arxiv: 2605.04427 · v1 · submitted 2026-05-06 · 🧮 math.NA · cs.NA

Recognition: unknown

Structure-Preserving and Pressure-Robust PINNs for Incompressible Oseen Problems

Shiv Mishra , Arbaz Khan

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Pith reviewed 2026-05-08 16:56 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords physics-informed neural networksOseen equationspressure-robust approximationsstructure-preserving PINNsoptimal error boundsincompressible flowsconsistent loss functionsBesov regularity

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The pith

New consistent PINNs for Oseen equations make velocity errors independent of pressure.

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

This paper develops physics-informed neural networks for the stationary Oseen equations using loss functions derived directly from the problem's stability properties. The approach yields two variants: standard consistent PINNs that improve on conventional methods and pressure-robust versions where the velocity approximation error depends only on the divergence-free component of the forcing term. Quantitative recovery estimates and optimal error bounds are established for velocity in the Sobolev H1 norm and pressure in the L2 norm under suitable regularity assumptions on the solution. The framework handles both divergence-free and general velocity networks in a unified way. Numerical results validate the theoretical guarantees for these incompressible flow approximations.

Core claim

What carries the argument

The consistent PINN (CPINN) framework with stability-consistent loss constructions that inherit the stability structure of the continuous Oseen problem

Load-bearing premise

The true solution satisfies suitable Besov regularity assumptions and the loss functions can be constructed to inherit the continuous problem's stability structure.

What would settle it

Run numerical tests on Oseen problems while adding large gradient forces to the right-hand side and measure whether the velocity error stays constant as the pressure-robust claim predicts.

Figures

Figures reproduced from arXiv: 2605.04427 by Arbaz Khan, Shiv Mishra.

**Figure 1.** Figure 1: Convergence plots for the solution using PINNs (left) and view at source ↗

**Figure 2.** Figure 2: Convergence plots for the solution using PINNs (left) and view at source ↗

**Figure 4.** Figure 4: Convergence plots for the solution using PINNs (left) and view at source ↗

read the original abstract

We develop a new class of physics-informed neural network approximations for the stationary Oseen equations based on stability-consistent loss constructions. In contrast to standard PINN formulations, which are typically heuristic, the proposed consistent PINN (CPINN) framework is systematically derived from the stability structure of the continuous problem. Within this setting, we introduce two fundamentally new approaches. First, we design standard CPINN formulations that exhibit clear improvements over conventional PINNs. Second, we propose pressure-robust CPINN formulations that provably eliminate the influence of gradient forces on the velocity approximation, yielding velocity errors that depend solely on the divergence-free component of the forcing and are independent of the pressure. The framework accommodates both exactly divergence-free architectures and unconstrained velocity approximations, providing a unified treatment of these two paradigms. Using techniques from optimal recovery theory, we establish, for the first time in the PINN setting for Oseen-type problems, quantitative recovery estimates and optimal error bounds for both velocity and pressure under suitable Besov regularity assumptions. In particular, we obtain optimal rates for the velocity in $\boldsymbol{H}^1(\Omega)$ and for the pressure in $L^2(\Omega)$. The proposed methodology introduces a pressure-robust CPINN paradigm for incompressible flows, combining structural consistency, robustness with respect to irrotational forces, and rigorous accuracy guarantees. Numerical experiments corroborate the theoretical findings and demonstrate the effectiveness of the approach.

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

The paper derives PINN losses for Oseen from continuous stability to get pressure-robust versions and optimal recovery rates, which is a clear step beyond heuristic residuals, though the discrete collocation step risks reintroducing pressure dependence.

