Pith. sign in

REVIEW 2 major objections 4 minor 49 references

Operator-split Bayesian posteriors for elliptic PDEs contract at near-minimax rates when interior and boundary data are unequal.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

An operator-split Bayesian posterior, which pushes independent neural-network posteriors for the source and boundary through the elliptic solution map, contracts around the true solution at a near-minimax rate.

T0 review reviewed 2026-08-02 challenge →

load-bearing objection The operator-split posterior construction is genuinely new and the main concentration theorem is likely correct as an application of existing BNN regression contraction, but the advertised near-minimax optimality is not proven and the single-chart assumption does not cover the square experiment. the 2 major comments →

arxiv 2607.14680 v1 pith:Q7JGHE6L submitted 2026-07-16 math.NA cs.NAmath.STstat.TH

Operator-Split Bayesian Learning for Elliptic PDEs with Unequal Interior and Boundary Data

Emmanuel E. Oguadimma This is my paper
classification math.NA cs.NAmath.STstat.TH
keywords Bayesian neural networkselliptic PDEsposterior contractionoperator splittinguncertainty quantificationDirichlet boundary conditionsimbalanced dataphysics-informed loss
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

The paper proposes a Bayesian way to learn the solution of a second-order elliptic Dirichlet problem when interior source measurements and boundary measurements are noisy, imbalanced in number, and treated as separate statistical problems. It assigns independent neural-network priors to the source term and to the boundary data, combines the two posteriors into a product, and pushes this distribution through the elliptic solution map. The central claim is that the resulting posterior contracts around the true solution at a rate that separates into an interior term depending on the d-dimensional sample size and a boundary term depending on the (d-1)-dimensional boundary sample size, matching the minimax lower bound up to logarithms. A sympathetic reader would care because this gives a rigorous uncertainty-quantification guarantee for a practical split PINN construction and shows how to allocate sampling effort between volume and boundary.

Core claim

Let u* solve Lu=f in Ω with u=g on ∂Ω. Theorem 4.2 asserts that, when f is β−2-Hölder and g is β-Hölder and a single boundary chart covers ∂Ω up to measure zero, the operator-split posterior satisfies Π_S({u : E(u) > M^2 R^2} | D) → 0 in true-data probability, where R^2 = n_Ω^{−2(β−2)/(d+2(β−2))} (log n_Ω)^{2γ} + λ n_∂^{−2β/(d−1+2β)} (log n_∂)^{2γ}. Because every posterior draw has E(u)=∥F−f∥²_{L²(Ω)}+λ∥G−g∥²_{L²(∂Ω)}, this means the learned source and boundary functions contract separately at their natural dimensional rates. Up to logarithmic factors, the radius matches the cited minimax lower bound, so the split construction is claimed to be near-optimal.

What carries the argument

The central object is the elliptic solution map S(F,G)=S₀F+S₁G, where S₀ solves Lv=F with zero boundary data and S₁ lifts boundary data G to a solution of Lw=0. The key identity is the operator-split loss E(S(F,G))=∥F−f∥²_{L²(Ω)}+λ∥G−g∥²_{L²(∂Ω)}, which converts posterior contraction for f and g directly into posterior contraction for u. Boundary observations are re-expressed in local coordinates through a chart φ:B⊂R^{d−1}→∂Ω, so the boundary regression lives on a (d−1)-dimensional space. The product posterior factorization and pushforward through S carry the two independent contraction guarantees to the solution space.

Load-bearing premise

The theorem assumes one smooth boundary chart maps an open subset of R^{d−1} onto the entire boundary up to a measure-zero set; for the square in the experiments and for topologically nontrivial or cornered boundaries no such single connected chart exists, so the central contraction guarantee does not literally cover those cases without a finite-atlas argument.

