Pith. sign in

REVIEW 3 major objections 6 minor 23 references

Physics-Informed Neural Networks for Solving the Two-Dimensional Shallow Water Equations with Terrain Topography and Rainfall Source Terms

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper argues that physics-informed neural networks can solve the two-dimensional shallow water equations with terrain and rainfall, and that the hybrid variable-conservation form is a more reliable choice than either the conservative…

desk verdict A useful empirical benchmark suite for PINNs on 2D SWE with rainfall and topography, but the claimed theoretical and empirical superiority of the variable-conservation form is not established. read the letter →

arxiv 2501.11372 v1 pith:DZNBRBRO submitted 2025-01-20 physics.flu-dyn

Yongfu Tian , Shan Ding , Guofeng Su , Lida Huang , Jianguo Chen This is my paper · ORCID
classification physics.flu-dyn
keywords physics-informedneuralnetworksshallowwaterequationsrainfallsourcetermsterraintopographywell-balancedschemevariable-conservationformentropyconditionfloodsimulation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Rainstorm floods are usually simulated with finite-volume or finite-difference codes; this paper asks whether physics-informed neural networks (PINNs) can do the same job as an unsupervised, mesh-free alternative. The authors seek to establish that PINNs, helped by a dimensional transformation of the equations and neuron-wise locally adaptive activation functions, can handle two-dimensional shallow water flow over uneven terrain with rainfall source terms. Through eleven test cases, they argue that the shallow water equations should be embedded in a hybrid variable-conservation form, which automatically weights mass and momentum residuals and gives better accuracy and stability than the primitive-variable form. They also report that adding an energy-conservation/entropy penalty does not improve results and can cause training to fail. A sympathetic reader would care because these findings give concrete guidance on how to build PINN flood models and where the method's accuracy currently stands.

What carries the argument

The load-bearing object is the residual-coupling matrix $A$ that turns primitive-variable residuals into variable-conservation residuals; the paper shows that the gradient dynamics of the hybrid form descend in characteristic coordinates with weights given by the eigenvalues of $A^T A$, approximately $(1,h^2,h^2)$ at rest, so deeper water automatically receives stronger momentum penalties. This self-weighting is the mechanism claimed to make the hybrid form more stable and less prone to local minima. The other machinery is N-LAAF, the Neuron-wise Local Adaptive Activation Function, which gives each neuron a learnable slope plus a gradient-recovery term, and a dimensional transformation used to rescale variables in the rain cases.

What would settle it

Reproduce case dDB8 (pseudo-2D dam break) with identical architecture, seed, and 30,000 training steps but swap the variable-conservation loss for the primitive-variables loss; if the primitive-variables version matches or beats the hybrid version, the paper's central claim about equation forms would be falsified. A second check: remove the undocumented dimensional transformation from cases sFR4 through sTR7 and keep all other settings unchanged; if accuracy does not degrade, the transformation is not load-bearing.

Watch

Extended reading notes

Core claim

The paper's central claim is that, for embedding the two-dimensional shallow water equations in a physics-informed neural network, the hybrid variable-conservation form is the right choice. In that form the residuals are $R = A r$, where $r$ are the primitive-variable residuals and $A$ is the $3\times 3$ matrix with rows $(1,0,0)$, $(u,h,0)$, and $(v,0,h)$; the paper shows that the resulting loss automatically changes into characteristic coordinates and weights each component by the eigenvalues of $A^T A$, approximately $(1,h^2,h^2)$ near static water. Because of this automatic pointwise weighting, the hybrid form is argued to avoid the local minima and instability that plague the primitive-variable form, and in the pseudo-2D dam-break benchmark it matches the exact Riemann solution more closely than a first-order HLL reference. The paper also claims that PINNs with this form, together with a dimensional transformation and N-LAAF activation, solve static water over nontrivial terrain to about $10^{-4}$ m, reproduce rainfall-driven water-level rise to millimeter accuracy, and handle circular dam breaks and tidal flows with and without rain, while adding an entropy-stability penalty gives no real benefit and can over-smooth shocks into training failure.

Load-bearing premise

The load-bearing premise is that the dimensional transformation applied to the equations and the per-case loss weights are part of a general recipe rather than tuned to each benchmark; the paper never defines the transformation or gives a rule for choosing the case-specific differential-equation weights, so if those choices were hand-tuned, the reported accuracies and the comparison between equation forms would not transfer automatically.