read the letter

The main contribution is a consistent PINN framework built directly from the Oseen stability structure instead of the usual residual sums. This yields two new formulations: standard CPINNs that improve on conventional ones, and pressure-robust variants where velocity errors depend only on the divergence-free forcing component and stay independent of pressure or gradient forces. The work also supplies the first quantitative recovery estimates for PINNs on these problems, giving optimal H1 rates for velocity and L2 rates for pressure under Besov regularity. It treats both exactly divergence-free networks and unconstrained ones in one setting, and the numerics back the claims. That systematic derivation and the error bounds are the parts that stand out as useful advances in the neural PDE literature. The soft spot is the discrete implementation. PINN losses are finite collocation sums rather than exact integrals, yet the robustness and recovery proofs appear to rest on the continuous weak form without explicit quadrature-error terms that remain independent of irrotational forces. If those terms are not controlled, the claimed pressure independence may not survive in practice. The Besov assumptions are standard for the rates but narrow the scope. The citations and foundations look external and non-circular. This is for people working on theoretical error analysis or structure-preserving methods for neural solvers in incompressible flows. Readers who want quantitative guarantees rather than just empirical fits will find the derivations and bounds worth their time. It deserves a serious referee because the claims are specific, the approach is grounded, and the experiments are present, even if the discrete robustness needs tightening in revision.

Referee Report

1 major / 2 minor

Summary. The manuscript develops a new class of physics-informed neural network approximations for the stationary Oseen equations based on stability-consistent loss constructions. In contrast to heuristic standard PINNs, the consistent PINN (CPINN) framework is systematically derived from the stability structure of the continuous problem. It introduces standard CPINN formulations with improvements over conventional PINNs and pressure-robust CPINN formulations that provably eliminate the influence of gradient forces on the velocity approximation, yielding velocity errors dependent only on the divergence-free component of the forcing. The framework accommodates both exactly divergence-free architectures and unconstrained velocity approximations. Using techniques from optimal recovery theory, it establishes quantitative recovery estimates and optimal error bounds for velocity in H^1(Omega) and pressure in L^2(Omega) under suitable Besov regularity assumptions. Numerical experiments corroborate the theoretical findings.

Significance. If the central claims hold, particularly the provable pressure-robustness and the first quantitative optimal error bounds for PINNs on Oseen-type problems, this would be a significant contribution to the rigorous analysis of structure-preserving neural network methods for incompressible flows. It addresses a key practical limitation of standard PINNs (pressure pollution of velocity errors) while providing a unified treatment of divergence-free and unconstrained architectures, backed by optimal recovery estimates. The emphasis on systematic derivation from continuous stability rather than ad-hoc losses strengthens the methodological foundation.

major comments (1)

[Framework and loss construction] Framework and loss construction (abstract and theoretical sections): The central claim of provable pressure-robustness (velocity error independent of irrotational forces/pressure) rests on the loss exactly inheriting the continuous Oseen stability structure via optimal recovery. However, PINN losses are realized as finite sums over collocation points rather than exact weak-form integrals. The abstract and framework do not indicate whether the recovery estimates or robustness proofs include explicit quadrature-error terms that remain independent of the gradient component of the forcing; if they assume exact integration, the discrete implementation central to the method risks losing the claimed independence. This is load-bearing for the pressure-robustness guarantee.

minor comments (2)

The precise Besov regularity assumptions (spaces and indices) required for the optimal rates should be stated explicitly in the theorem statements rather than referred to generically as 'suitable'.
Notation for the neural network function spaces and the distinction between the two paradigms (exactly divergence-free vs. unconstrained) could be introduced with a dedicated preliminary subsection for improved readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments on our manuscript. We address the major comment regarding the framework and loss construction below, providing clarifications on the theoretical assumptions and plans for revision to strengthen the connection to the discrete implementation.

read point-by-point responses

Referee: Framework and loss construction (abstract and theoretical sections): The central claim of provable pressure-robustness (velocity error independent of irrotational forces/pressure) rests on the loss exactly inheriting the continuous Oseen stability structure via optimal recovery. However, PINN losses are realized as finite sums over collocation points rather than exact weak-form integrals. The abstract and framework do not indicate whether the recovery estimates or robustness proofs include explicit quadrature-error terms that remain independent of the gradient component of the forcing; if they assume exact integration, the discrete implementation central to the method risks losing the claimed independence. This is load-bearing for the pressure-robustness guarantee.

