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Finite difference physics-informed neural networks enable improved solution accuracy of the Navier-Stokes equations

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Subdomain-refined neural nets recover corner vortices at Re=1000.

desk verdict A warm-started FD-PINN plus domain decomposition that recovers lid-driven-cavity corner vortices at Re=1000 without known-solution data; the result is plausible and useful but needs reproducibility and robustness work before it is fully convincing. read the letter →

arxiv 2501.00014 v1 pith:E6ONMR2G submitted 2024-12-14 physics.comp-ph physics.flu-dyn

Nityananda Roy , Robert Dürr , Andreas Bück , S.Sundar This is my paper · ORCID
classification physics.comp-phphysics.flu-dyn MSC 65N5565M0668T0776D05
keywords Navier-StokesequationsLid-drivencavityFinitedifferencemethodsPhysics-informedneuralnetworksDomaindecompositionSecondaryvorticesIncompressibleflowHighReynoldsnumber
topics Navier-Stokes Regularity
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

Physics-informed neural networks (PINNs) that rely on automatic differentiation have a hard time resolving the weak counter-rotating eddies that form in the lower corners of a lid-driven cavity at Reynolds number 1000. This paper argues that the failure lies in how derivatives are computed and in solving on one global grid, and it fixes both: replace automatic differentiation with second-order finite-difference stencils, then split the cavity into 16 subdomains and retrain a fresh network on each, seeded with the whole-domain network's weights and fed boundary values taken from that network. The paper reports that this two-stage procedure, which never uses the classical solution as training data, recovers accurate secondary vortices in both lower corners, while the standard FD-PINN on a single grid from 100×100 up to 300×300 loses the lower-left vortex. Compared with a 500×500-grid finite-difference reference solution, the refined solutions lower the mean-square error near the walls and place the vortex centers within the spread of published spectral and multigrid benchmarks. If the claim holds, it offers a simple, data-free way to push PINN solutions of the Navier–Stokes equations to higher Reynolds numbers near boundaries.

What carries the argument

The workhorse is the FD-PINN: a physics-informed neural network whose PDE residual is evaluated not by automatic differentiation but by second-order central finite-difference stencils on a rectangular grid, with the network's outputs reshaped to the grid and boundary values inserted before differencing. The loss is the sum of squared residuals of the velocity–pressure Navier–Stokes equations and the incompressibility constraint over interior grid points, minimized with the L-BFGS optimizer. The novel part is the two-stage domain decomposition: a whole-domain FD-PINN is trained at 100×100 points, and then the domain is cut into 16 rectangular subdomains, each fitted with its own network initialized from the whole-domain weights and biases; each subdomain's boundary conditions are extracted from the already-trained global solution, which removes any need for interface conditions between neighboring subdomains and lets the local networks refine structure (such as corner eddies) that the global network resolved poorly.

What would settle it

Run the two-stage scheme starting from a deliberately degraded whole-domain solution — for example, stop the L-BFGS optimization early on the 100×100 global grid, or train that grid to a higher loss tolerance — and check whether the 16 subdomain networks still produce the lower-left secondary vortex with its benchmark center. If the vortex is recovered despite incorrect inherited boundary data, the claim that subdomain accuracy is tied to global accuracy would be contradicted; if the vortex is missed or shifted, the dependency is confirmed.

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Extended reading notes

Core claim

The authors claim that the steady incompressible Navier–Stokes equations in a lid-driven cavity can be solved accurately at Re=1000 by an FD-PINN without any known-solution data, provided the problem is split across scales: first train one network on the whole domain with a moderate 100×100 grid, then divide the cavity into 16 rectangular subdomains and retrain an independent network in each, initialized from the whole-domain weights and biases and constrained on its boundaries by the whole-domain network's output. The loss in both stages is the squared PDE residual (momentum equations plus the divergence-free constraint) evaluated with central finite differences on the grid, so no interface conditions between subdomains are needed. On this scheme, the lower-left secondary vortex, which standard FD-PINNs produce only faintly at 100×100 and lose entirely at finer grids, appears clearly at all three subdomain grid resolutions, and the most accurate vortex centers come from the coarsest 25×25 subdomain grid. The authors quantify the gain with mean-square errors against a finite-difference reference solution, showing substantial error reduction in the wall-adjacent strips, and they demonstrate that the same refinement rescues the vanished lower-left corner vortex at Re=400. The central assertion is that the dominant error source in FD-PINNs is the overprediction that comes from training a single network on a fine global grid, and that local retraining on coarse grids eliminates that overprediction while preserving accuracy in the interior.