Editorial extensions

If this is right

  • Static water over bump, depression, and tidal terrain stays well balanced: $h+z$ remains at $0.3$ m and $u,v$ are near zero, with errors between $10^{-5}$ and $10^{-4}$ m.
  • Rainfall cases show that cumulative water-level rise tracks cumulative rainfall closely, with errors mostly in the $10^{-3}$ to $10^{-5}$ cm range, and the error does not accumulate over time the way time-stepping numerical errors do.
  • For the pseudo-2D dam break, the variable-conservation form is more accurate than the primitive-variable form and is competitive with a first-order HLL numerical solution, while the primitive form captures wave trends but with large velocity errors.
  • Adding the entropy condition is not recommended: in static cases it only slightly reduces micro-oscillations, and in the dam-break case it over-smooths the solution and leads to training failure.
  • Initial-condition accuracy sets the error floor: the approximation error at later times is generally no smaller than the initial-condition error, so high weights on initial conditions and better initial fitting are essential.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference: The matrix-$A$ weighting argument is not limited to shallow water; any hyperbolic system written in quasi-linear form inherits a similar $A$ and could benefit from a hybrid residual form, so the paper's main idea is portable.
  • Inference: A natural blind test is to apply the same recipe to a new watershed with a different terrain function and a multi-hour rainstorm; the paper's cases run from 0.5 seconds to 5 minutes, so long-horizon accuracy and stability remain open.
  • Inference: The negative result on the entropy condition is specific to adding it as a soft ReLU-penalized loss; enforcing energy balance through architecture or as a hard constraint might behave differently, so the conclusion should not be over-generalized.
  • Inference: Since error does not accumulate in time and initial-condition error seems to bound later error, periodic data-assimilation retraining at early times could be a promising route to longer flood forecasts.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper investigates the use of physics-informed neural networks (PINNs) for solving the two-dimensional shallow water equations (SWE) with terrain topography and rainfall source terms. The authors combine PINNs with a dimensional transformation technique and N-LAAF, and they introduce several loss components for non-negative water depth, dry-wet conditions, and an entropy stability condition. The manuscript reports eleven case studies: three static water cases on bump, depression, and tidal topographies; four static cases with rainfall; and four dynamic cases (pseudo-2D dam break, circular dam break, tidal flow, and tidal flow with rainfall). Errors are measured against exact solutions or high-resolution numerical references (HLL and entropy-stable schemes). The paper additionally argues, through a gradient-based theoretical analysis in Section 2.3 and the pseudo-2D dam break experiment, that a hybrid variable-conservation form of the SWE is preferable to the variables form and the conservative form. It also reports that adding an entropy condition does not improve accuracy and can cause training failure. An open-source module on the PINNacle platform is released.

Significance. If the central feasibility claim holds, this paper is a useful contribution to the growing literature on PINNs for 2D free-surface flows, especially because it includes rainfall and complex topography, which are absent from most prior PINN-SWE studies. Strengths include the breadth of benchmarks (eleven cases), the use of external numerical references (exact Riemann solution, HLL, ES1) rather than self-comparison, the consistently small reported MAE/RMSE in the static cases, and the open-source code with a data availability statement. However, the paper's headline claim of form superiority is not currently supported: the theoretical argument in Section 2.3 contains a logical error, and the only experimental form comparison (dDB8) omits the conservative form entirely. The undefined 'dimensional transformation technique' also prevents replication of the successful static rainfall cases. The conclusion about the entropy condition is interesting but rests on a limited set of experiments. Overall, the feasibility result is credible, but the main comparative and theoretical claims require substantial revision before the paper can be accepted.

