Pith. sign in

REVIEW 4 major objections 6 minor 30 references

Finite Volume Physical Informed Neural Network (FV-PINN) with Reduced Derivative Order for Incompressible Flows

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read This paper proposes FV-PINN, which applies Gauss's theorem to convert the Navier-Stokes momentum equation into boundary flux integrals evaluated at Gaussian quadrature points, reducing the required derivative order of velocity, and…

desk verdict A useful finite-volume weak-form PINN with a real derivative-order reduction, but the written loss function is ambiguous and the reported gains need quantitative support. read the letter →

arxiv 2411.17095 v1 pith:NH22C62H submitted 2024-11-26 physics.flu-dyn

Zijie Su , Yunpu Liu , Sheng Pan , Zheng Li , Changyu Shen This is my paper · ORCID
classification physics.flu-dyn
keywords Physics-informedneuralnetworksFinitevolumemethodNavier-StokesequationsIncompressibleflowSteady-stateproblemsGaussianquadratureWeakformStreamfunction
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

The paper proposes a finite-volume physics-informed neural network (FV-PINN) for steady, incompressible, laminar flow. The key move is to rewrite the Navier-Stokes momentum equation, using Gauss's theorem, as surface flux integrals over control-volume boundaries, and to evaluate these fluxes at Gaussian quadrature points rather than computing PDE residuals at scattered collocation points. This reduces the highest derivative of the network's stream-function output from third order to second order, because the viscous term now requires only first derivatives of velocity. The paper applies FV-PINN to the pipe-bend and double-pipe benchmark problems and reports that it predicts velocity and pressure fields that agree with a commercial finite-volume solver, while outperforming a traditional residual-based PINN in accuracy, sampling efficiency, and training speed. If the method holds up, it gives a simple recipe for lowering the automatic-differentiation cost of PINNs for incompressible flow.

What carries the argument

The central object is the divergence theorem, applied term-by-term to the steady incompressible Navier-Stokes equations to convert each volume integral into a surface flux integral over the boundary of a control volume. The network is trained by assigning a loss to the sum of convective, pressure, and viscous flux integrals at Gaussian quadrature points on each cell boundary. This is the mechanism that reduces the derivative order: the viscous term becomes an integral of $\nabla\mathbf{u}$, so the network's second derivatives (of the stream function) are the highest needed, eliminating third-derivative computations. The stream-function ansatz, which automatically enforces mass conservation, works together with the flux-form loss to define the optimization problem.

What would settle it

Train FV-PINN on a case with a known analytical solution (e.g., channel flow) and record the per-cell flux residual from Eq. (16) at convergence; if the signed sum across cells is near zero but the individual cell residuals are not, the loss as written is not equivalent to enforcing the equations cell by cell. Alternatively, re-run the paper's pipe-bend case with the loss defined as the squared sum of per-cell residuals and compare the predicted pressure field; a noticeable change would confirm that the sign convention matters.

Watch

Extended reading notes

Core claim

The paper's central claim is that converting the differential Navier-Stokes equations into boundary-flux integrals and using those integrals as the PINN loss yields more accurate velocity and pressure predictions for steady incompressible flows than using pointwise residuals, while needing fewer sampling points and less training time. In the FV-PINN, the network outputs the stream function and pressure; velocity is obtained by first derivatives of the stream function, which automatically satisfies the continuity equation. Gauss's theorem turns the inertial, pressure, and viscous terms of the momentum equation into surface integrals of fluxes across each control volume, and Gaussian quadrature on the cell boundaries discretizes those integrals. The viscous flux contains only first derivatives of velocity, so the highest derivative appearing in the loss is a second derivative of the stream function, in contrast to the third derivative needed when the residual is written in strong form. On the pipe-bend and double-pipe problems, the resulting fields are reported to match the commercial solver closely and to be markedly better than those of the traditional PINN.

Load-bearing premise

The main unstated assumption is that the loss actually sums the squares of the per-cell flux residuals, so that a small total loss means each cell's momentum flux is small; if instead the implementation sums raw signed fluxes, the reported accuracy could hide large local errors through cancellation.

Editorial extensions

If this is right

  • Training a PINN for steady incompressible flow no longer requires third derivatives of the network output, which lowers the cost and error accumulation of automatic differentiation.
  • The finite-volume flux form means the loss is assembled from surface integrals over cells, so a converged solution approximately satisfies momentum conservation in integral form on each cell, not only at scattered points.
  • On the two benchmark problems, FV-PINN used about 30% fewer sampling points and roughly 40% less training time than the traditional PINN while giving better agreement with the commercial solver.
  • The same divergence-theorem reduction could be applied to the momentum equations used in density-based fluid topology optimization, where accurate pressure fields are needed at many design iterations.

