pith. sign in

arxiv: 2605.11293 · v2 · pith:DYJH5DUBnew · submitted 2026-05-11 · ⚛️ physics.flu-dyn

Pressure reconstruction from error-embedded gradient measurements: a Gaussian-process generalization of Green's function integration

Zejian You , Mohamed Amine Abassi , Xiaofeng Liu , Qi Wang This is my paper

Pith reviewed 2026-06-30 22:08 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords pressure reconstructionGaussian process regressionGreen's function integrationturbulent flowsinverse problemsfluid dynamicsuncertainty quantification
0
0 comments X

The pith

Gaussian process regression reconstructs pressure fields from noisy gradient measurements and reduces to Green's function integration when noise vanishes.

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

The paper establishes that Gaussian process regression offers a probabilistic way to recover scalar pressure from error-embedded gradient data in fluid flows. It shows this framework eliminates the need for boundary conditions required by Poisson solvers or standard Green's function integration, while supplying built-in uncertainty estimates. A key result is that the classical Green's function integration method appears exactly as the zero-noise limit of the Gaussian process approach, with the kernel becoming logarithmic in two dimensions and inverse-distance in three. Validation on slices of homogeneous isotropic turbulence data confirms that an empirical mixture-of-Gaussians kernel fitted to the observed correlations yields performance at least as good as Green's function integration, and better when data are sparse or noisy.

Core claim

A central theoretical result of the present work is that GFI is the noiseless limit of GPR, which on the unbounded plane reduces to the well-known logarithmic kernel and in three dimensions to the inverse-distance kernel. With an empirical mixture-of-Gaussians kernel fitted directly to the pressure correlation function, GPR performs at least as well as GFI on turbulent flow data while delivering calibrated pointwise posterior uncertainty.

What carries the argument

Gaussian Process Regression (GPR) with an empirical mixture-of-Gaussians kernel, shown to have Green's function integration as its noiseless limit.

If this is right

  • GPR supplies tunable denoising and pointwise posterior-variance estimates without explicit boundary conditions.
  • On the unbounded plane the method recovers the logarithmic kernel; in three dimensions it recovers the inverse-distance kernel.
  • The framework extends to three dimensions via a tensor-product Kronecker solver with near O(N^3 log N) cost.
  • A closed-form error lower bound holds on a periodic cube, with the residual gap on finite domains attributed to boundary contamination.

Where Pith is reading between the lines

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

  • The same GPR construction could be applied to other linear inverse problems that recover scalars from noisy gradients, such as temperature or concentration fields.
  • Fitting the kernel directly from the noisy observations may allow the method to adapt to non-stationary statistics without separate calibration data.
  • The boundary-contamination gap identified on non-periodic domains suggests a possible extension that incorporates known boundary values when they are available.

Load-bearing premise

The pressure field can be treated as a Gaussian process whose covariance is accurately captured by an empirical mixture-of-Gaussians kernel fitted directly to the observed pressure correlation function from the same noisy dataset.

What would settle it

A direct test on held-out turbulence data where the standardized residuals of the GPR posterior fail to satisfy |z| < 2 over at least 95 percent of points, or where GPR reconstruction error exceeds that of Green's function integration under high noise.

Figures

Figures reproduced from arXiv: 2605.11293 by Mohamed Amine Abassi, Qi Wang, Xiaofeng Liu, Zejian You.