major comments (3)
  1. [Section 2.3, Eqs. (40)-(42)] The inference "Considering that r is not necessarily 0, this implies ∇θ*r = 0" is incorrect: the necessary condition (∇θ*r)^T r = 0 is a sum of three scalar products and does not force the Jacobian ∇θ*r to vanish. Consequently, the deduction that θ* is not a minimizer of the variable-conservation loss (Eq. 42) does not follow. Even if the two loss functions had different local minima, that would not imply that the variable-conservation form is 'less prone to getting trapped in local minima' without further properties of the loss landscape. Thus the theoretical support for the superiority claim in the abstract and in Section 2.3 is not established.
  2. [Section 4.3, Figure 3] The experimental comparison of equation forms in case dDB8 includes only the primitive variables form, the variable-conservation form, and the variable-conservation form with entropy stability. The conservative form is never run, despite being named in the abstract and Conclusion as one of the two alternatives. The only evidence against the conservative form is the unreported 'preliminary experiments' mentioned in Section 2.3. To support the claim that the variable-conservation form is superior to both the variables form and the conservative form, the authors should either run the conservative form in the same benchmark or substantially weaken the claim.
  3. [Section 3.2 and Section 3.4] The 'dimensional transformation technique' is stated to be used in cases sFR4, sBR5, sDR6, and sTR7 (Section 3.2) and is advertised in the abstract as part of the method, but its definition is never given. Without specifying the transformation of coordinates or variables and the inverse transformation, the experiments cannot be reproduced and the claimed benefit of this technique cannot be assessed. Please provide the exact transformation formulas and state which cases employ them.
minor comments (6)
  1. [Table 2 caption] The caption lists the four static rainfall cases as 'sFR5, sBR5, sDR6, and sTR7', but the row label in the table and Section 3.2 use sFR4. Please correct the case numbering for consistency.
  2. [References] The reference for PINNacle (Zhu et al., 2023) has a garbled author list ('Zhongkai Hao Zhu' and a lone 'Jun'). Please correct the author names and citation format.
  3. [Section 2.3, notation] The subsection uses F(uθ) for the residual after defining r and R; please clarify the relationship between F, r, and R to avoid confusion.
  4. [Equation (24)] The rainfall source term in the entropy inequality is written as g(h+b)p, but b is not defined anywhere. It should presumably be the topography z (or b should be defined explicitly).
  5. [Section 3.4] The loss weights are case-specific (e.g., 1,10,10,1 for static cases; 10,1,1 for circular dam break and tidal cases; 1,1,1 for dDB8), but no sensitivity analysis is provided. A brief discussion of robustness to these choices would strengthen the reproducibility of the approach.
  6. [Section 4.3, dDB8 discussion] The statement that the variable-conservation form PINN 'generally outperforms' the HLL scheme is not quantified. Reporting the MAE/RMSE of both the PINN and the HLL solution against the exact solution would substantiate this comparison.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the feasibility claims are checked against external benchmarks (exact Riemann solutions, HLL, ES1), and the theoretical form-comparison argument, although flawed, is not a circular reduction.

full rationale

The paper's central feasibility claim—that PINNs enhanced by dimensional transformation and N-LAAF can solve 2D shallow water equations with terrain and rainfall—is validated against external references: the pseudo-2D dam break is compared with an exact Riemann solution (Section 3.3, dDB8), the circular dam break with an HLL numerical solution (Section 3.3, dCDB9), and the tidal cases with the entropy-stable ES1 scheme (Section 3.3, dT10 and dTR11). These are independent benchmarks, so the predictions do not reduce to fitted inputs or to the paper's own assumptions. The form-selection claim in Section 2.3 contains a genuine logical error: from the necessary condition (∇_{θ*}r)^T r = 0, the paper infers ∇_{θ*}r = 0, which is false in general; this invalidates the theoretical argument that local minima of the two forms differ. That is a correctness defect, not circularity. Likewise, the paper never runs the conservative form in the dDB8 comparison despite claiming superiority over it, and the 'dimensional transformation' is never explicitly defined for the rainfall cases; these are reproducibility and evidentiary gaps, but no equation is shown to be equivalent to its own input by construction. The per-case loss weights are hand-chosen hyperparameters, not parameters fitted to the benchmark outputs and then relabeled as predictions. There is no load-bearing self-citation: the cited techniques (N-LAAF, Dazzi's augmented system, PINNacle) are external prior work. Accordingly, no circular step can be exhibited with the required specificity, and the honest finding is no significant circularity.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central feasibility claim rests on standard PINN approximation assumptions and on the accuracy of HLL/ES1 benchmarks; the form-superiority claim additionally relies on two ad hoc optimization assumptions in Section 2.3. No new physical entities are introduced. The main free choices are the per-case loss weights and the unspecified dimensional transformation.

