Pith. sign in

REVIEW 4 major objections 4 minor 1 cited by

Prediction error certification for PINNs: Theory, computation, and application to Stokes flow

T0 review · 4 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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

Pith's one-line read A semigroup-based a posteriori error estimator for PINNs is extended so its stability constants can be approximated numerically, yielding certified error bounds for Stokes flow around a cylinder.

desk verdict Theorem 3.1 is a real improvement over the authors' earlier ISS bound and the heat example is rigorously validated, but the Stokes 'certification' rests on guessed constants and a flawed projection argument, so the abstract overstates what is proven. read the letter →

arxiv 2508.07994 v1 pith:744N4LUM submitted 2025-08-11 math.NA cs.LGcs.NA

Birgit Hillebrecht , Benjamin Unger This is my paper · ORCID
classification math.NAcs.LGcs.NA MSC 65N1547D0635A3535F1641A65
keywords physics-informedneuralnetworksaposteriorierrorboundsemigrouptheoryTrotter-KatoapproximationStokesflowboundaryconditionsinput-to-statestabilitycertification
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) approximate PDE solutions by training a network to minimize residuals, but bounding the resulting prediction error has been limited to academic examples because the stability constants were hard to compute. This paper extends a semigroup-based a posteriori error estimator to problems on complicated domains by replacing analytically derived input-to-state stability functions with constants that can be approximated by standard discretization, such as finite differences or finite elements. The modified bound, built on the Fattorini trick for homogenizing boundary conditions, requires only the PINN residuals and numerically estimated semigroup growth bounds and operator norms. The paper proves the approximation strategy is justified, shows it recovers exact analytic constants for the 1D heat equation, and demonstrates a certified error upper bound for 2D Stokes flow around a cylinder.

What carries the argument

The Fattorini trick: homogenize the boundary condition by writing the error as D0 delta_b(t) plus a semigroup evolution whose source terms involve delta, delta_b_dot, and A D0 delta_b. This removes the need for input-to-state stability growth functions. The numerical constants are obtained via Trotter-Kato approximation of semigroups (Theorem 3.4) and Banach-Steinhaus bounds for ||D0|| and ||AD0||.

What would settle it

Compute verified enclosures of omega, ||D0||, and ||AD0|| on the Stokes domain (e.g., via interval-arithmetic eigenvalue bounds or a certified finite element error estimator) and check whether the limits reported in Figure 5 lie above them; or compute the PINN error against a reference solution with a proven error bound and check whether epsilon_tot exceeds it at all times. Either failure would show the certification is not an upper bound.

Watch

Extended reading notes

Core claim

The central claim is that the prediction error of a PINN for a linear evolution BIVP can be certified a posteriori by the explicit bound of Theorem 3.1, expressed in terms of the PINN residuals and four constants: the semigroup growth parameters M and omega, and the operator norms ||D0|| and ||AD0||. What makes the bound usable is Theorem 3.4, which shows these constants can be obtained numerically: if a sequence of discretized problems converges in the strong operator topology (in the sense of Trotter-Kato) and the discretized growth parameters converge, then the limits yield a valid growth bound for the continuous semigroup; the boundary operator norms are bounded from above via Banach-Ste

Load-bearing premise

For the Stokes certification to be rigorous, the numerical limits in Figure 5 must be true upper bounds for the growth rate and operator norms of the continuous problem, not merely approximations with unknown error.

Editorial extensions

If this is right

  • A trained PINN for a linear evolution PDE can be certified by computing residuals and a small set of numerical constants; no analytic stability analysis is needed.
  • The estimator applies to systems that are not stable, since the input-to-state stability growth function is no longer required.
  • Including the time derivative of the boundary residual in the PINN loss is motivated by the bound and is shown numerically not to degrade the estimator.
  • Any convergent discretization scheme (finite differences, finite elements) can supply the required constants, so existing simulation tools provide the certification machinery.
  • The Stokes example demonstrates certification on a two-dimensional domain with an obstacle and time-dependent boundary conditions, beyond toy benchmarks.