Load-bearing premise

The whole method rests on the accuracy of the boundary values that the whole-domain FD-PINN supplies to each subdomain; if that global solution is wrong in a corner, the local network inherits the error and cannot recover the true vortex on its own.

Editorial extensions

If this is right

  • At Re=1000, the subdomain-refined FD-PINN produces a clear secondary vortex in the lower-left corner at every subdomain grid tested (25×25, 50×50, 100×100), whereas standard FD-PINNs on 100×100, 120×120, 150×150, and 300×300 grids lose or distort that vortex.
  • The 25×25 subdomain grid gives the vortex-center locations closest to the reference values (x ≈ 0.0735, y ≈ 0.0753 for the lower left; x ≈ 0.8695, y ≈ 0.1210 for the lower right).
  • Along the horizontal line y=0.99, the mean-square error of the vertical velocity component drops from about 2.6×10^-3 for the standard FD-PINN to about 1.2×10^-5 for the present method with 100×100 subdomain grids, a gain of roughly two orders of magnitude near the moving lid.
  • The same refinement scheme restores the completely vanished lower-left corner vortex at Re=400 using only 25×25 subdomain grid points, so the benefit is not specific to Re=1000.
  • Because subdomains are trained independently with no interface conditions, the refinement step is embarrassingly parallel, so the approach can be distributed across cores without changing the algorithm.

Reading between the lines

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

  • The pattern of results — coarse grids winning in the low-intensity lower corners and finer grids winning near the lid — suggests an automatic adaptive strategy: let the network choose its own local grid density based on velocity magnitude or gradient, which could cut training cost below the fixed 16×16 split.
  • The two-stage scheme resembles a classical defect-correction or multigrid idea, with the whole-domain network acting as a coarse solver and subdomain retraining as local smoothing; if the analogy holds, iterating the global-refine cycle (re-train the global network from the refined subdomains) might push the method to still higher Reynolds numbers.
  • A caution implied by the method: because subdomain boundaries come from the global network without any interface constraint, a wrong global solution can be faithfully reproduced rather than corrected; a testable variant would add overlap or interface consistency to see whether the vortex centers shift toward the benchmarks.
  • The reference solution is itself numerical, not experimental; the next discriminating test for the claim that FD-PINNs resolve true corner physics would be comparison against physical experiments or a spectral solver at the same Reynolds number.
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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 a domain-decomposition extension of finite-difference physics-informed neural networks (FD-PINNs) for steady incompressible Navier-Stokes flow in a square lid-driven cavity. The method first trains a global FD-PINN on the whole cavity at 100×100 interior grid points, then divides the domain into 16 non-overlapping subdomains; each subdomain network is initialized with the global network weights and trained against the residual of the Navier-Stokes equations with Dirichlet velocity boundary data taken from the global solution. No inter-subdomain interface conditions are used. Results are presented for Re=400 and Re=1000, including streamline plots, vortex-center locations, and MSE tables against a 500×500 SOR reference solution, with emphasis on the lower corners where standard FD-PINNs fail to produce accurate secondary vortices. The central claim is that this two-stage procedure improves solution accuracy near the walls and generates correct secondary vortices without using the reference solution in the loss function.

Significance. If the results are reproducible, the paper makes a useful contribution: it shows that a simple subdomain refinement stage, initialized from a global FD-PINN, can recover corner vortices at Re=1000 where the plain FD-PINN fails, and it provides a transparent comparison against a validated reference solution (Ghia et al.). The 120×120 stress test in Section 4.3, where the global solution lacks the lower-left vortex yet the local solver recovers it, is a particularly persuasive piece of evidence. The method avoids interface conditions and is easy to implement. However, the quantitative claims are weakened by the absence of repeated runs, seeds, cost-controlled baselines, and an assessment of the sensitivity to the approximate subdomain Dirichlet data; these gaps currently make the headline accuracy improvements conditional.