free parameters (2)
  • Case-specific loss weights (omega_e, omega_I, omega_B, omega_En, omega_p, omega_d) = Static cases: 1/10/10/1; dynamic cases: 10/1/1 (plus initial-condition weight 100)
    Chosen by hand per case in Section 3.4; they directly control the balance between PDE residual, initial/boundary conditions, entropy condition, and dry-wet constraints, and are not derived from the theory.
  • Dimensional transformation scaling factors = Not specified
    Each rainfall/static and dynamic case is said to use 'the dimensional transformation technique' (Sections 3.2 and 3.4), but no formula or values are given; any scaling information is a hidden free choice affecting trainability and accuracy.
assumptions (6)
  • standard math Universal approximation theorem: a sufficiently wide neural network can approximate the smooth solution of the SWE.
    Invoked in Section 2.1 to justify network expressiveness; accepted background result.
  • domain assumption The shallow water equations are an adequate model for the flow, i.e., inviscid, hydrostatic-pressure, depth-averaged assumptions hold.
    Section 2.2 states SWE are derived from incompressible Navier-Stokes under these assumptions; all cases inherit this modeling choice.
  • domain assumption The HLL and ES1 numerical solutions used as benchmarks are accurate enough to stand in for the exact solution.
    Section 3.3 sets fine grids and low CFL numbers to make benchmarks 'sufficiently accurate', but the grade of accuracy is not quantified.
  • ad hoc to paper After initial training, the term r^T (nabla_theta A)^T A r in the gradient of the variable-conservation loss is a negligible higher-order term.
    Section 2.3 uses this to justify analyzing only the main gradient term; it is an unproved assumption about the training trajectory.
  • ad hoc to paper A difference in local minima between two loss functions implies the reweighted loss is less prone to getting trapped in local minima.
    Section 2.3 draws this inference from Eq. (42); it is not a theorem and is load-bearing for the claimed superiority of the variable-conservation form.
  • domain assumption Rainfall only increases water depth and carries negligible kinetic energy.
    Used in Section 2.2 to write the entropy inequality with rainfall source term -g(h+b)p; conventional for this rain-source parameterization.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Physics-Informed Neural Networks for Solving the Two-Dimensional Shallow Water Equations with Terrain Topography and Rainfall Source Terms." pith.science (2026). https://pith.science/paper/DZNBRBRO

@misc{pith2026250111372,
  author       = {Pith},
  title        = {Pith review of: Physics-Informed Neural Networks for Solving the Two-Dimensional Shallow Water Equations with Terrain Topography and Rainfall Source Terms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DZNBRBRO}},
  note         = {Machine review of arXiv:2501.11372}
}
read the original abstract

Solving the two-dimensional shallow water equations is a fundamental problem in flood simulation technology. In recent years, physics-informed neural networks (PINNs) have emerged as a novel methodology for addressing this problem. Given their advantages in parallel computing, the potential for data assimilation and parameter calibration, and the rapid advancement of artificial intelligence, it is crucial to investigate both the capabilities and limitations of PINNs. While current research has demonstrated the significant potential of PINNs, many aspects of this new approach remain to be explored. In this study, we employ PINNs enhanced by dimensional transformation and N-LAAF techniques to validate their effectiveness in solving two-dimensional free surface flow with rainfall on terrain topography. The shallow water equations primarily exist in two forms: the variables form and the conservative form. Through theoretical analysis and experimental validation, we demonstrate that a hybrid variable-conservation form offers superior performance. Additionally, we find that incorporating the energy conservation law, specifically the entropy condition, does not yield substantial improvements and may even lead to training failure. Furthermore, we have developed an open-source module on the PINNacle platform for solving shallow water equations using PINNs, which includes over ten case studies and various equation forms, to promote research and application in this field.

Figures

Figures reproduced from arXiv: 2501.11372 by the authors.

Figure 1
Figure 1. The static tidal case sT3. (a) and (e) water level, (b) and (f) water depth, (c) and (g) flow [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Comparison of cumulative water depth growth and cumulative rainfall for (a) sFR4, (b) [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. The pseudo 2D dam break case dDB8. This figure compares the approximate solutions of [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: 2D circular dam-break case dCDB9. Water depth and velocity at 0.0s, 0.4s, and 0.8s. And [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: Tidal problem case dT10. Water depth, velocity [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Tidal problem case dTR11. Water depth, water level, velocity [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 12 canonical work pages

  1. [1]