Reading between the lines

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

  • A natural testable extension is to feed the certified bound into the training loop as a weighted loss term; the paper only ties the bound to loss design heuristically (Remark 3.2).
  • The same constant-approximation strategy could certify PINNs for semilinear problems by linearizing around the network, but the paper does not address the additional linearization error.
  • Genuinely rigorous certification would require verified (interval) enclosures of the eigenvalue and norm limits rather than the visual convergence estimates used in Figure 5; the paper's Stokes claim is therefore best read as an indicator, not a proof, of rigor.
  • The reference error in the Stokes validation is the L2-distance to a FEM solution; comparing against a reference with its own proven error bound, or against an exact solution, would settle whether the displayed bound is indeed a bound on the exact error.
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 / 4 minor

Summary. The paper extends the authors' earlier semigroup-based a posteriori error estimation framework for PINNs. The main theoretical result, Theorem 3.1, is a Fattorini-trick error bound that avoids input-to-state stability assumptions; Theorem 3.4 shows how semigroup growth constants can be approximated via Trotter–Kato limits. The framework is first validated on a one-dimensional heat equation, where the required constants are shown analytically, and then applied to a two-dimensional Stokes flow around a cylinder using finite-element approximations of the stability constants and harmonic feature embeddings for the PINN. The paper claims that the Stokes example demonstrates certification of PINN predictions in a realistic scenario. The code is made available via a Zenodo DOI.

Significance. If fully substantiated, the paper would be a useful step toward making rigorous PINN error estimation practicable for realistic problems. The modified bound in Theorem 3.1 is clean and the idea of computing semigroup growth bounds via Trotter–Kato approximation is sensible. The heat example is backed by a genuine convergence proof (Lemma 4.1) and the repository provides reproducible code; those are concrete strengths. However, the advertised certification in a 'more realistic' scenario rests entirely on the Stokes example, and that example uses non-rigorously guessed constants and a projection-neglect step that is not justified. As a result, the main claim is not yet supported.

major comments (4)
  1. [§4.2.1 and §4.2.3] The Stokes 'certification' uses the non-rigorous route that the authors themselves distinguish in §3: after Theorem 3.4 they write that 'in case mathematical rigor is not required ... guess the corresponding limits'. In §4.2.1 the constants ω*, ||D0||, ||AD0|| are indeed obtained as 'estimated limit values' from visual inspection of Figure 5, with no proof of convergence of the sequences or of the required one-sided inequalities. Without such proof, ε(t) in Theorem 3.1 is not a certificate. Figure 8 therefore cannot demonstrate that the bound is a true upper bound for the Stokes problem.
  2. [§4.2, p. 14] The claim that neglecting the Helmholtz projection P yields upper bounds is not generally valid. The paper states: 'Since P is a projection, the computed quantities will be upper bounds on the desired values.' But the Helmholtz projection does not preserve boundary traces: if D0 is a right inverse of the full trace operator, then P D0 u generally does not satisfy D(P D0 u)=u, because the pressure correction changes the trace. Hence the computed norms ||Dn,0|| and ||AnDn,0|| need not bound the corresponding constants for the Stokes operator A=μPΔ. This is a load-bearing correctness issue for the Stokes demonstration.
  3. [§4.1, after Eq. (4.1); Corollary 4.2] The stated constant ||D0||_{L(U,Z)} ≤ 2/3 is incorrect. For D0(a,b) = (1-x)a + x b and U=R^2 with the Euclidean norm, the Gram matrix is [[1/3,1/6],[1/6,1/3]], whose largest eigenvalue is 1/2; hence the operator norm is 1/√2, not 2/3. Thus the bound in Corollary 4.2 is not a rigorous upper bound as written. This is a concrete error in the heat validation and must be corrected; the figure and text should be rerun with the correct constant.
  4. [§4.2.3, Figure 8; §5] The reference error ε_ref is defined as the L2-distance between the PINN prediction and the FEM solution, not to the exact solution. A plot showing ε_tot ≥ ε_ref only shows that the estimator exceeds an approximate error. It does not verify the statement in §5 that 'the error estimator is indeed a true upper bound on the prediction error'. The authors should either compare against an exact or certified reference solution, or explicitly reframe the Stokes comparison as a heuristic validation only.
minor comments (4)
  1. [Theorem 3.1] The assumption writes zb,δb ∈ W^{1,p}(T;Z), but these functions take values in the boundary space U; it should be W^{1,p}(T;U).
  2. [§4.2.1] The notation ||D0||_{L(Z,U)} and ||AD0||_{L(Z,U)} appears swapped; the operators map from U to Z, so the spaces should be L(U,Z).
  3. [Figure 5 caption] The caption lists 'ω, M, ||AD0||, ||D0||' but the panels show ωn, ||Dn,0||, and ||AnDn,0||; clarify that M is inferred from symmetry and not plotted.
  4. [Appendix A, Eq. (A.1)] The Neumann boundary operator formula is stated without a signed derivation; a brief explanation of the determinant sign would help reproducibility.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the new error bound is derived from the residual and operator constants independently of the target error; the Stokes example has rigor gaps but does not reduce to its own output.