What would settle it

Check whether the boundary of the unit square admits a single connected chart φ:B⊂R→∂Ω that is a diffeomorphism onto ∂Ω minus a measure-zero set. It does not: the smooth part of a square boundary has four connected components, while B is connected, so the theorem's single-chart hypothesis is false for the paper's own 2D experiment. A re-run of that experiment therefore cannot be validated by Theorem 4.2 as stated.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the theorem is correct, one can learn elliptic solutions without placing a prior directly on the solution, instead splitting the regression into source and boundary components and propagating uncertainty through a deterministic solver.
  • The two-term contraction radius yields an explicit boundary sampling condition n_∂ ≳ n_Ω^{κ(d,β)} under which the boundary contribution does not dominate, guiding allocation of unequal sampling budgets.
  • Under additional second-moment conditions, the posterior mean contracts at the same radius, and the physics-informed loss controls H^{1/2}(Ω) error via the cited stability estimate.
  • The reuse of one stiffness-matrix factorization across posterior draws makes the propagation step linear in ensemble size, as the numerical experiments report.
  • Up to logarithmic factors, the upper bound matches the two-sample minimax rate, so no statistical price is paid for the operator-split construction in the smooth setting considered.

Where Pith is reading between the lines

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

  • The single-chart assumption is the main constraint: for a square, a torus, or any boundary whose smooth part has several connected components, no one connected chart covers ∂Ω up to measure zero, so the theorem as stated does not cover the paper's own square experiment or general domains without a finite-atlas extension.
  • One testable extension would be to track chart-indexed error terms in a partition-of-unity proof; the rate structure suggests the same separated interior/boundary exponents would survive, but the proof would need explicit treatment of chart overlaps.
  • The separated loss components suggest an adaptive data-collection strategy: estimate the two empirical errors online and allocate new interior versus boundary samples according to the predicted rate imbalance, an idea the paper mentions but does not develop.
  • Because the solution map is linear, the same split-posterior construction may extend to Bayesian inverse problems where source or boundary data are recovered from noisy solution observations; the paper lists this as future work.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper proposes an operator-split Bayesian construction for elliptic Dirichlet problems with independent noisy interior-source and boundary observations. Separate Bayesian neural-network priors are placed on the source f and on a local-coordinate representation of the boundary data g; the product posterior is pushed forward through the linear elliptic solution operator. Theorem 4.2 asserts posterior contraction in the physics-informed loss at radius ε_{nΩ}^2 + λ ε_{n∂}^2, with separated d-dimensional interior and (d−1)-dimensional boundary rates. The proof uses the exact identity E(S(F,G)) = ||F−f||^2 + λ||G−g||^2, the external BNN contraction result Theorem 2.5, and a union bound. The paper further claims near-minimax optimality by comparison with the lower bound of [47, Thm 3.5], derives a boundary sampling budget condition, and reports 1D and 2D numerical experiments with finite-element propagation of posterior samples.

Significance. If Theorem 4.2 is read with its explicit single-chart assumption, the core derivation is clean and the separated-dimensional rate is a genuinely useful contribution: the split-loss identity (3.5) is exact, no fitted constants are used, and the proof is a valid combination of an external contraction theorem and a union bound. The numerical experiments usefully illustrate the two-sample budget trade-off and the stabilization of the boundary contribution as n∂ grows. However, two advertised conclusions go beyond what is proved: the near-minimax point-estimator claim and the scope of the theorem for general smooth boundaries. The paper is a solid conditional contribution, but these gaps need to be addressed before it can be accepted as stated.