Figure 1
Figure 1. Figure 1: (a) An example of one-dimensional GPR when observations of the values of a smooth function are available. Red [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: (a) An example of one-dimensional GPR when observations of the values of a smooth function are available. Red [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Noise-free reconstruction of the 256 × 256 JHTDB isotropic-turbulence pressure slice: (left) true field p; (center) GFI reconstruction (εGFI = 0.008); (right) full 2D GPR reconstruction using the empirical MoG-3 kernel (23) fitted to the true-field correlation function, with a tiny positive regularizer σε = 10−2 max |∇p| (εGPR = 0.019). Both methods recover the field to visual accuracy. with the expectatio… view at source ↗
Figure 2
Figure 2. Figure 2: Noise-free reconstruction of the 256 × 256 JHTDB isotropic-turbulence pressure slice: (left) true field p; (center) GFI reconstruction (εGFI = 0.008); (right) full 2D GPR reconstruction using the empirical MoG-3 kernel (23) fitted to the true-field correlation function, with a tiny positive regularizer σε = 10−2 max |∇p| (εGPR = 0.019). Both methods recover the field to visual accuracy. is everywhere non-n… view at source ↗
Figure 3
Figure 3. Figure 3: Empirical-kernel design for the 2D GPR reconstruction. (a) Azimuthally averaged correlation function [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: Empirical-kernel design for the 2D GPR reconstruction. (a) Azimuthally averaged correlation function [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Relative reconstruction RMSE ε versus gradient-observation noise amplitude η = ∆/ max |∇p| on the stride-4 sub￾sample of the JHTDB isotropic-turbulence slice (N = 64, L = π/2, the same operating point as Figs. 7 and 13(a)). Each marker is the mean of 15 independent uniform-noise realizations with the empirical MoG-3 kernel; error bars ±σ. Linear axes; η runs from 5% to 80%, covering the realistic PIV-noise… view at source ↗
Figure 4
Figure 4. Figure 4: Relative reconstruction RMSE ε versus gradient-observation noise amplitude η = ∆/ max |∇p| on the stride-4 sub￾sample of the JHTDB isotropic-turbulence slice (N = 64, L = π/2, the same operating point as Figs. 7 and 13(a)). Each marker is the mean of 15 independent uniform-noise realizations with the empirical MoG-3 kernel; error bars ±σ. Linear axes; η runs from 5% to 80%, covering the realistic PIV-noise… view at source ↗
Figure 5
Figure 5. Figure 5: Posterior uncertainty quantification on the 2D JHTDB benchmark at [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 5
Figure 5. Figure 5: Posterior uncertainty quantification on the 2D JHTDB benchmark at [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Impulse response of the GPR and GFI inverse operators to a unit-amplitude single-pixel [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 6
Figure 6. Figure 6: Impulse response of the GPR and GFI inverse operators to a unit-amplitude single-pixel [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Radial energy spectrum Ep(|k|) of the true pressure field (black), the GFI reconstruction (red dashed), and the GPR reconstruction (blue), computed by the annular-shell average of Eq. (25) on the stride-4 subsample of the JHTDB slice (N = 64, L = π/2). (a) Exact gradient observations (η = 0). (b) error-embedded observations at η = 80%. The GFI noise shelf is clearly visible above the truth at intermediate-… view at source ↗
Figure 7
Figure 7. Figure 7: Radial energy spectrum Ep(|k|) of the true pressure field (black), the GFI reconstruction (red dashed), and the GPR reconstruction (blue), computed by the annular-shell average of Eq. (25) on the stride-4 subsample of the JHTDB slice (N = 64, L = π/2). (a) Exact gradient observations (η = 0). (b) error-embedded observations at η = 80%. The GFI noise shelf is clearly visible above the truth at intermediate-… view at source ↗
Figure 8
Figure 8. Figure 8: Singular values λm of the GFI (black, solid) and MoG-3 GPR (blue, solid) operators on a 256 × 256 cell grid spanning L = π, at the benchmark operating point σε/σp ≈ 24 (equivalent to the η = 40% gradient-noise level). Light-blue dashed curves show MoG-3 at additional σε/σp ∈ {98, 49, 12, 6} to illustrate the convergence to the GFI spectrum as σε → 0. This is not coincidental: both operators admit closed-fo… view at source ↗
Figure 8
Figure 8. Figure 8: Singular values λm of the GFI (black, solid) and MoG-3 GPR (blue, solid) operators on a 256 × 256 cell grid spanning L = π, at the benchmark operating point σε/σp ≈ 24 (equivalent to the η = 40% gradient-noise level). Light-blue dashed curves show MoG-3 at additional σε/σp ∈ {98, 49, 12, 6} to illustrate the convergence to the GFI spectrum as σε → 0. noiseless, kernel-resolved limit, and departs from it on… view at source ↗
Figure 9
Figure 9. Figure 9: Leading twelve singular modes of the GFI (rows 1–2) and MoG- [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 9
Figure 9. Figure 9: Leading twelve singular modes of the GFI (rows 1–2) and MoG- [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The relative RMSE εrms vs gradient noise η on 643 isotropic1024coarse subcubes at two domain extents: (a) L = π/8, N = 128; (b) L = π/2, N = 256. Curves are plotted for plane-wise + LS (red squares), tensor-product 3D GPR (blue circles)—both with the same empirical MoG-3 kernel re-fitted to each cube—and dense 3D GFI (green triangles) as a hyperparameter-free reference. For each GPR method, the filled sol… view at source ↗
Figure 10
Figure 10. Figure 10: The relative RMSE εrms vs gradient noise η on 643 isotropic1024coarse subcubes at two domain extents: (a) L = π/8, N = 128; (b) L = π/2, N = 256. Curves are plotted for plane-wise + LS (red squares), tensor-product 3D GPR (blue circles)—both with the same empirical MoG-3 kernel re-fitted to each cube—and dense 3D GFI (green triangles) as a hyperparameter-free reference. For each GPR method, the filled sol… view at source ↗
Figure 11
Figure 11. Figure 11: Cube-surface comparison at 40% gradient noise on a JHTDB 643 isotropic1024coarse subcube using the empirical MoG-3 kernel for GPR. Each cube shows the pressure on the three visible outer faces. (a) true pressure, (b) 3D GFI reconstruc￾tion, (c) plane-wise + LS reconstruction, (d) tensor-product 3D GPR reconstruction, (e) full-volume error probability density functions for the three reconstructions. εrms d… view at source ↗
Figure 11
Figure 11. Figure 11: Cube-surface comparison at 40% gradient noise on a JHTDB 643 isotropic1024coarse subcube using the empirical MoG-3 kernel for GPR. Each cube shows the pressure on the three visible outer faces. (a) true pressure, (b) 3D GFI reconstruc￾tion, (c) plane-wise + LS reconstruction, (d) tensor-product 3D GPR reconstruction, (e) full-volume error probability density functions for the three reconstructions. εrms d… view at source ↗
Figure 12
Figure 12. Figure 12: Three-dimensional iso-surface rendering of the (a) GFI and (b) GPR (MoG- [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗
Figure 12
Figure 12. Figure 12: Three-dimensional iso-surface rendering of the (a) GFI and (b) GPR (MoG- [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Trend of reconstruction error with noise and density of observation. (a) Relative RMSE [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗
Figure 13
Figure 13. Figure 13: Trend of reconstruction error with noise and density of observation. (a) Relative RMSE [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
read the original abstract