Authors: We appreciate the referee's identification of this critical point. Our theoretical analysis in the manuscript derives the CPINN loss from the continuous stability structure of the Oseen equations using optimal recovery theory, leading to quantitative estimates that establish pressure-robustness for the velocity approximation in the continuous setting. Specifically, the recovery estimates show that the velocity error depends only on the divergence-free component of the forcing when the loss is exactly the continuous functional. However, as noted, the practical PINN implementation approximates this loss via finite collocation sums, introducing quadrature errors. The current proofs do not explicitly incorporate bounds on these quadrature errors while preserving independence from the gradient forces. We acknowledge that this assumption of exact integration means the discrete pressure-robustness is supported primarily by numerical evidence rather than fully rigorous bounds in the present version. To address this, we will revise the manuscript as follows: (1) Update the abstract and Section 2 to explicitly state that the provable pressure-robustness and optimal error bounds hold for the continuous loss functional, with the discrete version inheriting these properties approximately. (2) Add a new subsection or remark in the theoretical sections discussing quadrature error control, leveraging standard Monte Carlo integration error bounds under suitable regularity, and arguing that these errors can be controlled independently of the irrotational forcing by uniform sampling. (3) Enhance the numerical section to include experiments demonstrating that pressure-robustness is maintained for increasing numbers of collocation points. These changes will make the manuscript more precise. revision: yes

Circularity Check

0 steps flagged

No significant circularity; framework derived from external stability structure and optimal recovery theory

full rationale

The paper states that the CPINN framework is systematically derived from the stability structure of the continuous Oseen problem and that quantitative recovery estimates and optimal error bounds are established using techniques from optimal recovery theory under Besov regularity assumptions. No quoted steps reduce the central claims (pressure-robust velocity errors independent of gradient forces, optimal H1 velocity and L2 pressure rates) to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations whose content is unverified within the paper. The derivation chain remains self-contained against external mathematical foundations rather than tautological with its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the stability structure of the Oseen equations and optimal recovery theory; the abstract does not introduce new free parameters or invented entities.

axioms (2)

domain assumption The stationary Oseen equations possess a stability structure that can be used to construct consistent loss functions for neural approximations.
This is the foundation for the consistent PINN (CPINN) framework described in the abstract.
domain assumption Solutions satisfy suitable Besov regularity assumptions.
Invoked to establish the optimal error bounds for velocity and pressure.

pith-pipeline@v0.9.0 · 5550 in / 1155 out tokens · 34672 ms · 2026-05-08T16:56:45.556463+00:00 · methodology

discussion (0)

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Works this paper leans on

30 extracted references · 18 canonical work pages

[1]

Ahmed, G

N. Ahmed, G. R. Barrenechea, E. Burman, J. Guzm ´an, A. Linke, and C. Merdon , A pressure-robust discretization of oseen ’s equation using stabilization in the vorticity equa- tion, SIAM Journal on Numerical Analysis, 59 (2021), pp. 2746–27 74, https://doi.org/10. 1137/20M1351230. 27

2021
[2]

Badra and J.-P

M. Badra and J.-P. Raymond , Numerical approximation of the Oseen system in polyhedral or polygonal domains , Numer. Math., 157 (2025), pp. 2251–2289, https://doi.org/10.1007/ s00211-025-01496-1

2025
[3]

Boffi, F

D. Boffi, F. Brezzi, and M. Fortin , Mixed ﬁnite element methods and applications , vol. 44 of Springer Series in Computational Mathematics, Springer , Heidelberg, 2013, https://doi. org/10.1007/978-3-642-36519-5

work page doi:10.1007/978-3-642-36519-5 2013
[4]

Bonito, R

A. Bonito, R. DeVore, G. Petrova, and J. W. Siegel , Convergence and error control of consistent pinns for elliptic pdes , IMA Journal of Numerical Analysis, (2025), p. draf008, https://doi.org/10.1093/imanum/draf008

work page doi:10.1093/imanum/draf008 2025
[5]

