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arxiv: 2606.06268 · v1 · pith:NGT6CKYRnew · submitted 2026-06-04 · 🧮 math.NA · cs.NA

Error Analysis of Tr-PINNs Algorithm for 2D Incompressible Navier-Stokes Equations with Non-Homogeneous Boundary Conditions

Dongjie Liu , Xuebo Li , Rong Yang This is my paper

Pith reviewed 2026-06-28 00:14 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Tr-PINNsphysics-informed neural networksNavier-Stokes equationsnon-homogeneous boundary conditionserror analysisboundary value correction
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The pith

Tr-PINNs corrects boundary errors in PINNs to raise accuracy for 2D Navier-Stokes equations with non-homogeneous boundaries.

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

The paper proposes the Tr-PINNs algorithm to address the problem of inaccurate boundary simulation in conventional PINNs when dealing with non-homogeneous boundary conditions. Instead of relying solely on an L2-norm constraint for boundaries, Tr-PINNs corrects the boundary value error. It establishes an error analysis based on the nonhomogeneous Stokes problem and applies it to the Navier-Stokes equations. Numerical experiments demonstrate that this leads to an improvement in computational accuracy.

Core claim

By correcting the boundary value error, the Tr-PINNs algorithm enables the error analysis derived from the nonhomogeneous Stokes problem to carry over to the 2D incompressible Navier-Stokes equations, yielding higher prediction accuracy than standard PINNs.

What carries the argument

The boundary value correction mechanism in Tr-PINNs, which adjusts the network outputs to satisfy non-homogeneous conditions more accurately than L2-norm enforcement alone.

If this is right

  • The method provides a more reliable way to enforce non-homogeneous boundaries in neural network solvers for fluid dynamics.
  • Error estimates for Tr-PINNs follow from known results on the nonhomogeneous Stokes problem.
  • Numerical tests show improved accuracy on specific 2D incompressible flow problems with non-homogeneous boundaries.

Where Pith is reading between the lines

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

  • This correction approach may extend to other physics-informed neural network applications involving complex boundary conditions.
  • Similar boundary adjustments could be tested on time-dependent or three-dimensional Navier-Stokes problems.

Load-bearing premise

The error analysis from the nonhomogeneous Stokes problem applies without additional gaps to the nonlinear Navier-Stokes equations under the same boundary correction.

What would settle it

A direct comparison of computed errors in Tr-PINNs versus standard PINNs on a Navier-Stokes problem where the Stokes-based error bounds predict a specific accuracy gain, but the observed gain differs substantially.

Figures

Figures reproduced from arXiv: 2606.06268 by Dongjie Liu, Rong Yang, Xuebo Li.

Figure 4.1
Figure 4.1. Figure 4.1: The first case: L 2 Relative Error of u1 and u2 under different working conditions at t = 0.5. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_4_1.png] view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: The first case: Root Mean Square Error of [PITH_FULL_IMAGE:figures/full_fig_p013_4_2.png] view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: The first case: (a)(b) and (c)(d) show the predicted solutions and point-wise absolute [PITH_FULL_IMAGE:figures/full_fig_p013_4_3.png] view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: The second case: L 2 Relative Error of u1 and u2 under different working conditions at t = 0.5 [PITH_FULL_IMAGE:figures/full_fig_p015_4_4.png] view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: The second case: Root Mean Square Error of [PITH_FULL_IMAGE:figures/full_fig_p015_4_5.png] view at source ↗
Figure 4.6
Figure 4.6. Figure 4.6: The second case: (a)(b) and (c)(d) show the predicted solutions and point-wise [PITH_FULL_IMAGE:figures/full_fig_p016_4_6.png] view at source ↗
Figure 4.7
Figure 4.7. Figure 4.7: The third case: L 2 Relative Error of u1 and u2 under different working conditions at t = 0.5 [PITH_FULL_IMAGE:figures/full_fig_p018_4_7.png] view at source ↗
Figure 4.8
Figure 4.8. Figure 4.8: The third case: Root Mean Square Error of [PITH_FULL_IMAGE:figures/full_fig_p018_4_8.png] view at source ↗
Figure 4.9
Figure 4.9. Figure 4.9: The third case: (a)(b) and (c)(d) show the predicted solutions and point-wise [PITH_FULL_IMAGE:figures/full_fig_p019_4_9.png] view at source ↗
read the original abstract