major comments (2)
  1. [Corollary 4.5, Remark 4.3, Abstract] The near-minimax claim is not supported. Theorem 4.2 is a posterior concentration statement: for every M_n→∞, Π_S(E > M_n^2 R^2 | D) → 0. This does not control the first posterior moment, and in particular it does not imply E[E(\bar u)] = O(R^2) for the posterior mean. Remark 4.3 explicitly imposes unproved second-moment contractions to conclude E(\bar u) = O_p(R^2). Without such control, the posterior-mean risk could be M_n^2 R^2 with M_n→∞, which is not the claimed rate. Since Theorem 2.7 is a minimax lower bound for point estimators, Corollary 4.5 and the abstract's 'near-minimax upper bound' overstate the result. Please either prove the second-moment contraction or revise the optimality claim to a statement about the contraction radius only.
  2. [Theorem 4.2, §3.2, Remark 3.3, §5.2] The main theorem assumes a single boundary chart φ:B⊂R^{d−1}→∂Ω that covers ∂Ω up to surface measure zero. This is false for general smooth compact boundaries: for example, a sphere or torus does not admit such a single chart, and if B is required compact (as in Lemma 3.1), a chart cannot cover a connected closed boundary up to measure zero. The proof and the contraction statement are one-chart only; Remark 3.3 acknowledges the finite-atlas issue but does not supply a multi-chart version of Theorem 4.2. Moreover, the 2D numerical experiment is on the unit square with the piecewise parametrization γ:[0,4]→∂Ω, which is not a smooth chart and has corners where g is not C^β for β>2; the experiment therefore lies outside the theorem's hypotheses. Please either prove the finite-atlas extension or explicitly restrict the claims and experiments to domains satisfying the stated chart condition.
minor comments (4)
  1. [Theorem 2.7] The notation '(R^d)^{⊗nΩ} × R^{⊗nΩ} × (R^d)^{⊗n∂} × R^{⊗n∂}' is garbled and should be cleaned up.
  2. [§3.2 / Lemma 3.1] The paper alternates between an open set B⊂R^{d−1} in the chart definition and a compact B in Lemma 3.1. Since the single-chart covering assumption is central, the topological assumptions on B should be stated consistently.
  3. [§5.2] Figure 4 is discussed in terms of 's=2' but the horizontal axis is the boundary parameter t; please unify the notation.
  4. [References] References [27]–[30] appear only in general introduction material and are not related to the theoretical argument; they could be trimmed or moved to a broader discussion.

Circularity Check

0 steps flagged

No circularity: the main contraction theorem is a direct application of an external BNN contraction theorem plus the operator-split loss identity; self-citations are not load-bearing.

full rationale

The paper's central result, Theorem 4.2, is not circular. The posterior is defined by pushing forward independent BNN posteriors for the source F and boundary coordinate function eG through the solution operator S, and the proof applies the external nonparametric BNN contraction theorem [14, Theorem 2] separately to the interior regression (F with smoothness beta-2 in dimension d) and the boundary regression (eG with smoothness beta in dimension d-1). The contraction in the physics-informed loss then follows from the exact identity E(u(F,G)) = ||F-f||^2_L2(Omega) + lambda ||G-g||^2_L2(dOmega) (Eq. 3.5), together with the inclusion argument in Eq. (4.6). No fitted constants, no data-dependent quantities, and no self-citations are used to obtain the rates; the rates are inherited from the externally proved [14] result and the external lower bound [47]. The authors' own prior works, [27]-[30], appear only in the introduction/related-work context and are not load-bearing for the derivation. The single-chart assumption, while restrictive and not satisfied by the square experiment, is acknowledged in Remark 3.3 as a removable finite-atlas extension, and is a scope/assumption issue rather than a circularity. Similarly, Remark 4.3 explicitly concedes that the second-moment conditions needed to conclude posterior-mean risk O_p(R^2) are not proved; this is a genuine gap in the advertised near-minimax point-estimator claim, but it is not a circular use of inputs. Corollary 4.5 compares the obtained contraction radius with the external minimax lower bound of [47, Theorem 3.5]; matching an external lower bound is not circular. Overall, the derivation chain is self-contained relative to its cited external theorems, and no prediction reduces by construction to a fitted input or to the paper's own definitions.

Axiom & Free-Parameter Ledger

3 free parameters · 7 axioms · 0 invented entities

The central claim rests on two imported theorems (BNN contraction from [14], minimax lower bound from [47]) and on a paper-specific single-chart assumption. The loss weight lambda and log-exponent gamma are free but rate-neutral. No invented entities.