full rationale

Theorem 3.1 is proved in the paper from the explicit mild-solution formula for the error e = z̃ − z (Eq. (3.1)); the bound's right-hand side combines the boundary-extension term D0δb, the semigroup-evolved initial residual, and integrated terms involving δ, δb, δ̇b, with constants ||D0||, ||AD0||, M, ω. None of these quantities is defined through the error being bounded, and the constants are properties of the PDE/semigroup, not of the PINN. In the Stokes example the constants are estimated from FE discretizations before the PINN residual is evaluated, so ε(t) is not fitted to the reference error εref. The projection-neglect step and the guessed limits are genuine rigor gaps: the assertion 'Since P is a projection, the computed quantities will be upper bounds on the desired values' (Sec. 4.2) is not justified because the Helmholtz projection does not preserve the trace right-inverse property, and the Fig. 8 validation against an FEM reference rather than the exact solution cannot establish a rigorous certificate. The manuscript itself flags the non-rigorous route: 'In case mathematical rigor is not required... guess the corresponding limits from the discretized problem' (Sec. 3). These are correctness/rigor concerns, not circularity. The self-citations to [24] supply the prior framework and implementation but are not load-bearing: Theorem 3.1 and Theorem 3.4 are proved here with standard semigroup estimates and an external Trotter-Kato reference [25]. Hence no circular step is identified.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central bound depends on standard semigroup theory assumptions and the numerical approximation of operator constants. The Stokes example additionally depends on hand-estimated limits from finite element convergence data, which are not proven to be rigorous upper bounds. No new physical entities are introduced.

free parameters (3)
  • Stokes growth bound omega* = 0.014
    Estimated from numerical convergence of affine FE discretization in Figure 5, no proof of convergence or that it bounds the exact semigroup growth bound.
  • Stokes D0 operator norm = 0.19
    Estimated from numerical limits in Figure 5, used as if an exact upper bound.
  • Stokes AD0 operator norm = 0.001
    Estimated from numerical limits in Figure 5, used as if an exact upper bound.
assumptions (5)
  • domain assumption Assumption 2.1: A|ker D generates a C0 semigroup on Z and D has a bounded right-inverse D0.
    Required for well-posedness and for the Fattorini trick in Theorem 3.1.
  • domain assumption Boundary perturbation regularity: zb, delta_b in W^{1,p}(T;Z) and delta in L^p(T;Z).
    Needed for the mild solution formula used in Theorem 3.1.
  • domain assumption The discretized operators satisfy SOT convergence and (M_n), (omega_n) are Cauchy sequences.
    Needed for Theorem 3.4 to bound M and omega by limits.
  • domain assumption For Stokes, omitting the Helmholtz projection P in the constant estimation gives upper bounds.
    Since P is a projection, ||P Delta D0|| <= ||Delta D0||; used in Section 4.2.1.
  • domain assumption The PINN is sufficiently smooth (e.g., tanh activation) so residuals delta, delta_b, delta_0 are well-defined.
    Standard PINN smoothness assumption, stated in Section 1.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Prediction error certification for PINNs: Theory, computation, and application to Stokes flow." pith.science (2026). https://pith.science/paper/744N4LUM