Reading between the lines

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

  • The order reduction is a consequence of the weak form itself, not of the finite-volume discretization, so the same integration-by-parts step should work for other second-order PDEs solved with PINNs, replacing third derivatives with second derivatives.
  • The loss definition carries a hidden choice: if the implementation follows Eq. (16) literally and sums raw per-cell flux residuals, positive and negative residuals from neighboring cells could cancel and leave large local errors hidden in a small total. The paper does not state whether the loss is a sum of residuals or a sum of their squares, so code inspection would be needed to settle this.
  • The reported advantage over the traditional PINN is shown only for two relatively simple steady benchmarks; the claim that FV-PINN generalizes to unsteady flows or evolving fluid-solid interfaces would need separate tests, since the time derivative would reset the derivative-order accounting.
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

4 major / 6 minor

Summary. The paper proposes FV-PINN, a physics-informed neural network for steady incompressible laminar flow. Instead of evaluating strong-form Navier-Stokes residuals at collocation points, the method applies the divergence theorem to convert momentum equation residuals over finite-volume cells into boundary flux integrals, which are then evaluated at Gaussian quadrature points on cell boundaries. A stream function output enforces mass conservation, and the total loss combines a physical loss with a boundary-condition loss. The method is tested on a pipe-bend problem and a double-pipe problem and compared qualitatively with COMSOL, with one comparison against a traditional strong-form PINN. The paper claims improved prediction accuracy, faster convergence, and a 40% reduction in training time.

Significance. If the formulation is corrected and the reported gains are reproducible, the paper makes a modest but useful contribution: it demonstrates a finite-volume-style weak-form loss that reduces the automatic-differentiation derivative order for incompressible Navier-Stokes problems and can be trained with fewer sampling points. Validation against COMSOL and FEM is a positive feature, and the use of Gauss's theorem is standard. However, the contribution is incremental relative to existing weak/variational PINN literature (Refs. 15-19), and the current manuscript does not provide quantitative accuracy metrics, code, or a precise statement of the optimized loss, so the central claims are not yet substantiated.

major comments (4)
  1. [§2.4, Eq. (16) and Eq. (13)] The written physical loss is not the MSE defined in Eq. (13). Eq. (13) defines the loss as 1/N multiplied by a sum of squared residuals, but Eq. (16) defines L_G as a sum of per-cell flux integrals weighted by quadrature weights, with no square and no 1/N factor. Minimizing a raw sum can let residuals of opposite signs cancel across cells, so large local violations of the momentum equation could coexist with a small L_G. In addition, the second integrand in the quadrature sum in Eq. (16) repeats the x-momentum expression (with u1, du1/dx1, du1/dx2, p n1) instead of the y-momentum expression from Eq. (10), and the pressure flux terms omit the 1/rho factor carried through in Eq. (6). Because the training objective is the central object of the method, this must be corrected, and the exact loss actually minimized should be stated unambiguously, preferably with code or pseudocode.
  2. [§3, Figures 4 and 6] The claim that FV-PINN 'significantly improves the prediction accuracy' is not supported by quantitative evidence. Figures 4 and 6 show contour plots, and the text states that results 'closely resemble' COMSOL, but no error norms (for example, relative L2 error) are reported for velocity or pressure, and no error is reported for the traditional PINN baseline in Figure 4(c). Without quantitative comparisons, the relative accuracy claim cannot be assessed.
  3. [§3.1 and Table 2] The training-time comparison is incomplete. Table 2 gives training time and number of sampling points for FV-PINN and traditional PINN but does not report the error level achieved by each method, the final loss values, or whether both runs used the same stopping criteria. The 40% speedup is therefore not an apples-to-apples comparison; it should be presented as an accuracy-versus-training-time curve or as error at matched epochs.
  4. [§3.1, pressure-field discussion] The explanation that pressure differences arise from the arbitrary constant is not consistent with the problem setup, which specifies a zero-pressure outlet (p=0) at the bottom boundary. This Dirichlet condition fixes the pressure level, so the observed pressure mismatch should be quantified and explained rather than attributed to indeterminacy. A likely contributor is the very small BC weight beta=0.0001 in Eq. (15), which makes the pressure boundary condition only weakly enforced.
minor comments (6)
  1. [Abstract, §2.3, Table 1] The method is called both FV-PINN and FVI-PINN in different parts of the manuscript; please use one name consistently throughout.
  2. [Eqs. (9), (10), (16)] The kinematic viscosity is written as mu/rho in Eqs. (9)-(10) but as v in Eq. (16); please define nu=mu/rho and use a single symbol to avoid confusion with the velocity component u1.
  3. [Eq. (3)] The stream-function relation should be written with explicit components, u1=phi_y and u2=-phi_x, and the sign convention should be checked against the vorticity definition or the momentum equation to ensure consistency.
  4. [Section headings] The text refers to 'Chapter 2' and 'Chapter 3'; these should be 'Section 2' and 'Section 3' in a journal article.
  5. [Figure 5] The loss curves in Figure 5 are unlabeled and the axes are not described; please identify which curve corresponds to which method and state whether the plotted quantity is L_PINN or one of its components.
  6. [Reproducibility] The manuscript does not include a data or code availability statement; for a methods paper, providing the code or pseudocode for the loss computation is strongly recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: FV-PINN is a textbook weak-form reformulation benchmarked against independent COMSOL solutions.

