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arxiv: 2606.12050 · v1 · pith:227TCAPEnew · submitted 2026-06-10 · 💻 cs.LG · math.DS

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds

Ismail Huseynov , Arzu Ahmadova , Agamirza Bashirov This is my paper

Pith reviewed 2026-06-27 10:54 UTC · model grok-4.3

classification 💻 cs.LG math.DS
keywords PINNsa posteriori error boundsordinary differential equationslocalized strong monotonicityone-sided Lipschitz conditionresidual-based certificatesphysics-informed neural networks
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The pith

Computable lower and upper a posteriori bounds certify PINN errors for ODEs without the exact solution.

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

The paper derives explicit lower bounds on the error of a physics-informed neural network approximation to an ordinary differential equation, using only the network output and the residual on domains where a localized strong monotonicity condition holds. These lower bounds are paired with complementary upper bounds that require only a one-sided Lipschitz condition, which is weaker than the global Lipschitz assumption common in earlier work. Both families of estimates are expressed solely in terms of the approximation, the residual, and local constants that can be verified without knowing the true solution. Explicit formulas are supplied for linear time-invariant and time-varying systems in terms of the extreme eigenvalues of the symmetric part of the system matrix. The work also examines how the choice between soft and hard enforcement of initial conditions affects the informativeness of the lower certificate and proposes a signed-residual finite-probe alternative for the linear case.

Core claim

Under a localized strong monotonicity condition on certified state-space domains, the pointwise error of a PINN solution to an ODE admits a computable a posteriori lower bound that depends only on the residual and local growth constants; this lower bound can be combined with an upper bound obtained from a one-sided Lipschitz condition to produce two-sided, fully computable error enclosures that require no access to the exact solution.

What carries the argument

Localized strong monotonicity condition (for lower bounds) paired with one-sided Lipschitz condition (for upper bounds) on certified state-space domains, yielding residual-based a posteriori error estimates.

If this is right

  • Two-sided error enclosures become available for PINN solutions of ODEs on domains where the monotonicity and Lipschitz conditions can be checked.
  • For linear systems the bounds reduce to explicit expressions involving only the minimal and maximal eigenvalues of the symmetric part of the system matrix.
  • A certificate-informed training procedure can use the propagated upper bound as an auxiliary regularizer while retaining lower certificates as post-training diagnostics.
  • Exact enforcement of initial conditions can render the scalar lower certificate uninformative, but a signed-residual finite-probe certificate recovers nontrivial lower information in the linear setting.

Where Pith is reading between the lines

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

  • The same residual-plus-monotonicity strategy could be tested on classes of nonlinear ODEs where the required local constants remain computable from the vector field alone.
  • If the certified domains can be enlarged adaptively during training, the framework might produce progressively tighter error bands without retraining from scratch.
  • The distinction between soft and hard initial-condition enforcement suggests analogous trade-offs may appear when PINNs are applied to boundary-value problems.
  • The eigenvalue formulas for linear systems invite direct comparison with classical a posteriori estimators used in finite-element or finite-difference schemes for the same ODEs.

Load-bearing premise

The ODE must satisfy a localized strong monotonicity condition on the chosen domains for the lower bound and a one-sided Lipschitz condition for the upper bound, and both conditions must be verifiable from the data without the exact solution.

What would settle it

For an ODE whose exact solution is known, compute the PINN approximation, evaluate the derived lower and upper bounds on a certified domain, and check whether the true error lies strictly outside the interval formed by those bounds.

Figures

Figures reproduced from arXiv: 2606.12050 by Agamirza Bashirov, Arzu Ahmadova, Ismail Huseynov.

Figure 1
Figure 1. Figure 1: Nonlinear radial-growth plus rotation system. Left: exact and surrogate [PITH_FULL_IMAGE:figures/full_fig_p032_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Diagnostic sharpness comparison for the nonlinear radial-growth plus [PITH_FULL_IMAGE:figures/full_fig_p033_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Selected components of the 96-dimensional stiff system. The surrogate [PITH_FULL_IMAGE:figures/full_fig_p034_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Certificates for the 96-dimensional stiff system. The coordinate-unit [PITH_FULL_IMAGE:figures/full_fig_p035_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Certificate-informed training on the unstable oscillatory system. The [PITH_FULL_IMAGE:figures/full_fig_p036_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Post-training certificate evaluation for the unstable oscillatory system. [PITH_FULL_IMAGE:figures/full_fig_p036_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Two-stage training history. Stage one uses the standard PINN loss. Stage [PITH_FULL_IMAGE:figures/full_fig_p036_7.png] view at source ↗
read the original abstract