@misc{pith2026250807994,
  author       = {Pith},
  title        = {Pith review of: Prediction error certification for PINNs: Theory, computation, and application to Stokes flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/744N4LUM}},
  note         = {Machine review of arXiv:2508.07994}
}
read the original abstract

Rigorous error estimation is a fundamental topic in numerical analysis. With the increasing use of physics-informed neural networks (PINNs) for solving partial differential equations, several approaches have been developed to quantify the associated prediction error. In this work, we build upon a semigroup-based framework previously introduced by the authors for estimating the PINN error. While this estimator has so far been limited to academic examples - due to the need to compute quantities related to input-to-state stability - we extend its applicability to a significantly broader class of problems. This is accomplished by modifying the error bound and proposing numerical strategies to approximate the required stability parameters. The extended framework enables the certification of PINN predictions in more realistic scenarios, as demonstrated by a numerical study of Stokes flow around a cylinder.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. A Hybrid Framework for Healing Semigroups with Machine Learning

    math.RA 2025-09 conditional novelty 5.0 of 10

    A hybrid random-forest-plus-deterministic method heals corrupted finite semigroup tables, restoring associativity in 95% of small cases and 60% at n=10.

Reference graph

Works this paper leans on

52 extracted references · 51 canonical work pages · cited by 1 Pith paper

  1. [1]

    I. A. Baratta, J. P. Dean, J. S. Dokken, M. Habera, J. Hale, C. N. Richardson, M. E. Rognes, M. W. Scroggs, N. Sime, and G. N. Wells . DOLFINx: the next generation FEniCS problem solving environment, 2023

    work page 2023

  2. [2]

    Berrone, C

    S. Berrone, C. Canuto, and M. Pintore . Solving PDEs by variational physics-informed neural networks: an a posteriori error analysis.Ann. Univ. Ferrara Sez. VII Sci. Mat., 68(2):575–595, 2022

    work page 2022

  3. [3]

    Berrone, C

    S. Berrone, C. Canuto, and M. Pintore . Variational physics informed neural networks: the role of quadratures and test functions.J. Sci. Comput., 92(3):100, 2022

    work page 2022

  4. [4]

    H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Springer-Verlag, first edition, 2011

    work page 2011

  5. [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(12):1727–1738, 2021

    work page 2021

  6. [6]

    A. J. Chorin and J. E. Marsden . A mathematical introduction to fluid mechanics. Texts Appl. Math. Springer-Verlag, third edition, 1993

    work page 1993

  7. [7]

    Cuomo, V

    S. Cuomo, V. S. Di Cola, F. Giampaolo, G. Rozza, M. Raissi, and F. Piccialli . Scientific machine learning through physics–informed neural networks: Where we are and what’s next.J. Sci. Comput., 92(3):88, 2022

    work page 2022

  8. [8]

    R. F. Curtain and H. Zwart . Introduction to Infinite-Dimensional Systems Theory: A State-Space Approach. Texts in Applied Mathematics. Springer-Verlag, 2020

    work page 2020

Show all 52 references
  1. [9]

    De Ryck, A

    T. De Ryck, A. D. Jagtap, and S. Mishra . Error estimates for physics-informed neural networks approximating the Navier–Stokes equations.IMA J. Numer. Anal., 44(1):83–119, 2024

    2024

  2. [10]

    De Ryck and S

    T. De Ryck and S. Mishra . Generic bounds on the approximation error for physics-informed (and) operator learning.Adv. Neural. Inf. Process. Syst., 35:10945–10958, 2022

    2022

  3. [11]

    De Ryck and S

    T. De Ryck and S. Mishra . Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning.Acta Numer., 33:633–713, July 2024

    2024

  4. [12]

    Engel and R

    K.-J. Engel and R. Nagel . One-parameter semigroups for linear evolution equations. Graduate Texts in Mathematics. Springer-Verlag, first edition, 2000

    2000

  5. [13]