Reconstructing scalar fields from error-embedded gradient measurements is a fundamental linear inverse problem with broad applications in computational physics. Conventional approaches, such as Poisson-based solvers and the Green's Function Integration (GFI) method, require explicit boundary conditions extracted from the same error-embedded observations. In this study we assess the accuracy of a Gaussian Process Regression (GPR) framework for reconstructing pressure fields in turbulent flows from error-embedded pressure-gradient data derived from kinematic measurements. The probabilistic nature of GPR inherently provides tunable denoising, eliminates the need for boundary conditions, and produces a pointwise posterior-variance error estimate. A central theoretical result of the present work is that GFI is the noiseless limit of GPR, which on the unbounded plane reduces to the well-known logarithmic kernel and in three dimensions to the inverse-distance kernel. The framework is validated on two-dimensional slices and three-dimensional subdomains of a forced homogeneous isotropic turbulence from the Johns Hopkins Turbulence Database. With an empirical mixture-of-Gaussians (MoG-$3$) kernel fitted directly to the pressure correlation function, GPR performs at least as well as GFI. In situations with under-resolved data or high noise, GPR outperforms GFI, while delivering a calibrated pointwise posterior uncertainty whose standardized residuals satisfy $|z|<2$ over $95\%$ of grid points. The framework extends to three dimensions through a tensor-product Kronecker solver coupled to conjugate gradients with close to $\mathcal{O}(N^3\log N)$ cost. A closed-form error lower bound on a periodic cube is derived for the GPR operator, with the residual gap attributable to boundary contamination on non-periodic finite domains.

