procedures. This demonstrates consistency of the inverse solver but does not test robustness to model mismatch (non-hydrostatic pressure, bottom friction, or 3D effects), which is required to support the abstract's claim of applicability to real shallow-water flows.
[Abstract] Abstract and results: No quantitative error metrics (L2 norms, relative errors, or convergence rates for topography and velocity) or explicit noise/sparsity levels are supplied, making the statements of 'successful reconstruction' and 'robustness up to reasonable levels' difficult to assess objectively.

minor comments (2)

[Introduction] The introduction would benefit from additional citations to prior adjoint-based inversions for shallow-water or free-surface problems to better situate the contribution.
Notation for the surface deformation data and the recovered velocity field should be defined more explicitly at first use to improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: [Numerical experiments] Numerical experiments section: Synthetic data are generated from the identical shallow-water equations used inside both inversion procedures. This demonstrates consistency of the inverse solver but does not test robustness to model mismatch (non-hydrostatic pressure, bottom friction, or 3D effects), which is required to support the abstract's claim of applicability to real shallow-water flows.

Authors: We agree that the synthetic tests use data generated from the exact same shallow-water model employed in the inversion procedures. This verifies that both the PINN and adjoint-state methods can recover the bottom topography and surface velocity when the forward model is perfectly matched, which is a necessary first step to establish that the inverse problem is well-posed. We recognize, however, that these experiments do not directly probe robustness to model discrepancies such as non-hydrostatic pressure, bottom friction, or three-dimensional effects that can occur in real shallow-water flows. To address this, we will add a new paragraph in the Discussion section that explicitly states the modeling assumptions, discusses the potential effects of such mismatches on reconstruction accuracy, and outlines future validation steps using laboratory or field data. This revision clarifies the current scope without overstating applicability. revision: yes
Referee: [Abstract] Abstract and results: No quantitative error metrics (L2 norms, relative errors, or convergence rates for topography and velocity) or explicit noise/sparsity levels are supplied, making the statements of 'successful reconstruction' and 'robustness up to reasonable levels' difficult to assess objectively.

Authors: We acknowledge that the abstract and result descriptions currently use qualitative phrasing. Although the manuscript presents visual comparisons in the figures for different noise and sparsity cases, we agree that explicit quantitative metrics would make the performance claims more objective and easier to evaluate. In the revised manuscript we will update the abstract to report specific relative L2 errors for the reconstructed topography and velocity, and we will state the exact noise amplitudes (e.g., 1–10 % Gaussian) and data-sparsity ratios tested. We will also add a concise summary table in the Results section that compiles these error metrics across the 1D and 2D cases. These changes will allow readers to assess the reported robustness directly from the text. revision: yes

Circularity Check

0 steps flagged

No circularity: standard inverse problem solved via PINN and adjoint methods on matching synthetic data

full rationale

The paper formulates bottom-topography estimation as a standard inverse problem: surface deformation data are used to recover topography (and velocity) by minimizing a loss that enforces the shallow-water PDE residual plus data fidelity. Synthetic data are generated from the identical forward operator, which tests consistency and well-posedness of the inverse map but does not reduce any claimed result to a fitted parameter by construction. No self-definitional equations, no renaming of known results, and no load-bearing self-citations that close the derivation loop are present. The methods are externally verifiable against the known forward model; success therefore constitutes independent evidence of numerical recovery rather than tautological re-expression of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the shallow-water equations being an adequate model and on the numerical inverse solvers converging to the true bottom given surface data generated from the same model.

axioms (1)

domain assumption The flow obeys the shallow-water equations
Invoked as the forward model for both inversion techniques.

pith-pipeline@v0.9.0 · 5359 in / 1118 out tokens · 43980 ms · 2026-05-10T18:06:49.401949+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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Anagnostopoulos, S. J., Toscano, J. D., Stergiopulos, N., & Karniadakis, G. E. (2024, March).Learning in PINNs: Phase transition, total diffusion, and gen- eralization(No. arXiv:2403.18494). arXiv. doi: 10.48550/arXiv.2403.18494 Cai, S., Mao, Z., & Wang, Z. (2021). Physics-informed neural networks (pinns) for fluid mechanics: a review.Acta Mechanica Sinic...

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J., Toscano, J

Anagnostopoulos, S. J., Toscano, J. D., Stergiopulos, N., & Karniadakis, G. E. (2024, March).Learning in PINNs: Phase transition, total diffusion, and gen- eralization(No. arXiv:2403.18494). arXiv. doi: 10.48550/arXiv.2403.18494 Cai, S., Mao, Z., & Wang, Z. (2021). Physics-informed neural networks (pinns) for fluid mechanics: a review.Acta Mechanica Sinic...

work page doi:10.48550/arxiv.2403.18494 2024

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Retrieved fromhttps://doi.org/10.1140/epje/s10189-023-00276-9 doi: 10.1140/epje/s10189-023-00276-9 Cleary, A., Wang, Q., & Zaki, T. A. (2025, December).Latent-space variational data assimilation in two-dimensional turbulence(No. arXiv:2512.15470). arXiv. doi: 10.48550/arXiv.2512.15470 Cuomo, S., Di Cola, V. S., Giampaolo, F., Rozza, G., Raissi, M., & Picc...

work page doi:10.1140/epje/s10189-023-00276-9 2025

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Cuomo, V.S

Retrieved 2022-09-14, fromhttps://doi.org/10.1007/s10915-022-01939-z doi: 10.1007/s10915-022-01939-z Degueldre, H., Metzger, J. J., Geisel, T., & Fleischmann, R. (2016). Random focus- ing of tsunami waves.Nature Physics,12, 259–262. doi: https://doi.org/10 .1038/nphys3557 Dimet, F.-x. L., & Talagrand, O. (1986). Variational algorithms for analysis and ass...

work page doi:10.1007/s10915-022-01939-z 2022

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Retrieved fromhttps://journals.ametsoc.org/view/journals/ phoc/45/3/jpo-d-14-0154.1.xmldoi: 10.1175/JPO-D-14-0154.1 Hasanuzzaman, G., Eivazi, H., Merbold, S., Egbers, C., & Vinuesa, R. (2022). En- hancement of PIV measurements via physics-informed neural networks.Mea- surement Science and Technology. Retrieved fromhttp://iopscience.iop .org/article/10.108...

work page doi:10.1175/jpo-d-14-0154.1 2022

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doi: 10.3390/geosciences8020063 Ohara, Y., Moteki, D., Muramatsu, S., Hayasaka, K., & Yasuda, H. (n.d.). Physics-informed neural networks for inversion of river flow and geometry with shallow water model. ,36(10), 106633. Retrieved 2026-03-23, from https://doi.org/10.1063/5.0232852doi: 10.1063/5.0232852 Pedlosky, J. (1987).Geophysical fluid dynamics. Spri...

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Retrieved fromhttps://journals.ametsoc.org/ view/journals/clim/24/3/2010jcli3745.1.xmldoi: 10.1175/2010JCLI3745 .1 Sitzmann, V., Martel, J. N. P., Bergman, A. W., Lindell, D. B., & Wetzstein, G. (2020).Implicit neural representations with periodic activation functions. Retrieved fromhttps://arxiv.org/abs/2006.09661 Smith, W., Marks, K., & Schmitt, T. (201...

work page doi:10.1175/2010jcli3745 2020