major comments (4)
  1. [§4.3, Algorithm 1, §5] The subdomain Dirichlet data are taken from the global FD-PINN solution, and Section 5 concedes that subdomain accuracy degrades if that global solution is inaccurate. The paper neither reports the error of the global solution along the artificial interfaces nor a perturbation study of these boundary values. The 120×120 experiment (Figure 12) is a single qualitative example; while it mitigates the concern, it does not quantify the dependence. Please add an interface-error table and a test in which the subdomain boundary data are perturbed or replaced by slightly different global solutions.
  2. [§4, L-BFGS setting paragraph] The text states that about one-third of the subdomain optimizations stopped early at local minima. No repeated runs, random seeds, or statistics are reported for the MSE tables or vortex centers. Given the small differences in Table 4.1 (e.g., x-coordinate 0.0782 vs 0.0830), the observed improvements may not be robust across initializations. Please provide mean ± standard deviation over several seeds, or a deterministic seed with a justification.
  3. [§4, Tables 4.2–4.7] The accuracy comparison is not cost-controlled. The present method trains one global network plus 16 independent subdomain networks, each with up to 2×10^5 L-BFGS evaluations, whereas the standard FD-PINNs baseline is a single 100×100 training run. The claimed improvement in solution accuracy should be accompanied by wall-clock times and parameter counts, or a baseline with equivalent total compute, so that the reader can distinguish a genuine algorithmic gain from a larger training budget.
  4. [§4.3, Table 4.1, Figure 8] The vortex-center locations for the present method are reported for subdomain grids 25×25, 50×50, and 100×100, but the selection of these three grid sizes is presented as a result of 'computational trials' without a criterion. Since the method's accuracy varies strongly with subdomain resolution (Tables 4.3–4.4 show different rankings for different lines), the paper should either propose a principled way to choose the subdomain resolution or report the sensitivity of the headline vortex-center claim to this choice.
minor comments (6)
  1. [Figure 12 caption] The caption contains a duplicated word: 'with with 120 × 120 grid points'.
  2. [Figure 7 caption] The caption reads 'lop left corner'; it should be 'top left corner'.
  3. [Reference [17]] The journal title is misspelled as 'Journal of Marchine Learning Research'; it should be 'Journal of Machine Learning Research'.
  4. [Section 5] The phrase 'we emphasizing' should be 'we emphasize'.
  5. [Equation (3.2)] The notation with double vertical bars and subscripts i,j is ambiguous; please clarify that the norm is evaluated at each grid point and summed over the interior points.
  6. [Algorithm 1] The step 'Reshape the output variables to a matrix form and add the boundary of the velocity components' is vague; specify how the boundary conditions are imposed (hard replacement of network outputs or soft penalty) and how the finite-difference stencils treat points adjacent to the boundary.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the FD-PINN + DDM solution is trained from the PDE residual and physical boundary conditions; the reference solution enters only in post-hoc MSE evaluation.

full rationale

The central derivation chain is self-contained. The whole-domain FD-PINN is trained by minimizing the finite-difference discretized Navier-Stokes residual (Eq. 3.2) with only the physical lid and wall boundary conditions; no classical or reference solution values appear in the loss. Algorithm 1 then uses the trained whole-domain network only to supply initial weights and Dirichlet boundary traces for independent sub-domain solves, and each sub-domain residual is again minimized from the PDE (Eqs. 2.1-2.2). The reference solution of Section 4.1 is used exclusively for computing reported MSE and vortex-center comparisons; it is not a training input. Section 5's admission that sub-domain accuracy degrades if the whole-domain solution is inaccurate is a numerical dependency shared by Dirichlet-domain-decomposition methods, not a reduction of the prediction to its inputs, because the global solution is itself generated from the PDE rather than from the target data. The approximately one-third of sub-domain optimizations stopping at local minima is an optimization robustness concern, not circularity. No fitted parameter is renamed as a prediction, and no load-bearing self-citation or imported uniqueness claim appears; the cited prior works (e.g., FD-PINNs [23], DDM variants [26-31]) are external and their use is not a substitute for the present residual-based derivation.

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

The central claim rests on no new physical entities. It does rest on hand-chosen hyperparameters (grid resolutions, subdomain count, architecture, optimizer settings) and on the assumption that the whole-domain FD-PINN solution provides sufficiently accurate subdomain boundary data.