Physics-informed neural networks (PINNs) combine machine learning with physical laws to solve differential equations. While existing results provide rigorous \emph{a posteriori} upper bounds for PINN prediction errors, complete certification also requires complementary lower information in order to obtain computable two-sided error enclosures. In this paper, we derive computable \emph{a posteriori} lower bounds for PINN errors in ordinary differential equations on suitable certified state-space domains under a localized strong monotonicity condition. We combine these estimates with complementary localized upper bounds under a one-sided Lipschitz condition, which is weaker than the global Lipschitz assumption used in previous work and can yield sharper upper error bands. The resulting bounds depend only on the neural-network approximation, the ODE residual, and local monotonicity and growth constants, and therefore do not require access to the exact solution. For linear time-invariant and time-varying systems, we further derive explicit formulas in terms of the minimal and maximal eigenvalues of the symmetric part of the system matrix. We also discuss the distinction between soft and hard enforcement of initial conditions in PINNs and explain why exact enforcement can make the scalar lower certificate uninformative. To recover nontrivial lower information in the linear setting, we use a signed-residual finite-probe certificate based on coordinate unit vectors. We also formulate a certificate-informed training strategy in which the propagated upper certificate is used as an auxiliary regularizer, while lower certificates remain post-training diagnostics. Altogether, the proposed framework provides rigorous and practically computable error certificates for PINN approximations of ODEs, while making explicit the domains and model classes for which the assumptions can be verified.

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

2 major / 1 minor

Summary. The manuscript derives computable a posteriori lower bounds on PINN approximation errors for ODEs on certified state-space domains under a localized strong monotonicity condition, paired with complementary upper bounds under a one-sided Lipschitz condition (weaker than global Lipschitz). Explicit eigenvalue formulas are given for linear time-invariant and time-varying systems; the work also distinguishes soft versus hard initial-condition enforcement, introduces a signed-residual finite-probe certificate for nontrivial lower bounds in the linear case, and proposes using the propagated upper certificate as an auxiliary regularizer during training while treating lower certificates as post-training diagnostics. All bounds are claimed to depend only on the network, the ODE residual, and local constants without requiring the exact solution.

Significance. If the localized monotonicity and growth constants can be certified a posteriori as asserted, the framework supplies the first rigorous two-sided error enclosures for PINNs on ODEs under assumptions weaker than those in prior work. The explicit linear-system formulas and the certificate-informed training strategy are concrete strengths that could improve reliability assessment in scientific machine learning. The explicit delineation of model classes and domains where the assumptions are verifiable is also a positive contribution.

major comments (2)
  1. [Abstract / §3] Abstract and §3 (localized strong monotonicity construction): the central claim that the lower bounds are 'computable' and 'do not require access to the exact solution' depends on the existence of a general, non-circular procedure to locate certified state-space domains and evaluate the monotonicity/growth constants from the residual alone. For nonlinear ODEs no such algorithm is supplied; any search over state space risks either depending on a priori knowledge comparable to solving the ODE or producing empty certificates. This assumption is load-bearing for the 'reliable error estimation' title claim.
  2. [Abstract] Abstract (one-sided Lipschitz upper bounds): while the weakening from global to one-sided Lipschitz is correctly noted as potentially sharper, the manuscript must still demonstrate that the local growth constants remain computable without the exact solution on the same certified domains used for the lower bound; otherwise the two-sided enclosure is not fully a posteriori.
minor comments (1)
  1. [Abstract] The distinction between soft and hard enforcement of initial conditions and the signed-residual probe are clearly motivated, but the precise statement of the finite-probe certificate (coordinate unit vectors) should be given as an explicit proposition or algorithm rather than only in prose.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and for highlighting the strengths of the framework. We address the two major comments below, clarifying the scope of our a posteriori claims while acknowledging limitations in the current presentation for nonlinear systems.

read point-by-point responses

  1. Referee: [Abstract / §3] Abstract and §3 (localized strong monotonicity construction): the central claim that the lower bounds are 'computable' and 'do not require access to the exact solution' depends on the existence of a general, non-circular procedure to locate certified state-space domains and evaluate the monotonicity/growth constants from the residual alone. For nonlinear ODEs no such algorithm is supplied; any search over state space risks either depending on a priori knowledge comparable to solving the ODE or producing empty certificates. This assumption is load-bearing for the 'reliable error estimation' title claim.

    Authors: We agree that the manuscript does not supply a general algorithm for locating certified domains in nonlinear cases, and that this step is essential for the bounds to be fully practical. The core contribution is the derivation of the error estimates themselves: once a domain is certified (by any means) where localized strong monotonicity holds, the lower bound is then computable from the residual and network weights alone, without the exact solution. For linear systems the domains and eigenvalue-based constants are explicit. We will revise the abstract, §3, and add a short subsection discussing practical certification approaches (e.g., residual-guided sampling combined with interval arithmetic or Lipschitz-constant verification on candidate boxes) to make the scope and limitations explicit. We do not claim a universal black-box search procedure, and will adjust wording to avoid overstating generality. revision: partial

  2. Referee: [Abstract] Abstract (one-sided Lipschitz upper bounds): while the weakening from global to one-sided Lipschitz is correctly noted as potentially sharper, the manuscript must still demonstrate that the local growth constants remain computable without the exact solution on the same certified domains used for the lower bound; otherwise the two-sided enclosure is not fully a posteriori.