free parameters (3)
  • boundary penalty weight lambda = lambda=1 in experiments; arbitrary fixed lambda>0 in theory
    Appears linearly in the contraction radius R^2 and in the numerical loss; not fitted but hand-chosen and affects constants only.
  • log-exponent gamma>2 = unspecified (any gamma>2)
    Introduced in Theorem 2.5 and inherited in rates (4.1); a free theoretical choice, not fitted to data.
  • architecture constants C1,C2 in (2.8) = unspecified positive constants
    Define the width/depth schedule in (2.8); existence is asserted by [14], not tuned in this paper.
axioms (7)
  • standard math Unique strong solution and Schauder estimates for Lu=f in Omega, u=g on dOmega with smooth coefficients and beta>2 (Grisvard; Gilbarg-Trudinger)
    Used to define S0, S1 and to ensure u(F,G) in C^beta; invoked in Sec 2.2 and Sec 3.1.
  • domain assumption Theorem 2.5 of [14]: fully-connected BNNs with priors satisfying Assumptions 2.3-2.4 contract in L2(P_X) at n^{-alpha/(2alpha+d)} log^gamma n
    Engine of Corollary 4.1; paper does not prove or verify the assumptions beyond citing [14].
  • domain assumption Theorem 2.7 of [47]: two-sample minimax lower bound for the physics-informed loss
    Used for the near-minimax comparison in Corollary 4.5; assumes the bound is valid for unequal nOmega,nDelta.
  • ad hoc to paper Single boundary chart phi covers dOmega up to surface measure zero
    Imposed in Theorem 4.2; not true for general boundaries; no multi-chart proof is provided despite Remark 3.3.
  • domain assumption X_i~U(Omega), Y_j~U(dOmega), independent Gaussian noises with variances sigma_Omega0^2, sigma_delta0^2; independent likelihoods
    Data model in Sec 2.4 and Sec 3.3; required to apply Theorem 2.5.
  • domain assumption Prior densities on network parameters and variances satisfy Assumptions 2.3 and 2.4
    Assumed in Theorem 4.2; Gaussian priors likely satisfy, but the HMC/SGLD samplers used numerically are not shown to sample the exact posterior satisfying these assumptions.
  • standard math Boundary chart nondegenerate with bounded Jacobian, giving L2(B) ~ L2(dOmega) equivalence (Lemma 3.1)
    Used to convert eg-contraction in L2(P_Z) to G-contraction in L2(dOmega); standard change-of-variables.

reviewed 2026-08-02 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Operator-Split Bayesian Learning for Elliptic PDEs with Unequal Interior and Boundary Data." pith.science (2026). https://pith.science/paper/Q7JGHE6L

@misc{pith2026260714680,
  author       = {Pith},
  title        = {Pith review of: Operator-Split Bayesian Learning for Elliptic PDEs with Unequal Interior and Boundary Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q7JGHE6L}},
  note         = {Machine review of arXiv:2607.14680}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

We propose an operator-split Bayesian learning framework for second-order uniformly elliptic Dirichlet problems with unequal numbers of interior and boundary observations. The data consist of noisy measurements of the source in the domain and noisy measurements of the boundary values. Independent Bayesian neural-network (BNN) priors are assigned to these two quantities, and the resulting product posterior is pushed forward through the elliptic solution operator. We prove that the posterior induced by this construction contracts around the true solution. The contraction radius separates a domain contribution, governed by the second-order elliptic operator, from a boundary contribution, governed by the intrinsic dimension of the boundary. Together with the minimax lower bound of \cite{ZhaoLu2026}, this yields a near-minimax upper bound up to logarithmic factors. Our numerical experiments illustrate the propagation of source and boundary uncertainty and the effects of unequal sampling budgets on the posterior reconstruction.

Figures

Figures reproduced from arXiv: 2607.14680 by Emmanuel E. Oguadimma.

Figure 1
Figure 1. Figure 1: Operator-split Bayesian learning for elliptic problem. Interior observations are used to [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Posterior approximation for the 1D Poisson problem. The source posterior is shown in the [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Interior and boundary sampling. Blue points denote interior observations in [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Boundary-data approximation with nΩ = 800 and n∂ = 200. The figure shows the target data ge(s), the posterior mean Ge(s), and the pointwise 95% credible band. As shown in [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Interior source approximation with nΩ = 800 and n∂ = 200. The panels show the exact source f, the posterior mean F¯, the relative error, and the posterior standard deviation σF [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Samplewise error decomposition with nΩ = 800 and n∂ = 200. The horizontal axis shows the source error E (s) F , and the vertical axis shows the boundary error E (s) G [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Posterior approximation of the solution with [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