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 / 0 minor

Summary. The manuscript proposes a Gaussian Process Regression (GPR) framework for reconstructing pressure fields from error-embedded gradient measurements in turbulent flows. It claims that Green's Function Integration (GFI) is the noiseless limit of GPR, reducing to the logarithmic kernel on the unbounded plane (2D) and the inverse-distance kernel (3D). Validation on 2D slices and 3D subdomains from the Johns Hopkins Turbulence Database uses an empirical MoG-3 kernel fitted to the pressure correlation function; GPR matches or exceeds GFI performance (especially under noise) while providing calibrated posterior uncertainty with |z|<2 on 95% of points. A closed-form error lower bound is derived for periodic domains, and a Kronecker-based 3D solver is presented.

Significance. If the central equivalence holds and the uncertainty calibration generalizes, the work offers a boundary-condition-free probabilistic alternative to Poisson solvers and GFI with built-in error estimates, which would be useful for inverse problems in fluid dynamics. Strengths include the efficient 3D implementation and the closed-form error bound on periodic cubes. The data-driven kernel fitting, however, limits claims of broad superiority.

major comments (2)
  1. [Abstract] Abstract (central theoretical result): The claim that GFI is the noiseless limit of GPR reducing to the logarithmic kernel (2D) and inverse-distance kernel (3D) is load-bearing but requires explicit handling. These kernels are only conditionally positive definite with spectral density ~1/|k|^2, non-integrable at k=0, and yield formally infinite pointwise variance; standard GPR posterior formulas assume a proper positive-definite kernel with finite K(0). The manuscript does not specify modifications (e.g., polynomial null space, increments, or fixed-mean constraints) needed to make the noiseless limit well-posed within the GPR framework used elsewhere.
  2. [Abstract] Validation paragraph (abstract): The MoG-3 kernel (3 weights, 3 means, 3 variances) is fitted directly to the pressure correlation function extracted from the same JHTDB slices used for validation. This makes the reported performance advantage of GPR over GFI in under-resolved or high-noise regimes dependent on a data-driven covariance that is not independent of the test data, undermining generalizability claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and valuable comments on our manuscript. We address the major comments point by point below, and have revised the manuscript accordingly where appropriate.

read point-by-point responses

  1. Referee: [Abstract] Abstract (central theoretical result): The claim that GFI is the noiseless limit of GPR reducing to the logarithmic kernel (2D) and inverse-distance kernel (3D) is load-bearing but requires explicit handling. These kernels are only conditionally positive definite with spectral density ~1/|k|^2, non-integrable at k=0, and yield formally infinite pointwise variance; standard GPR posterior formulas assume a proper positive-definite kernel with finite K(0). The manuscript does not specify modifications (e.g., polynomial null space, increments, or fixed-mean constraints) needed to make the noiseless limit well-posed within the GPR framework used elsewhere.

    Authors: We appreciate the referee pointing out this subtlety in the theoretical foundation. The central claim is that as the noise variance approaches zero, the GPR posterior mean converges to the GFI solution obtained via the Green's function corresponding to those kernels. In the revised manuscript, we will explicitly state that the logarithmic and inverse-distance kernels are employed in the sense of generalized functions or with a suitable null-space handling (e.g., by subtracting the mean or using increments) to address their conditional positive definiteness. We will add a brief discussion in the methods section clarifying how the noiseless limit is taken within the GPR framework, ensuring consistency with standard GPR assumptions for finite noise levels. revision: yes

  2. Referee: [Abstract] Validation paragraph (abstract): The MoG-3 kernel (3 weights, 3 means, 3 variances) is fitted directly to the pressure correlation function extracted from the same JHTDB slices used for validation. This makes the reported performance advantage of GPR over GFI in under-resolved or high-noise regimes dependent on a data-driven covariance that is not independent of the test data, undermining generalizability claims.