    Authors: The local one-sided Lipschitz growth constants are properties of the known vector field f on the certified domain and can be computed directly from f (via its Jacobian or finite differences) restricted to that domain, without any reference to the unknown exact solution. Because the same certified domains are used for both bounds, and the constants depend only on f and the domain geometry, the upper bound remains a posteriori with respect to the PINN approximation. We will add an explicit statement and a short example in the revised manuscript confirming that these constants are evaluated solely from the ODE right-hand side on the certified region. revision: yes

Circularity Check

0 steps flagged

No circularity: bounds derived mathematically from residuals and explicit assumptions

full rationale

The paper derives computable a posteriori lower and upper bounds directly from the neural network residual, the ODE, and local monotonicity/growth constants under stated conditions (localized strong monotonicity and one-sided Lipschitz). For linear cases, explicit eigenvalue formulas are provided. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain. The assumptions are external and verifiable in principle (e.g., via eigenvalues for linear systems), making the central claims independent of the target error quantities. This is the standard non-circular case for a mathematical error-estimation paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on domain assumptions of localized strong monotonicity and one-sided Lipschitz continuity on certified state-space regions; these are not derived inside the paper but taken as given for the ODE class under consideration. No free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption The ODE satisfies a localized strong monotonicity condition on suitable certified state-space domains
    Invoked to obtain the lower error bound; stated in the abstract as the condition under which the lower bound holds.
  • domain assumption The ODE satisfies a one-sided Lipschitz condition (weaker than global Lipschitz) on the same domains
    Invoked to obtain the complementary upper bound; explicitly contrasted with the stronger global Lipschitz assumption of prior work.

pith-pipeline@v0.9.1-grok · 5827 in / 1528 out tokens · 27806 ms · 2026-06-27T10:54:14.066813+00:00 · methodology

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Reference graph

Works this paper leans on

28 extracted references · 16 canonical work pages

  1. [1]

    Hillebrecht, B

    B. Hillebrecht, B. Unger, Certified machine learning: a posteriori error estimation for physics-informed neural networks, in: International Joint Conference on Neural Networks, IEEE, 2022, pp. 1–8

    2022

  2. [2]

    Cuomo, et al., Scientific machine learning through physics-informed neural networks: where we are and what’s next, Journal of Scientific Computing 92 (2022) 88

    S. Cuomo, et al., Scientific machine learning through physics-informed neural networks: where we are and what’s next, Journal of Scientific Computing 92 (2022) 88. 38

    2022

  3. [3]

    Raissi, P

    M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of ComputationalPhysics378(2019)686–707.doi:10.1016/j.jcp.2018. 10.045

    work page doi:10.1016/j.jcp.2018 2019

  4. [4]

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, L. Yang, Physics-informed machine learning, Nature Reviews Physics 3 (6) (2021) 422–440

    2021

  5. [5]

    Cortés-Ciriano, A

    I. Cortés-Ciriano, A. Bender, Deep confidence: a computationally effi- cient framework for calculating reliable prediction errors for deep neural networks, Journal of Chemical Information and Modeling 59 (3) (2019) 1269–1281

    2019

  6. [6]

    Minakowski, T

    P. Minakowski, T. Richter, Error estimates for neural network solutions ofpartialdifferential equations, arXivpreprintArXiv:2107.11035(2021)

    work page arXiv 2021

  7. [7]

    van der Meer, C

    R. van der Meer, C. W. Oosterlee, A. Borzi, Goal-oriented er- ror estimation for physics-informed neural networks, arXiv preprint- ArXiv:2203.04247 (2022)

    work page arXiv 2022

  8. [8]

    J. S. Hesthaven, G. Rozza, B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations, Springer, 2016

    2016

  9. [9]

    E.Hairer, S.P.Nørsett, G.Wanner, SolvingOrdinaryDifferentialEqua- tions I: Nonstiff Problems, Springer, 2008

    2008

  10. [10]

    T. D. Ryck, S. Mishra, Generic bounds on the approximation error for physics-informedandoperatorlearning, AdvancesinNeuralInformation Processing SystemsArXiv:2205.11393 (2022)

    work page arXiv 2022

  11. [11]

    Haugen, A

    E. Haugen, A. Stepanenko, A. C. Hansen, Trustworthy AI in numerics: on verification algorithms for neural network-based PDE solvers, arXiv preprintArXiv:2509.26122 (2025)

    work page arXiv 2025

  12. [12]