49 extracted references · 3 canonical work pages

  1. [1]

    Weight uncertainty in neural network

    Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. Weight uncertainty in neural network. InProceedings of the 32nd International Conference on Machine Learning, volume 37 ofProceedings of Machine Learning Research, pages 1613–1622. PMLR, 2015

    2015

  2. [2]

    Fox, and Carlos Guestrin

    Tianqi Chen, Emily B. Fox, and Carlos Guestrin. Stochastic gradient hamiltonian monte carlo. In Proceedings of the 31st International Conference on Machine Learning, volume 32 ofProceedings of Machine Learning Research, pages 1683–1691. PMLR, 2014

    2014

  3. [3]

    Error analysis for physics-informed neural networks ap- proximating kolmogorov pdes.Advances in Computational Mathematics, 48(6):79, 2022

    Tim De Ryck and Siddhartha Mishra. Error analysis for physics-informed neural networks ap- proximating kolmogorov pdes.Advances in Computational Mathematics, 48(6):79, 2022. doi: 10.1007/s10444-022-09985-9

    doi:10.1007/s10444-022-09985-9 2022

  4. [4]

    The deep ritz method: A deep learning-based numerical algorithm for solving variational problems.Communications in Mathematics and Statistics, 6(1):1–12, 2018

    Weinan E and Bing Yu. The deep ritz method: A deep learning-based numerical algorithm for solving variational problems.Communications in Mathematics and Statistics, 6(1):1–12, 2018. doi: 10.1007/s40304-018-0127-z

    doi:10.1007/s40304-018-0127-z 2018

  5. [5]

    Posterior and variational inference for deep neural networks with heavy-tailed weights.Journal of Machine Learning Research, 26(122):1–58, 2025

    Paul Egels and Ismaël Castillo. Posterior and variational inference for deep neural networks with heavy-tailed weights.Journal of Machine Learning Research, 26(122):1–58, 2025

    2025

  6. [6]

    Evans.Partial Differential Equations, volume 19 ofGraduate Studies in Mathematics

    Lawrence C. Evans.Partial Differential Equations, volume 19 ofGraduate Studies in Mathematics. American Mathematical Society, 2 edition, 2010. ISBN 9780821849743

    2010

  7. [7]

    Convergence ratesof posterior distributions for non-i.i.d

    SubhashisGhosal andAad vander Vaart. Convergence ratesof posterior distributions for non-i.i.d. observations.The Annals of Statistics, 35(1):192–223, 2007. doi: 10.1214/009053606000001172

    doi:10.1214/009053606000001172 2007

  8. [8]

    van der Vaart.Fundamentals of Nonparametric Bayesian Inference, volume 44 ofCambridge Series in Statistical and Probabilistic Mathematics

    Subhashis Ghosal and Aad W. van der Vaart.Fundamentals of Nonparametric Bayesian Inference, volume 44 ofCambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, 2017

    2017

  9. [9]

    Trudinger.Elliptic Partial Differential Equations of Second Order, volume 224 ofClassics in Mathematics

    David Gilbarg and Neil S. Trudinger.Elliptic Partial Differential Equations of Second Order, volume 224 ofClassics in Mathematics. Springer, Berlin, Heidelberg, 2001. doi: 10.1007/ 978-3-642-61798-0. Reprint of the 1998 edition. 25

    2001

  10. [10]

    Michael B. Giles. Multilevel monte carlo methods.Acta Numerica, 24:259–328, 2015. doi: 10. 1017/S096249291500001X

    2015

  11. [11]

    Bayesian physics-informed neural net- works for the subsurface tomography based on the eikonal equation.IEEE Transactions on Geo- science and Remote Sensing, 61:1–12, 2023

    Rui Gou, Yijun Zhang, Xianfeng Zhu, and Jinghuai Gao. Bayesian physics-informed neural net- works for the subsurface tomography based on the eikonal equation.IEEE Transactions on Geo- science and Remote Sensing, 61:1–12, 2023. doi: 10.1109/TGRS.2023.3245665

    arXiv 2023

  12. [12]