full rationale

The claimed derivation chain is self-contained. The stream-function representation in Eq. (3) is the standard way to enforce incompressibility; Eqs. (5)-(7) apply Gauss's theorem to the inertial, pressure, and viscous terms, which are textbook identities rather than assumptions that presuppose the target result. The loss function in Eq. (16) is a discrete Gaussian-quadrature approximation of those boundary fluxes, and the validation compares against COMSOL, an external finite-volume solver, so the reported accuracy claims do not reduce to a fitted parameter or to a self-citation. The evident typographical inconsistency in Eq. (16), where the y-momentum bracket is printed with u1 instead of u2 and the squared-residual form from Eq. (13) is not reflected, is a correctness/documentation concern about the implemented objective, not a circularity. No self-citation is load-bearing, no uniqueness theorem is imported from the authors' prior work, and no prediction is equivalent by construction to an input. The paper's central content is therefore independent of any circular dependence.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The derivation rests on standard calculus identities and standard PINN training assumptions. The hand-chosen hyperparameters are not fitted to data in the circularity sense, but the absence of sensitivity analysis weakens the empirical claims. No new physical entities are postulated.

free parameters (4)
  • beta (BC loss weight) = 0.0001
    Chosen by hand to balance physics and boundary losses; no sensitivity study.
  • Learning rate = 0.003
    Adam optimizer hyperparameter; no tuning details.
  • Network architecture = [2,40,40,40,40,40,40,40,40,2]
    Fixed depth and width; no ablation.
  • Mesh and quadrature details = not reported
    Number of cells and Gaussian points per edge are not stated, so the reported 29280 sampling points cannot be reproduced.
assumptions (5)
  • standard math Divergence theorem (Gauss's theorem) converts volume integrals of convective, pressure, and viscous terms into boundary flux integrals.
    Used in Eqs. (4)-(8).
  • standard math For incompressible flow, the identity (u dot grad)u = div(u tensor u) holds when div u = 0, justifying Eq. (5).
    Incompressibility is enforced by the stream function, but the identity is invoked implicitly.
  • domain assumption Gaussian quadrature on cell boundaries gives sufficiently accurate flux integrals.
    Section 2.4; the quadrature order is not specified.
  • standard math The stream function parametrization in 2D automatically satisfies continuity.
    Eq. (3); stated in Section 2.1.
  • domain assumption Steady, laminar, incompressible Navier-Stokes is the target physics.
    Governing equations in Section 2.1.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Finite Volume Physical Informed Neural Network (FV-PINN) with Reduced Derivative Order for Incompressible Flows." pith.science (2026). https://pith.science/paper/NH22C62H

@misc{pith2026241117095,
  author       = {Pith},
  title        = {Pith review of: Finite Volume Physical Informed Neural Network (FV-PINN) with Reduced Derivative Order for Incompressible Flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NH22C62H}},
  note         = {Machine review of arXiv:2411.17095}
}
read the original abstract