S. Cai, Z. Mao, Z. W ang, M. Yin, and G. E. Karniadakis , Physics-informed neural networks (PINNs) for ﬂuid mechanics: a review , Acta Mech. Sin., 37 (2021), pp. 1727–1738, https:// doi.org/10.1007/s10409-021-01148-1

work page doi:10.1007/s10409-021-01148-1 2021
[6]

Cuomo, V

S. Cuomo, V. S. Di Cola, F. Giampaolo, G. Rozza, M. Raissi, and F . Piccialli , Sci- entiﬁc machine learning through physics–informed neural n etworks: Where we are and what’s next , Journal of Scientiﬁc Computing, 92 (2022), p. 88, https://doi.org/10.1007/ s10915-022-01939-z

2022
[7]

Dahlke, E

S. Dahlke, E. Novak, and W. Sickel , Optimal approximation of elliptic problems by linear and nonlinear mappings. I , J. Complexity, 22 (2006), pp. 29–49, https://doi.org/10.1016/ j.jco.2005.06.005

2006
[8]

R. A. DeVore, R. How ard, and C. Micchelli , Optimal nonlinear approximation , Manuscripta Math., 63 (1989), pp. 469–478, https://doi.org/10.1007/BF01171759

work page doi:10.1007/bf01171759 1989
[9]

R. A. DeVore and G. G. Lorentz , Constructive approximation , vol. 303 of Grundlehren der mathematischen Wissenschaften [Fundamental Principl es of Mathematical Sciences], Springer-Verlag, Berlin, 1993

1993
[10]

R. A. DeVore and R. C. Sharpley , Besov spaces on domains in Rd, Trans. Amer. Math. Soc., 335 (1993), pp. 843–864, https://doi.org/10.2307/2154408

work page doi:10.2307/2154408 1993
[11]

H. C. Elman, D. J. Silvester, and A. J. W athen , Finite elements and fast iterative solvers: with applications in incompressible ﬂuid dynamics , Numerical Mathematics and Scientiﬁc Computation, Oxford University Press, New York, 2005, https://doi.org/10.1093/acprof: oso/9780199678792.001.0001

work page doi:10.1093/acprof: 2005
[12]

G. Galdi , An introduction to the mathematical theory of the Navier-St okes equations: Steady-state problems, Springer Science & Business Media, 2011, https://doi.org/10.1007/ 978-0-387-09620-9

2011
[13]

Gazoulis, I

D. Gazoulis, I. Gkanis, and C. G. Makridakis , On the stability and convergence of physics informed neural networks , IMA Journal of Numerical Analysis, (2025), https://doi.org/10. 1093/imanum/draf090

2025
[14]

Girault and P

V. Girault and P. Raviart , Finite element methods for navier-stokes equations: Theor y and algorithms((book)), Berlin and New York, Springer-Verlag(Springer Series in C ompu- tational Mathematics., 5 (1986)

1986
[15]

V. John, A. Linke, C. Merdon, M. Neilan, and L. G. Rebholz , On the divergence constraint in mixed ﬁnite element methods for incompressible ﬂows , SIAM review, 59 (2017), pp. 492– 544, https://doi.org/10.1137/15M1047696

work page doi:10.1137/15m1047696 2017
[16]

G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. W ang, and L. Yang, Physics- informed machine learning , Nature Reviews Physics, 3 (2021), pp. 422–440, https://doi. org/10.1038/s42254-021-00314-5

work page doi:10.1038/s42254-021-00314-5 2021
[17]

Khan, K.-A

A. Khan, K.-A. Mardal, and S. Mishra , Mixed consistent pinns for elliptic obstacle problems with stability analysis , 2026, https://arxiv.org/abs/2604.01719

work page arXiv 2026
[18]

Krieg, E

D. Krieg, E. Novak, and M. Sonnleitner , Recovery of Sobolev functions restricted to iid sampling, Math. Comp., 91 (2022), pp. 2715–2738, https://doi.org/10.1090/mcom/3763

work page doi:10.1090/mcom/3763 2022
[19]