    Audusse, F

    E. Audusse, F. Bouchut, M. O. Bristeau, R. Klein, and B. Perthame. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. Siam Journal on Scientific Computing, 25(6):2050–2065,

    work page

  2. [4]

    doi: 10.1186/ s40323-022-00226-8

    ISSN 2213-7467. doi: 10.1186/ s40323-022-00226-8. URL https://doi.org/10.1186/s40323-022-00226-8 . Jian Chen, Yaowei Li, and Changhui Zhang. The effect of design rainfall patterns on urban flooding based on the chicago method. International Journal of Environmental Research and Public Health, 20(5):4245,

    work page doi:10.1186/s40323-022-00226-8

  3. [10]

    doi: ARTN2020033410.1098/rspa.2020.0334

    ISSN 1364-5021. doi: ARTN2020033410.1098/rspa.2020.0334. URL <GotoISI>://WOS:000553098000001 . E. Kharazmi, Z. Q. Zhang, and G. E. M. Karniadakis. -vpinns: Variational physics-informed neural networks with domain decomposition. Computer Methods in Applied Mechanics and Engineering, 374,

    work page arXiv 2020

  4. [13]

    URL <GotoISI>://WOS: 000779208200258

    doi: 10.1109/Icmla52953.2021.00266. URL <GotoISI>://WOS: 000779208200258. Randall J. LeVeque. Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Ap- plied Mathematics. Cambridge University Press, Cambridge,

    arXiv 2021

  5. [15]

    doi: ARTN10547810.1016/j

    ISSN 1364-8152. doi: ARTN10547810.1016/j. envsoft.2022.105478. URL <GotoISI>://WOS:000848653100008 . X. Qi, G. A. M. de Almeida, and S. Maldonado. Physics-informed neural networks for solving flow problems modeled by the 2d shallow water equations without labeled data. Journal of Hydrology, 636,

    arXiv 2022

  6. [16]

    doi: 10.1016/j.jhydrol.2024.131263

    ISSN 0022-1694. doi: 10.1016/j.jhydrol.2024.131263. URL <GotoISI>://WOS: 001241443100001. M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics , 378:686–707,

    work page arXiv 2024

  7. [18]

    URL -http://www.applmathmech.cn/en/article/id/ b8951eb1-be11-4cb5-83a9-8d54dc2fa5f4

    doi: -10.21656/1000-0887.420063. URL -http://www.applmathmech.cn/en/article/id/ b8951eb1-be11-4cb5-83a9-8d54dc2fa5f4 . Eleuterio F. Toro. The HLL and HLLC Riemann Solvers, pages 293–311. Springer Berlin Heidelberg, Berlin, Heidelberg,

    work page arXiv

  8. [20]

    doi: ARTN11076810

    ISSN 0021-9991. doi: ARTN11076810. 1016/j.jcp.2021.110768. URL <GotoISI>://WOS:000723567300001 . Chenxi Wu, Min Zhu, Qinyang Tan, Yadhu Kartha, and Lu Lu. A comprehensive study of non- adaptive and residual-based adaptive sampling for physics-informed neural networks. Com- puter Methods in Applied Mechanics and Engineering , 403:115671,

    arXiv 2021

Show all 23 references
  1. [23]

    URL https://arxiv.org/abs/2306.08827. 24

    arXiv

  2. [1397]

    URL <GotoISI>://WOS:000949529900008

    doi: 10.1029/2022wr033168. URL <GotoISI>://WOS:000949529900008 . U. S. Fjordholm, S. Mishra, and E. Tadmor. Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography. Journal of Computational Physics , 230(14): 5587–5609,

    doi

  3. [1954]

    doi: DOI10.1002/cpa.3160070112

    ISSN 0010-3640. doi: DOI10.1002/cpa.3160070112. URL <GotoISI>://WOS:A1954XS80800011 . R. Leiteritz, M. Hurler, and D. Pflüger. Learning free-surface flow with physics-informed neural networks. 20th Ieee International Conference on Machine Learning and Applications (Icmla 2021)...

  4. [1997]

    doi: 10.1007/978-3-662-03490-3_10

    ISBN 978-3-662-03490-3. doi: 10.1007/978-3-662-03490-3_10. URL https://doi.org/10.1007/978-3-662-03490-3_10 . Eleuterio F. Toro. The Shallow Water Equations, pages 1–13. Springer Nature Switzerland, Cham, 2024a. ISBN 978-3-031-61395-1. doi: 10.1007/978-3-031-61395-1_1. URL htt...