Physics-Informed Neural Networks (PINN) has evolved into a powerful tool for solving partial differential equations, which has been applied to various fields such as energy, environment, en-gineering, etc. When utilizing PINN to solve partial differential equations, it is common to rely on Automatic Differentiation (AD) to compute the residuals of the governing equations. This can lead to certain precision losses, thus affecting the accuracy of the network prediction. This paper pro-poses a Finite Volume Physics-Informed Neural Network (FV-PINN), designed to address steady-state problems of incompressible flow. This method divides the solution domain into mul-tiple grids. Instead of calculating the residuals of the Navier-Stokes equations at collocation points within the grid, as is common in traditional PINNs, this approach evaluates them at Gaussian in-tegral points on the grid boundaries using Gauss's theorem. The loss function is constructed using the Gaussian integral method, and the differentiation order for velocity is reduced. To validate the effectiveness of this approach, we predict the velocity and pressure fields for two typical examples in fluid topology optimization. The results are compared with commercial software COMSOL, which indicates that FVI-PINN significantly improves the prediction accuracy of both the velocity and pressure fields while accelerating the training speed of the network.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

30 extracted references · 29 canonical work pages

  1. [1]

    This integration enhances model interpretability and ensure s that predictions are more consistent with physical laws

    Introduction In recent years, Physics -Informed Neural Network (PINN) (1) has emerged as a promising numerical method that, unlike traditional data -driven machine learning tech- niques, directly incorporates governing equations and boundary conditions into the loss function. This integration enhances model interpretability and ensure s that predictions a...

    work page

  2. [2]

    Finite Volume Physical-Informed Neural Network (FV-PINN) 2.1 Governing equation In this paper, we present FV-PINN, which is used to predict the velocity and pressure fields for incompressible laminar flow under steady-state conditions. Let us first consider the governing equations of this problem in the steady-state case: ∇ ⋅ 𝐮 = 0 (1) (𝐮 ⋅ ∇)𝐮 = − 1 𝜌 ∇𝑝...

    work page

  3. [3]

    This transformation facilitates the handling of complex bound- aries and discontinuities

    2 Integral Formulation of the Navier-Stokes Equations Using the Finite Volume Method In solving the Navier -Stokes equations, the Finite Volume Method applies integra- tion of the governing equations over discrete control volumes, converting the differential form into an integral form. This transformation facilitates the handling of complex bound- aries a...

    work page

  4. [4]

    Here, 𝑥 and 𝑦 are the horizontal and vertical coordinates of the flow field integration points

    3 Implementation of FV-PINN Figure 1 Schematic of FV-PINN. Here, 𝑥 and 𝑦 are the horizontal and vertical coordinates of the flow field integration points. 𝜽{𝑤, 𝑏} represents the weights and biases of FV-PINN, which are continuously updated during training. In the present work , the proposed FV -PINN takes a standard multilayer fully con- nected neural net...

    work page

  5. [5]

    Model Validation and Results Analysis The Double Pipe and Pipe Bend are classic examples in fluid topology optimization, first introduced by Borrvall and Peterson in 2003(24). In the following chapter, we will use the proposed FV -PINN to predict the steady -state velocity and pressure fields for these two cases and compare the results with the commercial...

    work page 2003

  6. [6]

    Conclusions This paper presents FV -PINN, a novel Physics -Informed Neural Network inspired by the Finite Volume Method. By leveraging the divergence theorem to reformulate the residuals of Navier-Stokes equations, FV-PINN reduces the reliance on high-order deriv- atives and achieves enhanced prediction accuracy and faster convergence. The proposed method...

    work page

  7. [7]

    Physics -informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

    Raissi M, Perdikaris P, Karniadakis GE. Physics -informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys. 2019;378:686 –707

    work page 2019

  8. [8]

    Tackling the curse of dimensionality with physics -informed neural networks

    Hu Z, Shukla K, Karniadakis GE, Kawaguchi K. Tackling the curse of dimensionality with physics -informed neural networks. Neural Netw. 2024;176:106369

    work page 2024

Show all 30 references
  1. [9]

    A physics -informed variational DeepONet for predicting crack path in quasi-brittle materials

    Goswami S, Yin M, Yu Y, Karniadakis GE. A physics -informed variational DeepONet for predicting crack path in quasi-brittle materials. Comput Methods Appl Mech Eng. 2022;391:114587

    2022

  2. [10]

    PINN Model of Diffusion Coefficient Identification Problem in Fick’s Laws

    Li D, Yan B, Gao T, Li G, Wang Y. PINN Model of Diffusion Coefficient Identification Problem in Fick’s Laws. ACS Omega. 2024;9(3):3846–57

    2024

  3. [12]

    Physics -informed Neural Networks (PINN) for computational solid mechanics: Numerical frameworks and applications

    Hu H, Qi L, Chao X. Physics -informed Neural Networks (PINN) for computational solid mechanics: Numerical frameworks and applications. Thin-Walled Struct. 2024;112495

    2024

  4. [13]