    Practical variational inference for neural networks

    Alex Graves. Practical variational inference for neural networks. InAdvances in Neural Informa- tion Processing Systems, volume 24, 2011

    2011

  13. [13]

    SIAM, 2011

    Pierre Grisvard.Elliptic Problems in Nonsmooth Domains. SIAM, 2011

    2011

  14. [14]

    Posterior concentrations of fully-connected bayesian neural net- works with general priors on the weights.Journal of Machine Learning Research, 26(94):1–60, 2025

    Insung Kong and Yongdai Kim. Posterior concentrations of fully-connected bayesian neural net- works with general priors on the weights.Journal of Machine Learning Research, 26(94):1–60, 2025

    2025

  15. [15]

    Masked Bayesian neural networks: Theoretical guarantee and its posterior inference

    Insung Kong, Dongyoon Yang, Jongjin Lee, Ilsang Ohn, Gyuseung Baek, and Yongdai Kim. Masked Bayesian neural networks: Theoretical guarantee and its posterior inference. InProceed- ings of the 40th International Conference on Machine Learning, volume 202 ofProceedings of Machine Learning Research, pages 17462–17491. PMLR, 2023

    2023

  16. [16]

    Asymptotic properties for bayesian neural network in besov space

    Kyeongwon Lee and Jaeyong Lee. Asymptotic properties for bayesian neural network in besov space. InAdvances in Neural Information Processing Systems, volume 35, pages 5641–5653, 2022

    2022

  17. [17]

    Posterior contraction for sparse neural networks in besov spaces with intrinsic dimensionality, 2025

    Kyeongwon Lee, Lizhen Lin, Jaewoo Park, and Seonghyun Jeong. Posterior contraction for sparse neural networks in besov spaces with intrinsic dimensionality, 2025

    2025

  18. [18]

    Bayesian physics-informed neural networks for real-world nonlinear dynamical systems

    Kevin Linka, Amelie Schäfer, Xuhui Meng, Zongren Zou, George Em Karniadakis, and Ellen Kuhl. Bayesian physics-informed neural networks for real-world nonlinear dynamical systems. Computer Methods in Applied Mechanics and Engineering, 402:115346, 2022. doi: 10.1016/j.cma. 2022.115346

    arXiv 2022

  19. [19]

    Machine learning for elliptic PDEs: Fast rate generalization bound, neural scaling law and minimax optimality

    Yiping Lu, Haoxuan Chen, Jianfeng Lu, Lexing Ying, and Jose Blanchet. Machine learning for elliptic PDEs: Fast rate generalization bound, neural scaling law and minimax optimality. In International Conference on Learning Representations, 2022

    2022

  20. [20]

    David J. C. MacKay. A practical bayesian framework for backpropagation networks.Neural Computation, 4(3):448–472, 1992. doi: 10.1162/neco.1992.4.3.448

    doi:10.1162/neco.1992.4.3.448 1992

  21. [21]

    David J. C. MacKay. Probable networks and plausible predictions: A review of practical bayesian methods for supervised neural networks.Network: Computation in Neural Systems, 6(3):469–505,

  22. [22]

    Estimates on the generalization error of physics- informed neural networks for approximating pdes.IMA Journal of Numerical Analysis, 43(1): 1–43, 2023

    Siddhartha Mishra and Roberto Molinaro. Estimates on the generalization error of physics- informed neural networks for approximating pdes.IMA Journal of Numerical Analysis, 43(1): 1–43, 2023. doi: 10.1093/imanum/drab093

    doi:10.1093/imanum/drab093 2023

  23. [23]

    Molnar and Samuel J

    Joseph P. Molnar and Samuel J. Grauer. Flow field tomography with uncertainty quantification using a bayesian physics-informed neural network.Measurement Science and Technology, 33(6): 065305, 2022. doi: 10.1088/1361-6501/ac4e36

    work page doi:10.1088/1361-6501/ac4e36 2022

  24. [24]