    Authors: The referee is correct that fitting the kernel to the correlation function from the validation dataset introduces a dependence that affects the interpretation of generalizability. This setup demonstrates the potential of GPR when the covariance structure is known from the flow statistics, as is common in turbulence studies. However, to strengthen the manuscript, we will revise the abstract and discussion to qualify that the performance comparison is for this specific dataset with kernel fitted to its statistics, and note that for new flows the kernel would need to be determined independently (e.g., from separate simulations or theory). We do not claim broad superiority beyond the tested conditions. revision: partial

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The central theoretical result that GFI is the noiseless limit of GPR (reducing to the logarithmic kernel in 2D and inverse-distance kernel in 3D) is presented as a derived mathematical equivalence independent of data. The empirical MoG-3 kernel is fitted to the pressure correlation function as a methodological choice for validation on the JHU turbulence database, but this does not reduce the theoretical claim or performance statements to the inputs by construction. No self-citations, self-definitional steps, or fitted parameters renamed as predictions are quoted in the provided text that would force the result. The derivation chain is self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The framework rests on a Gaussian-process prior whose covariance is supplied by an empirical three-component Gaussian mixture fitted to pressure correlations; no new physical entities are postulated. The only free parameters are the three mixture weights, means, and variances of the MoG-3 kernel.

free parameters (1)
  • MoG-3 kernel parameters (3 weights, 3 means, 3 variances)
    Fitted directly to the empirical pressure correlation function extracted from the turbulence database; these parameters control the denoising behavior and are not derived from first principles.
axioms (2)
  • domain assumption The pressure field is a realization of a Gaussian process with stationary covariance given by the chosen kernel.
    Invoked when the GPR posterior is written; standard in GPR literature but not proved for turbulent pressure.
  • domain assumption The noise in the gradient measurements is additive, zero-mean, and uncorrelated with the signal.
    Required for the posterior variance formula and the claim of calibrated uncertainty.

pith-pipeline@v0.9.1-grok · 5837 in / 1777 out tokens · 27166 ms · 2026-06-30T22:08:06.100226+00:00 · methodology

Review history (2 revisions) →

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 · 5 canonical work pages · 1 internal anchor

  1. [1]

    X. Liu, J. Katz, Measurements of pressure distribution by integrating the material acceleration, in: Cav03-GS-14-001, Fifth international symposium on cavitation (CAV2003), Osaka, Japan, 2003, pp. 1–4

    2003

  2. [2]

    X. Liu, J. Katz, Instantaneous pressure and material acceleration measurements using a four-exposure piv system, Experiments in fluids 41 (2006) 227–240

    2006

  3. [3]

    D. A. Buchta, J. B. Freund, The near-field pressure radiated by planar high-speed free-shear-flow turbulence, Journal of Fluid Mechanics 832 (2017) 383–408

    2017

  4. [4]

    S. F. Wu, Methods for reconstructing acoustic quantities based on acoustic pressure measurements, The Journal of the Acoustical Society of America 124 (2008) 2680–2697

    2008

  5. [5]

    R. A. Gramann, D. S. Dolling, Detection of turbulent boundary-layer separation using fluctuating wall pressure signals, AIAA journal 28 (1990) 1052–1056

    1990

  6. [6]

    Cerretelli, K

    C. Cerretelli, K. Kirtley, Boundary layer separation control with fluidic oscillators (2009)

    2009

  7. [7]

    Yeung, D

    P. Yeung, D. Donzis, K. Sreenivasan, Dissipation, enstrophy and pressure statistics in turbulence simulations at high reynolds numbers, Journal of Fluid Mechanics 700 (2012) 5–15

    2012

  8. [8]

    M. A. Abassi, Q. Wang, X. Liu, Adjoint-based data assimilation in a subdomain using omnidirectional- integration-enabled pressure dirichlet boundary conditions, Physics of Fluids 37 (2025)

    2025

  9. [9]

    T. A. Zaki, Turbulence from an observer perspective, Annual Review of Fluid Mechanics 57 (2025) 311–334

    2025

  10. [10]