    Hillebrecht, B

    B. Hillebrecht, B. Unger, Rigorous a posteriori error bounds for PDE- defined physics-informed neural networks, IEEE Transactions on Neural Networks and Learning Systems 36 (1) (2025) 1583–1593

    2025

  13. [13]

    Hillebrecht, B

    B. Hillebrecht, B. Unger, Prediction error certification for pinns: Theory, computation, and application to stokes flow, arXiv preprint arXiv:2508.07994 (2025). 39

    work page arXiv 2025

  14. [14]

    Eiras, A

    F. Eiras, A. Bibi, R. R. Bunel, K. D. Dvijotham, P. Torr, M. P. Kumar, Efficient error certification for physics-informed neural networks, in: In- ternational Conference on Machine Learning, Vol. 235 of Proceedings of Machine Learning Research, 2024, pp. 12318–12347

    2024

  15. [15]

    Fanaskov, A

    V. Fanaskov, A. Rudikov, I. Oseledets, Neural functional a posteriori error estimates, arXiv preprintArXiv:2402.05585 (2024)

    work page arXiv 2024

  16. [16]

    M. Guo, E. Haghighat, Energy-based error bound of physics-informed neural network solutions in elasticity, arXiv preprintArXiv:2010.09088 (2020)

    work page arXiv 2010

  17. [17]

    Chowell, L

    G. Chowell, L. Sattenspiel, S. Bansal, C. Viboud, Mathematical models to characterize early epidemic growth: A review, Physics of Life Reviews 18 (2016) 66–97.doi:10.1016/j.plrev.2016.07.005

    work page doi:10.1016/j.plrev.2016.07.005 2016

  18. [18]

    Diekmann, J

    O. Diekmann, J. A. P. Heesterbeek, M. G. Roberts, The construction of next-generation matrices for compartmental epidemic models, Journal of the Royal Society Interface 7 (47) (2010) 873–885.doi:10.1098/ rsif.2009.0386

    work page arXiv 2010

  19. [19]

    M. T. Hoang, M. J. Ehrhardt, Differential equation models for infec- tious diseases: Mathematical modeling, qualitative analysis, numeri- cal methods and applications, SeMA Journal (2025).doi:10.1007/ s40324-025-00404-9

    2025

  20. [20]

    Schuster, What is special about autocatalysis?, Monat- shefte für Chemie – Chemical Monthly (2019).doi:10.1007/ s00706-019-02437-z

    P. Schuster, What is special about autocatalysis?, Monat- shefte für Chemie – Chemical Monthly (2019).doi:10.1007/ s00706-019-02437-z

    2019

  21. [21]

    A. I. Hanopolskyi, V. A. Smaliak, A. I. Novichkov, S. N. Semenov, Autocatalysis: Kinetics, mechanisms and design, ChemSystemsChem (2020).doi:10.1002/syst.202000026

    work page doi:10.1002/syst.202000026 2020

  22. [22]

    D. V. Kriukov, J. Huskens, A. S. Y. Wong, Exploring the programmabil- ityofautocatalyticchemicalreactionnetworks, NatureCommunications (2024).doi:10.1038/s41467-024-52649-z

    work page doi:10.1038/s41467-024-52649-z 2024

  23. [23]

    L. Lu, R. Pestourie, W. Yao, Z. Wang, F. Verdugo, S. G. Johnson, Physics-informed neural networks with hard constraints for inverse de- sign, SIAM Journal on Scientific Computing 43 (6) (2021) B1105– B1132.doi:10.1137/21M1397908. 40

    work page doi:10.1137/21m1397908 2021

  24. [24]

    B. Hao, U. Braga-Neto, C. Liu, L. Wang, M. Zhong, Stability in train- ing pinns for stiff pdes: Why initial conditions matter, arXiv preprint arXiv:2404.16189 (2024).doi:10.48550/arXiv.2404.16189

    work page doi:10.48550/arxiv.2404.16189 2024

  25. [25]

    V. I. Arnold, Ordinary Differential Equations, MIT Press, 1978

    1978

  26. [26]

    L. B. Rall, Automatic Differentiation: Techniques and Applications, Vol. 120 of Lecture Notes in Computer Science, Springer, 1981

    1981

  27. [27]

    J. E. Marsden, T. S. Ratiu, Introduction to Mechanics and Symme- try: A Basic Exposition of Classical Mechanical Systems, Vol. 17 of Texts in Applied Mathematics, Springer, New York, 1999.doi: 10.1007/978-0-387-21792-5

    work page doi:10.1007/978-0-387-21792-5 1999

  28. [28]

    Y.Nutku, Hamiltonianstructureofthelotka-volterraequations, Physics Letters A 145 (1) (1990) 27–28.doi:10.1016/0375-9601(90)90270-X. 41

    work page doi:10.1016/0375-9601(90)90270-x 1990