Physics-informed neural networks (PINNs) have been widely applied to solve Navier-Stokes equations by enforcing outputs and gradients of deep models to satisfy target equations. However, conventional PINNs only constrain the boundary terms by means of the $L^2$-norm when addressing the equations with non-homogeneous boundary conditions. This single constraint strategy may cause inaccurate boundary simulation, further resulting in the decline of prediction accuracy. To resolve this critical issue, this paper proposes an improved physics-informed neural network by correcting the error of the boundary value, which is called Tr-PINNs. Based on the results of nonhomogeneous Stokes problem, the algorithm error analysis of Tr-PINNs is established. The efficacy of the Tr-PINNs algorithm is demonstrated via numerical experiments, which further demonstrate that the Tr-PINNs algorithm achieves a remarkable improvement in computational accuracy.

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.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces Tr-PINNs, a boundary-error-corrected variant of physics-informed neural networks for the 2D incompressible Navier-Stokes equations with non-homogeneous boundary conditions. It asserts that an error analysis is established by transferring results from the nonhomogeneous Stokes problem and that numerical experiments confirm a remarkable improvement in accuracy over standard PINNs.

Significance. If the claimed error analysis can be made rigorous for the nonlinear system, the work would supply a concrete theoretical handle on boundary enforcement in PINNs for fluid problems, addressing a documented practical weakness of the standard L2-boundary penalty approach.

major comments (1)
  1. [Abstract] Abstract: the error analysis is stated to rest on 'the results of nonhomogeneous Stokes problem.' The full Navier-Stokes system contains the convective term (u·∇)u, which is absent from the linear Stokes problem. No explicit small-data assumption, a-priori bound, or fixed-point argument is referenced that would absorb this term once the neural-network approximation and boundary correction are introduced. Because the central claim is the establishment of error analysis for the NS equations, this transfer step is load-bearing.
minor comments (1)
  1. The abstract would be clearer if it stated the precise form of the error bound obtained (e.g., in which norm and under what regularity assumptions on the data).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the important observation on the rigor of the error analysis. We address the comment below and will revise the manuscript to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the error analysis is stated to rest on 'the results of nonhomogeneous Stokes problem.' The full Navier-Stokes system contains the convective term (u·∇)u, which is absent from the linear Stokes problem. No explicit small-data assumption, a-priori bound, or fixed-point argument is referenced that would absorb this term once the neural-network approximation and boundary correction are introduced. Because the central claim is the establishment of error analysis for the NS equations, this transfer step is load-bearing.

    Authors: We agree that the transfer of error estimates from the linear non-homogeneous Stokes problem to the nonlinear Navier-Stokes equations requires explicit justification to control the convective term. The current manuscript states the analysis is based on Stokes results but does not detail the small-data assumption or fixed-point argument needed for the nonlinear case. In the revised version we will add this justification, including the precise small-data regime and the fixed-point construction that absorbs the nonlinear contribution after the boundary correction is applied, thereby making the central claim fully rigorous. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives its error analysis for Tr-PINNs explicitly from prior results on the nonhomogeneous Stokes problem and validates performance via separate numerical experiments. No equations, parameters, or claims reduce by construction to fitted inputs or self-referential definitions; no load-bearing self-citations or uniqueness theorems imported from the authors' prior work appear in the provided text. The central accuracy claim rests on external Stokes analysis plus empirical tests, making the derivation self-contained against the listed circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the transfer of Stokes-problem error estimates to the Navier-Stokes case and on the assumption that the boundary correction term improves accuracy without introducing new instabilities; no free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption Error analysis of Tr-PINNs is established based on results of the nonhomogeneous Stokes problem
    Explicitly stated in the abstract as the foundation for the algorithm error analysis.

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Reference graph

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