    Ernst and K

    L. Ernst and K. Urban . A certified wavelet-based physics-informed neural network for the solution of parameterized partial differential equations.IMA J. Numer. Anal., 45(1):494–515, 2024

    2024

  6. [14]

    L. C. Evans. Partial Differential Equations, volume 19 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2010

    2010

  7. [15]

    H. O. F attorini. Boundary control systems.SIAM J. Cont., 6(3):349–385, 1968

    1968

  8. [16]

    Gagliardo

    E. Gagliardo. Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili. Rend. Sem. Mat. Univ. Padova, 27:284–305, 1957

    1957

  9. [17]

    Gao and Y

    J. Gao and Y. Zakharian . PINNs error estimates for nonlinear equations in R-smooth Banach spaces. ArXiv e-print 2305.11915, 2024

    2024 arXiv

  10. [18]

    G. Greiner. Perturbing the boundary conditions of a generator.Houston J. Math., 13(2):213–229, 1987. 20 BIRGIT HILLEBRECHT ⋆ AND BENJAMIN UNGER†

    1987

  11. [19]

    Grohs and F

    P. Grohs and F. Voigtlaender . Proof of the theory-to-practice gap in deep learning via sampling complexity bounds for neural network approximation spaces.Found. Comput. Math., 24:1085–1143, 2024

    2024

  12. [20]

    T. G. Grossmann, U. J. Komorowska, J. Latz, and C.-B. Schönlieb . Can physics-informed neural networks beat the finite element method?IMA J. Numer. Anal., 89(1):143–174, 2024

    2024

  13. [21]

    Haghighat, M

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

    2021

  14. [22]

    N. J. Higham . Functions of matrices: theory and computation. Other Titles in Applied Mathematics. Society for Industrial and Applied Mathematics, first edition, 2008

    2008

  15. [23]

    Hillebrecht and B

    B. Hillebrecht and B. Unger . Certified machine learning: A posteriori error estimation for physics- informed neural networks. In2022 International Joint Conference on Neural Networks (IJCNN), pages 1–8, 2022

    2022

  16. [24]

    Hillebrecht and B

    B. Hillebrecht and B. Unger . Rigorous a posteriori error bounds for PDE-defined PINNs.IEEE Trans. Neural Netw. Learn. Syst., 36(1):1583–1593, 2025

    2025

  17. [25]

    Ito and F

    K. Ito and F. Kappel . The Trotter-Kato theorem and approximation of PDEs.Math. Comp., 67(221):21–44, 1998

    1998

  18. [26]

    Jacob, R

    B. Jacob, R. Nabiullin, J. R. Partington, and F. L. Schwenninger . Infinite-dimensional input-to-state stability and Orlicz spaces.SIAM J. Cont. Optim., 56(2):868–889, 2018

    2018

  19. [27]

    A. D. Jagtap and G. E. Karniadakis . Extended physics-informed neural networks (xPINNs): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations.Commun. Comput. Phys., 28(5):2002–2041, 2020

    2002

  20. [28]

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. W ang, and L. Yang . Physics- informed machine learning.Nat. Rev. Phys., 3(6):422–440, 2021

    2021

  21. [29]

    Kast and J

    M. Kast and J. S. Hesthaven . Positional embeddings for solving PDEs with evolutional deep neural networks. J. Comput. Phys., 508:112986, 2024

    2024

  22. [30]

    Kharazmi, Z

    E. Kharazmi, Z. Zhang, and G. E. Karniadakis . Variational physics-informed neural networks for solving partial differential equations.ArXiv e-print 1912.00873, 2019

    1912 arXiv

  23. [31]

    D. P. Kingma and J. Ba . Adam: A method for stochastic optimization. InInternational Conference on Learning Representations, 2015

    2015

  24. [32]

    T. G. Kurtz . A general theorem on the convergence of operator semigroups.Trans. Amer. Math. Soc., 148(1):23–32, 1970

    1970

  25. [33]

    G. Leoni. A First Course in Sobolev Spaces, volume 181 ofGraduate studies in mathematics. American Mathematical Society, 2 edition, 2017

    2017

  26. [34]