    Notes on exact boundary values in residual minimisation

    Johannes Müller and Marius Zeinhofer. Notes on exact boundary values in residual minimisation. InProceedings of Mathematical and Scientific Machine Learning, volume 190 ofProceedings of Machine Learning Research, pages 231–240. PMLR, 2022. 26

    2022

  25. [25]

    Neal.Bayesian Learning for Neural Networks

    Radford M. Neal.Bayesian Learning for Neural Networks. PhD thesis, University of Toronto, 1995

    1995

  26. [26]

    Neal.Bayesian Learning for Neural Networks, volume 118 ofLecture Notes in Statis- tics

    Radford M. Neal.Bayesian Learning for Neural Networks, volume 118 ofLecture Notes in Statis- tics. Springer, New York, 1996. doi: 10.1007/978-1-4612-0745-0

    doi:10.1007/978-1-4612-0745-0 1996

  27. [27]

    Structure-preservingphysics-informedneuralnetwork for the korteweg–de vries (kdv) equation.arXiv preprint arXiv:2511.00418, 2025

    VictoryObiekeandEmmanuelOguadimma. Structure-preservingphysics-informedneuralnetwork for the korteweg–de vries (kdv) equation.arXiv preprint arXiv:2511.00418, 2025

    Pith/arXiv arXiv 2025

  28. [28]

    Structure- informed neural operators for long-time prediction of parametric hamiltonian pdes.arXiv preprint arXiv:2606.14913, 2026

    Victory C Obieke, Christopher Chukwuemeka, and Emmanuel E Oguadimma. Structure- informed neural operators for long-time prediction of parametric hamiltonian pdes.arXiv preprint arXiv:2606.14913, 2026

    Pith/arXiv arXiv 2026

  29. [29]

    Oguadimma, Mohamed A

    Emmanuel E. Oguadimma, Mohamed A. F. Elbarkawy, Dominic O. Oranugo, Heba E. Salem, Mustafa Bayram, and Okechukwu J. Obulezi. A foundational review of ordinary differential equation solution methods and their inherent symmetries.Boletim da Sociedade Paranaense de Matemática, 44(8):1–27, 2026. doi: 10.5269/bspm.80626

    work page doi:10.5269/bspm.80626 2026

  30. [30]

    Oguadimma, Victory C

    Emmanuel E. Oguadimma, Victory C. Obieke, and Xueying Yu. Operator learning for cubic nonlinear schrödinger equation on periodic domains.arXiv preprint arXiv:2606.27459, 2026

    Pith/arXiv arXiv 2026

  31. [31]

    Polson and Veronika Ročková

    Nicholas G. Polson and Veronika Ročková. Posterior concentration for sparse deep learning. In Advances in Neural Information Processing Systems, volume 31, pages 938–949, 2018

    2018

  32. [32]

    Springer, 2 edition, 2007

    Alfio Quarteroni, Riccardo Sacco, and Fausto Saleri.Numerical Mathematics, volume 37 ofTexts in Applied Mathematics. Springer, 2 edition, 2007. doi: 10.1007/b98885

    doi:10.1007/b98885 2007

  33. [33]

    Springer, 2016

    Alfio Quarteroni, Andrea Manzoni, and Federico Negri.Reduced Basis Methods for Par- tial Differential Equations: An Introduction, volume 92 ofUNITEXT. Springer, 2016. doi: 10.1007/978-3-319-15431-2

    doi:10.1007/978-3-319-15431-2 2016

  34. [34]

    Maziar Raissi, Paris Perdikaris, and George Em Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.Journal of Computational Physics, 378:686–707, 2019. doi: 10.1016/j.jcp. 2018.10.045

    doi:10.1016/j.jcp 2019

  35. [35]

    Carl Edward Rasmussen and Christopher K. I. Williams.Gaussian Processes for Machine Learn- ing. Adaptive Computation and Machine Learning. MIT Press, Cambridge, MA, 2006

    2006

  36. [36]

    Accelerated training of physics-informed neural networks (PINNs) using meshless discretizations

    Ramansh Sharma and Varun Shankar. Accelerated training of physics-informed neural networks (PINNs) using meshless discretizations. InAdvances in Neural Information Processing Systems, volume 35, 2022