Linke and C

A. Linke and C. Merdon , Pressure-robustness and discrete helmholtz projectors in mixed ﬁnite element methods for the incompressible navier–stoke s equations, Computer Methods in Applied Mechanics and Engineering, 311 (2016), pp. 304–3 26, https://doi.org/10.1016/ j.cma.2016.08.018

2016
[20]

ETNA, 63–92 (2026) https: //doi.org/10.1553/etna vol65s63

A. Linke, C. Merdon, and M. Neilan , Pressure-robustness in quasi-optimal a priori estimates for the Stokes problem , Electron. Trans. Numer. Anal., 52 (2020), pp. 281–294, https:// doi.org/10.1553/etna vol52s281

work page doi:10.1553/etna 2020
[21]

Linke, C

A. Linke, C. Merdon, M. Neilan, and F. Neumann , Quasi-optimality of a pressure-robust nonconforming ﬁnite element method for the stokes-problem , Mathematics of Computa- tion, 87 (2018), pp. 1543–1566, https://doi.org/10.1090/mcom/3344

work page doi:10.1090/mcom/3344 2018
[22]

Y. Lu, H. Chen, J. Lu, L. Ying, and J. Blanchet , Machine learning for elliptic pdes: Fast rate generalization bound, neural scaling law and mini max optimality , International Conference on Learning Representations, https://par.nsf.gov/biblio/10337713. 28

work page arXiv
[23]

Mishra and A

S. Mishra and A. Khan , A priori error analysis of consistent pinns for parabolic pd es, arXiv preprint arXiv:2506.17614, (2025), https://arxiv.org/abs/2506.17614

work page arXiv 2025
[24]

Mishra and A

S. Mishra and A. Khan , Consistent pinns for higher-order elliptic pdes , International Journal for Numerical Methods in Engineering, 127 (2026), p. e70320 , https://doi.org/10.1002/ nme.70320

2026
[25]

Mishra and R

S. Mishra and R. Molinaro , Estimates on the generalization error of physics-informed neural networks for approximating PDEs , IMA J. Numer. Anal., 43 (2023), pp. 1–43, https://doi. org/10.1093/imanum/drab093

work page doi:10.1093/imanum/drab093 2023
[26]

Novak and H

E. Novak and H. Triebel , Function spaces in Lipschitz domains and optimal rates of co n- vergence for sampling , Constr. Approx., 23 (2006), pp. 325–350, https://doi.org/10.1007/ s00365-005-0612-y

2006
[27]

Raissi, P

M. Raissi, P. Perdikaris, and G. E. Karniadakis , Physics-informed neural networks: a deep learning framework for solving forward and inverse problem s involving nonlinear partial diﬀerential equations , J. Comput. Phys., 378 (2019), pp. 686–707, https://doi.org/10.1016/ j.jcp.2018.10.045

2019
[28]

Y. Shin, J. Darbon, and G. E. Karniadakis , On the convergence of physics informed neural networks for linear second-order elliptic and parabolic ty pe PDEs , Commun. Comput. Phys., 28 (2020), pp. 2042–2074, https://doi.org/10.4208/cicp.oa-2020-0193

work page doi:10.4208/cicp.oa-2020-0193 2020
[29]

Y. Shin, Z. Zhang, and G. E. Karniadakis , Error estimates of residual minimization using neural networks for linear pdes , Journal of Machine Learning for Modeling and Computing, 4 (2023), https://doi.org/10.1615/JMachLearnModelComput.2023050411

work page doi:10.1615/jmachlearnmodelcomput.2023050411 2023
[30]

Zeinhofer, R

M. Zeinhofer, R. Masri, and K. Mardal , A uniﬁed framework for the error analy- sis of physics-informed neural networks , IMA Journal of Numerical Analysis, (2024), p. 2988–3025, https://doi.org/10.1093/imanum/drae081. 29

work page doi:10.1093/imanum/drae081 2024