free parameters (5)
  • Global grid resolution = 100 x 100 interior points
    Selected by 'computational experience' and Jiang et al. [23]; central to the whole-domain pre-training step.
  • Subdomain grid resolution = 25 x 25, 50 x 50, 100 x 100 per subdomain
    Varies across runs; the paper reports accuracy for all three and notes that 25x25 works best near lower corners but worse in the interior.
  • Number of subdomains = 16
    Chosen without systematic study; affects how boundary conditions are inherited from the global solution.
  • Network architecture = 3 hidden layers, 50 neurons each, tanh activation
    Chosen after 'several computational trials'; no theory is given.
  • L-BFGS iterations and function evaluations = 2e5 maximum
    Stopping criterion; about one third of subdomains stop early on step-size tolerance, possibly at local minima.
assumptions (4)
  • domain assumption The steady lid-driven cavity Navier-Stokes problem at Re=400 and Re=1000 has a unique steady solution, and the reference FDM solution is accurate.
    Section 4.1 validates the reference solution against Ghia et al. [39] and Botella-Peyret [41], but the method assumes closeness to the true solution as ground truth.
  • standard math Central-difference approximations with the chosen grid spacings are accurate enough for the neural-network training loss to guide convergence.
    Section 3 defines second-order finite differences for all derivatives; the loss becomes zero only if these discrete residuals are zero.
  • ad hoc to paper Boundary conditions for each subdomain can be taken from the whole-domain FD-PINN solution and are accurate enough to train local networks.
    Algorithm 1 and Section 4.2 use the global solution for subdomain Dirichlet data; the authors note in Section 5 that subdomain accuracy is limited by global accuracy.
  • domain assumption A 3-layer, 50-neuron tanh network can represent the cavity solution in each subdomain.
    No approximation theory is provided; the paper relies on empirical convergence.

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Cite this review

Pith. "Pith review of Finite difference physics-informed neural networks enable improved solution accuracy of the Navier-Stokes equations." pith.science (2026). https://pith.science/paper/E6ONMR2G

@misc{pith2026250100014,
  author       = {Pith},
  title        = {Pith review of: Finite difference physics-informed neural networks enable improved solution accuracy of the Navier-Stokes equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E6ONMR2G}},
  note         = {Machine review of arXiv:2501.00014}
}
read the original abstract

Generating an accurate solution of the Navier--Stokes equations using physics--informed neural networks (PINNs) for higher Reynolds numbers in the corners of a lid--driven cavity problem is challenging. In this paper, we improve the solution accuracy of the incompressible Navier--Stokes equations in the region near the walls significantly and generate accurate secondary vortices in the corners of the lid--driven cavity by solving the governing equations using finite difference--based PINNs (FD--PINNs) without employing the known solution. We adopt the domain decomposition method (DDM) and combine it with the FD--PINNs to solve the lid--driven cavity problem for the Reynolds numbers Re = 400 and Re=1000. A comparison of the mean square error (MSE) between the presented and standard FD--PINNs using the reference solution is exhibited, showing the accuracy and effectiveness of the new approach.

Figures

Figures reproduced from arXiv: 2501.00014 by the authors.

Figure 1
Figure 1. Geometry of a 2D square lid–driven cavity. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Neural networks with four hidden layers and eight neurons in each. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Framework for second–order central difference scheme [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Pictorial representation of the present technique [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: a.) Fluid streamlines using the reference solution for the Reynolds number Re=1000, b.) Comparison of the [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Fluid streamlines for the Reynolds number Re=1000 using the standard FD–PINNs with a.) 100 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: a–d.) Error in velocity magnitude generated by the standard FD–PINNs for the Reynolds number Re=1000 with [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: a.) Fluid streamlines using the standard FD–PINNs with 100 [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: a.) Fluid streamlines using the standard FD–PINNs with 100 [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: a.) Fluid streamlines using the standard FD–PINNs with 100 [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]
Figure 11
Figure 11. Figure 11: a–d.) Error in the velocity magnitude for the Reynolds number Re=1000 generated by the standard FD–PINNs [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 12
Figure 12. Figure 12: a.) Fluid streamlines using the standard FD–PINNs in the lower left corner of the cavity with 120 [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]
Figure 13
Figure 13. Figure 13: a.) Fluid streamlines for the Reynolds number Re=400 using the standard FD–PINNs with 100 [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]
Figure 14
Figure 14. Figure 14: Final loss of the present FD–PINNs with 25 [PITH_FULL_IMAGE:figures/full_fig_p017_14.png]

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Derivative Computation in PINNs: Automatic Differentiation, Finite Differences and Beyond

    cs.LG 2026-08 conditional novelty 6.0 of 10

    With a carefully chosen step size, finite-difference derivatives match automatic differentiation in accuracy and beat it in speed and memory for MLP-based physics-informed neural networks.

  2. Solving Hamiltonian Constraint Equation with Physics-Informed Neural Networks

    gr-qc 2026-07 conditional novelty 5.5 of 10

    PINNs with specialized techniques solve the nonlinear Hamiltonian constraint for generic binary black hole initial data, matching traditional NR accuracy.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.