    2022

  37. [37]

    Error estimates of residual minimization using neural networks for linear PDEs.Journal of Machine Learning for Modeling and Computing, 4(4):73–101, 2023

    Yeonjong Shin, Zhongqiang Zhang, and George Em Karniadakis. Error estimates of residual minimization using neural networks for linear PDEs.Journal of Machine Learning for Modeling and Computing, 4(4):73–101, 2023. doi: 10.1615/JMachLearnModelComput.2023050411

    doi:10.1615/jmachlearnmodelcomput.2023050411 2023

  38. [38]

    Uncertainty quantification in PINNs for turbulent flows: Bayesian inference and repulsive ensem- bles, 2026

    Khemraj Shukla, Zongren Zou, Theo Kaeufer, Michael Triantafyllou, and George Em Karniadakis. Uncertainty quantification in PINNs for turbulent flows: Bayesian inference and repulsive ensem- bles, 2026

    2026

  39. [39]

    Dgm: A deep learning algorithm for solving partial differential equations.Journal of Computational Physics, 375:1339–1364, 2018

    Justin Sirignano and Konstantinos Spiliopoulos. Dgm: A deep learning algorithm for solving partial differential equations.Journal of Computational Physics, 375:1339–1364, 2018. doi: 10. 1016/j.jcp.2018.08.029. 27

    2018

  40. [40]

    Smith, Petter E

    Barry F. Smith, Petter E. Bjørstad, and William D. Gropp.Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, 1996

    1996

  41. [41]

    Andrew M. Stuart. Inverse problems: A bayesian perspective.Acta Numerica, 19:451–559, 2010. doi: 10.1017/S0962492910000061

    doi:10.1017/s0962492910000061 2010

  42. [42]

    On the estimation rate of bayesian PINN for inverse problems, 2024

    Yi Sun, Debarghya Mukherjee, and Yves Atchadé. On the estimation rate of bayesian PINN for inverse problems, 2024

    2024

  43. [43]

    Learning specialized activation functions for physics-informed neural networks.Communications in Computational Physics, 34(4):869–906,

    Honghui Wang, Lu Lu, Shiji Song, and Gao Huang. Learning specialized activation functions for physics-informed neural networks.Communications in Computational Physics, 34(4):869–906,

  44. [44]

    Bayesian learning via stochastic gradient langevin dynamics

    Max Welling and Yee Whye Teh. Bayesian learning via stochastic gradient langevin dynamics. In Proceedings of the 28th International Conference on Machine Learning, pages 681–688, 2011

    2011

  45. [45]

    B-pinns: Bayesian physics-informed neural networks for forward and inverse pde problems with noisy data.Journal of Computational Physics, 425:109913, 2021

    Liu Yang, Xuhui Meng, and George Em Karniadakis. B-pinns: Bayesian physics-informed neural networks for forward and inverse pde problems with noisy data.Journal of Computational Physics, 425:109913, 2021. doi: 10.1016/j.jcp.2020.109913

    arXiv 2021

  46. [46]

    Cyclical stochastic gradient MCMC for bayesian deep learning

    Ruqi Zhang, Chunyuan Li, Jianyi Zhang, Changyou Chen, and Andrew Gordon Wilson. Cyclical stochastic gradient MCMC for bayesian deep learning. InInternational Conference on Learning Representations, 2020

    2020

  47. [47]

    Posteriorconcentrationofbayesianphysics-informedneuralnetworks for elliptic pdes

    YuxuanZhaoandYulongLu. Posteriorconcentrationofbayesianphysics-informedneuralnetworks for elliptic pdes. arXiv preprint arXiv:2605.08672, 2026. 28

    Pith/arXiv arXiv 2026

  48. [1995]

    doi: 10.1088/0954-898X_6_3_011

    doi:10.1088/0954-898x_6_3_011

  49. [2023]

    doi: 10.4208/cicp.OA-2023-0058

    work page doi:10.4208/cicp.oa-2023-0058 2023

This paper was first reviewed by deepseek-v4-flash on August 2, 2026.