    R. J. LeVeque . Finite Difference Methods for Ordinary and Partial Differential Equations. Society for Industrial and Applied Mathematics, 2007

    2007

  27. [35]

    D. C. Liu and J. Nocedal . On the limited memory BFGS method for large scale optimization.Math. Prog., 45:503–528, 1989

    1989

  28. [36]

    L. Maly, N. Shanmugalingam, and M. Snipes . Trace and extension theorems for functions of bounded variation.ArXiv e-print 1511.04503, 2015

    2015 arXiv

  29. [37]

    M. D. McKay, R. J. Beckman, and W. J. Conover . A comparison of three methods for selecting values of input variables in the analysis of output from a computer code.Technometrics, 21(2):239–245, 1979

    1979

  30. [38]

    H. N. Mhaskar, E. Tsoukanis, and A. D. Jagtap . An approximation theory perspective on machine learning.ArXiv e-print 2506.02168, 2025

    2025

  31. [39]

    Mojgani, M

    R. Mojgani, M. Balajewicz, and P. Hassanzadeh . Kolmogorov n-width and Lagrangian physics- informed neural networks: a causality-conforming manifold for convection-dominated PDEs.Comput. Meth. Appl. Mech. Eng., 404:115810, 2023

    2023

  32. [40]

    Moler and C

    C. Moler and C. V an Loan . Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later.SIAM Rev., 45(1):3–49, 2003

    2003

  33. [41]

    R. Namba. Rate of convergence in Trotter’s approximation theorem and its applications.Tokyo J. Math., 1(1), 2023

    2023

  34. [42]

    A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer Science & Business Media, first edition, 1989. PREDICTION ERROR CERTIFICATION FOR PINNS 21

    1989

  35. [43]

    J. Peetre. A counterexample connected with Gagliardo’s trace theorem.Comment. Math. Special Issue, 2:277–282, 1979

    1979

  36. [44]

    Radin, S

    N. Radin, S. Klinkel, and O. Altay . Effects of variational formulations on physics-informed neural network performance in solid mechanics.Proc. Appl. Math. Mech., 23(4):e202300222, 2023

    2023

  37. [45]

    Raissi, P

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

    2019

  38. [46]

    Rojas, P

    S. Rojas, P. Maczuga, J. Muñoz-Matute, D. Pardo, and M. Paszyński . Robust variational physics-informed neural networks.Comput. Meth. Appl. Mech. Eng., 425:116904, 2024

    2024

  39. [47]

    Schäfer, S

    M. Schäfer, S. Turek, F. Durst, E. Krause, and R. Rannacher . Benchmark computations of laminar flow around a cylinder. InE. Hirschel, editor,Flow Simulation with High-Performance Computers II, volume 48 ofNotes on Numerical Fluid Mechanics (NNFM), pages 547–566. Vieweg + Teub...

    1996

  40. [48]

    J. D. Schmid, P. Bauerschmidt, C. Gurbuz, M. Eser, and S. Marburg . Physics-informed neural networks for acoustic boundary admittance estimation.Mech. Sys. Signal Proc., 215:111405, 2024

    2024

  41. [49]

    F. L. Schwenninger . Input-to-state stability for parabolic boundary control: linear and semilinear systems. InControl theory of infinite-dimensional systems, pages 83–116. Springer-Verlag, 2020

    2020

  42. [50]

    Seifert, S

    C. Seifert, S. Trostorff, and M. W aurick . Evolutionary Equations: Picard’s Theorem for Partial Differential Equations, and Applications. Operator Theory: Advances and Applications. Birkhäuser, 2022

    2022

  43. [51]

    E. D. Sontag . Smooth stabilization implies coprime factorization.IEEE Trans. Automat. Control, 34(4):435–443, 1989

    1989

  44. [52]

    O. C. Zienkiewicz and R. L. Taylor . The finite element method set: Its basis and fundamentals. Elsevier, 6 edition, 2013. Appendix A. Derivation of the boundary operator matrices We here formalize the step of deriving Dn and Dn,0 for the Stokes example from Section 4.2. For e...

    2013

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.