  5. [2002]

    doi: DOI:10.1017/CBO9780511791253

    ISBN 9780521009249. doi: DOI:10.1017/CBO9780511791253. URL https://www.cambridge.org/core/product/ 97D5D1ACB1926DA1D4D52EAD6909E2B9. P. P. Luo, M. T. Luo, F. Y . Li, X. G. Qi, A. D. Huo, Z. H. Wang, B. He, K. Takara, D. Nover, and Y . H. Wang. Urban flood numerical simulation:...

    doi

  6. [2004]

    doi: 10.1137/S1064827503431090

    ISSN 1064-8275. doi: 10.1137/S1064827503431090. URL <GotoISI>://WOS:000224118700011 . A. Bihlo and R. O. Popovych. Physics-informed neural networks for the shallow-water equations on the sphere. Journal of Computational Physics, 456,

    doi

  7. [2005]

    doi: 10.1016/j.advwatres.2005.03.007

    ISSN 0309-1708. doi: 10.1016/j.advwatres.2005.03.007. URL <GotoISI>://WOS:000231572900010 . 22 A. D. Jagtap, K. Kawaguchi, and G. E. Karniadakis. Locally adaptive activation functions with slope recovery for deep and physics-informed neural networks. Proceedings of the Royal S...

    2005 doi

  8. [2011]

    doi: 10.1016/j.jcp.2011.03.042

    ISSN 0021-9991. doi: 10.1016/j.jcp.2011.03.042. URL <GotoISI>://WOS: 000291374000007. Moritz Hauck, Vadym Aizinger, Florian Frank, Hennes Hajduk, and Andreas Rupp. Enriched galerkin method for the shallow-water equations. GEM - International Journal on Geomathematics, 11(1): 31,

    2011 doi

  9. [2019]

    doi: 10.1016/j.jcp.2018.10.045

    ISSN 0021-9991. doi: 10.1016/j.jcp.2018.10.045. URL <GotoISI>://WOS:000453776000028 . D. E. Rumelhart, G. E. Hinton, and R. J. Williams. Learning representations by back-propagating errors. Nature, 323(6088):533–536,

    2018 doi

  10. [2020]

    doi: 10.1007/s13137-020-00167-7

    ISSN 1869-2680. doi: 10.1007/s13137-020-00167-7. URL https://doi.org/10. 1007/s13137-020-00167-7 . N. M. Hunter, M. S. Horritt, P. D. Bates, M. D. Wilson, and M. G. F. Werner. An adaptive time step solution for raster-based storage cell modelling of floodplain inundation. Adva...

    doi

  11. [2021]

    doi: ARTN11354710.1016/j.cma.2020.113547

    ISSN 0045-7825. doi: ARTN11354710.1016/j.cma.2020.113547. URL <GotoISI>://WOS:000608741800007 . A. S. Krishnapriyan, A. Gholami, S. D. Zhe, R. M. Kirby, and M. W. Mahoney. Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information...

    2020

  12. [2022]

    doi: ARTN11102410

    ISSN 0021-9991. doi: ARTN11102410. 1016/j.jcp.2022.111024. URL <GotoISI>://WOS:000762445000001 . Sebastián Cedillo, Ana-Gabriela Núñez, Esteban Sánchez-Cordero, Luis Timbe, Esteban Samaniego, and Andrés Alvarado. Physics-informed neural network water surface predictability for...

    2022

  13. [2023]

    doi: https://doi.org/10.1016/j.cma.2022.115671

    ISSN 0045-7825. doi: https://doi.org/10.1016/j.cma.2022.115671. URL https://www.sciencedirect.com/ science/article/pii/S0045782522006260. Z. D. Yin, J. M. Shi, L. L. Bian, W. H. Campbell, S. R. Zanje, B. C. Hu, and A. S. Leon. Physics- informed neural network approach for solv...

    2022

  14. [2024]

    doi: ARTNe2023WR03658910.1029/2023WR036589

    ISSN 0043-1397. doi: ARTNe2023WR03658910.1029/2023WR036589. URL <GotoISI>://WOS:001335841600001 . D. Y . Feng, Z. L. Tan, and Q. Z. He. Physics-informed neural networks of the saint-venant equations for downscaling a large-scale river model. Water Resources Research, 59(2),

  15. [2025]

    doi: Artn0402406010.1061/Jhend8.Hyeng-13572

    ISSN 0733-9429. doi: Artn0402406010.1061/Jhend8.Hyeng-13572. URL <GotoISI>://WOS:001365144800013 . Zhongkai Hao Zhu, Jiachen Yao, Chang Su, Hang Su, Ziao Wang, Fanzhi Lu, Zeyu Xia, Yichi Zhang, Songming Liu, Lu Lu, and Jun. Pinnacle: A comprehensive benchmark of physics- infor...

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.