    On physics-informed neural networks for quantum computers

    Markidis S. On physics-informed neural networks for quantum computers. Front Appl Math Stat. 2022;8:1036711

    2022

  5. [14]

    Physics -informed deep learning for incompressible laminar flows

    Rao C, Sun H, Liu Y. Physics -informed deep learning for incompressible laminar flows. Theor Appl Mech Lett. 2020;10(3):207–12

    2020

  6. [15]

    NSFnets (Navier -Stokes flow nets): Physics -informed neural networks for the incompressible Navier-Stokes equations

    Jin X, Cai S, Li H, Karniadakis GE. NSFnets (Navier -Stokes flow nets): Physics -informed neural networks for the incompressible Navier-Stokes equations. J Comput Phys. 2021;426:109951

    2021

  7. [16]

    Flow over an espresso cup: inferring 3 -D velocity and pressure fields from tomographic background oriented Schlieren via physics -informed neural networks

    Cai S, Wang Z, Fuest F, Jeon YJ, Gray C, Karniadakis GE. Flow over an espresso cup: inferring 3 -D velocity and pressure fields from tomographic background oriented Schlieren via physics -informed neural networks. J Fluid Mech. 2021;915:A102

    2021

  8. [17]

    Physics-informed neural networks for high-speed flows

    Mao Z, Jagtap AD, Karniadakis GE. Physics-informed neural networks for high-speed flows. Comput Methods Appl Mech Eng. 2020;360:112789

    2020

  9. [18]

    A physics -informed deep learning framework for inversion and surrogate modeling in solid mechanics

    Haghighat E, Raissi M, Moure A, Gomez H, Juanes R. A physics -informed deep learning framework for inversion and surrogate modeling in solid mechanics. Comput Methods Appl Mech Eng. 2021;379:113741

    2021

  10. [19]

    Automatic differentiation in pytorch

    Paszke A, Gross S, Chintala S, Chanan G, Yang E, DeVito Z, et al. Automatic differentiation in pytorch. 2017

    2017

  11. [20]

    DeepXDE: A deep learning library for solving differential equations

    Lu L, Meng X, Mao Z, Karniadakis GE. DeepXDE: A deep learning library for solving differential equations. SIAM Rev. 2021;63(1):208–28

    2021

  12. [21]

    The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems

    Yu B. The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Commun Math Stat. 2018;6(1):1–12

    2018

  13. [22]

    Variational physics-informed neural networks for solving partial differential equations

    Kharazmi E, Zhang Z, Karniadakis GE. Variational physics-informed neural networks for solving partial differential equations. ArXiv Prepr ArXiv191200873. 2019

    2019

  14. [23]

    MIM: A deep mixed residual method for solving high -order partial differential equations

    Lyu L, Zhang Z, Chen M, Chen J. MIM: A deep mixed residual method for solving high -order partial differential equations. J Comput Phys. 2022;452:110930

    2022

  15. [24]

    wPINNs: Weak physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws

    De Ryck T, Mishra S, Molinaro R. wPINNs: Weak physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws. SIAM J Numer Anal. 2024;62(2):811 –41

    2024

  16. [25]

    Weak adversarial networks for high-dimensional partial differential equations

    Zang Y, Bao G, Ye X, Zhou H. Weak adversarial networks for high-dimensional partial differential equations. J Com- put Phys. 2020;411:109409

    2020

  17. [26]

    Mathematics of classical and quantum physics

    Byron FW, Fuller RW. Mathematics of classical and quantum physics. Courier Corporation; 2012

    2012

  18. [27]

    Adam: A method for stochastic optimization

    Kingma DP. Adam: A method for stochastic optimization. ArXiv Prepr ArXiv14126980. 2014

    2014

  19. [28]

    On the limited memory BFGS method for large scale optimization

    Liu DC, Nocedal J. On the limited memory BFGS method for large scale optimization. Math Program. 1989;45(1):503– 28

    1989

  20. [29]

    Numerical analysis

    Gautschi W. Numerical analysis. Springer Science & Business Media; 2011

    2011

  21. [30]

    Topology optimization of fluids in Stokes flow

    Borrvall T, Petersson J. Topology optimization of fluids in Stokes flow. Int J Numer Methods Fluids. 2003;41(1):77 – 107

    2003

  22. [31]

    A detailed introduction to density -based topology optimisation of fluid flow problems with imple- mentation in MATLAB

    Alexandersen J. A detailed introduction to density -based topology optimisation of fluid flow problems with imple- mentation in MATLAB. Struct Multidiscip Optim. 2023;66(1):12

    2023

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.