    X. Liu, J. R. Moreto, S. Siddle-Mitchell, Instantaneous pressure reconstruction from measured pressure gradient using rotating parallel ray method, in: 54th AIAA Aerospace Sciences Meeting, 2016, p. 1049

    2016

  11. [11]

    X. Liu, J. Katz, Pressure–rate-of-strain, pressure diffusion, and velocity–pressure-gradient tensor mea- surements in a cavity flow, AIAA Journal 56 (2018) 3897–3914. 23

    2018

  12. [12]

    J. R. Moreto, W. J. Reeder, R. Budwig, D. Tonina, X. Liu, Experimentally mapping water surface elevation, velocity, and pressure fields of an open channel flow around a stalk, Geophysical Research Letters 49 (2022) e2021GL096835

    2022

  13. [13]

    X. Liu, J. R. Moreto, Error propagation from the piv-based pressure gradient to the integrated pressure by the omnidirectional integration method, Measurement Science and Technology 31 (2020) 055301

    2020

  14. [14]

    Q. Wang, X. Liu, Green’s function integral method for pressure reconstruction from measured pressure gradient and the interpretation of omnidirectional integration, Physics of Fluids 35 (2023)

    2023

  15. [15]

    Zigunov, J

    F. Zigunov, J. J. Charonko, One-shot omnidirectional pressure integration through matrix inversion, Measurement Science and Technology 35 (2024) 125301

    2024

  16. [16]

    C. E. Rasmussen, C. K. I. Williams, Gaussian Processes for Machine Learning, MIT Press, 2006

    2006

  17. [17]

    Schulz, M

    E. Schulz, M. Speekenbrink, A. Krause, A tutorial on gaussian process regression: Modelling, exploring, and exploiting functions, Journal of Mathematical Psychology 85 (2018) 1–16

    2018

  18. [18]

    V. Mons, Q. Wang, T. A. Zaki, Kriging-enhanced ensemble variational data assimilation for scalar- source identification in turbulent environments, Journal of Computational Physics 398 (2019) 108856

    2019

  19. [19]

    Z. You, Q. Wang, X. Liu, Pressure reconstruction from the measured pressure gradient using gaussian process regression, in: AIAA SCITECH 2023 Forum, 2023, p. 0414

    2023

  20. [20]

    Solak, R

    E. Solak, R. Murray-Smith, W. E. Leithead, D. J. Leith, C. E. Rasmussen, Derivative observations in Gaussian process models of dynamic systems, in: Advances in Neural Information Processing Systems 15, 2003, pp. 1057–1064

    2003

  21. [21]

    Wahba, Spline Models for Observational Data, volume 59 ofCBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 1990

    G. Wahba, Spline Models for Observational Data, volume 59 ofCBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 1990

    1990

  22. [22]

    Hastie, R

    T. Hastie, R. Tibshirani, J. Friedman, et al., The elements of statistical learning, 2009

    2009

  23. [23]

    Kimeldorf, G

    G. Kimeldorf, G. Wahba, Some results on Tchebycheffian spline functions, Journal of Mathematical Analysis and Applications 33 (1971) 82–95

    1971

  24. [24]

    Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences

    M. Kanagawa, P. Hennig, D. Sejdinovic, B. K. Sriperumbudur, Gaussian processes and kernel methods: A review on connections and equivalences, arXiv preprint arXiv:1807.02582 (2018)

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  25. [25]

    A. J. Smola, B. Schölkopf, Learning with kernels, volume 4, GMD-Forschungszentrum Informationstech- nik Berlin, Germany, 1998

    1998

  26. [26]

    Graepel, Solving noisy linear operator equations by gaussian processes: Application to ordinary and partial differential equations, in: ICML, volume 3, 2003, pp

    T. Graepel, Solving noisy linear operator equations by gaussian processes: Application to ordinary and partial differential equations, in: ICML, volume 3, 2003, pp. 234–241

    2003

  27. [27]

    Cockayne, C

    J. Cockayne, C. Oates, T. Sullivan, M. Girolami, Probabilistic numerical methods for pde-constrained bayesian inverse problems, in: AIP Conference Proceedings, volume 1853, AIP Publishing LLC, 2017, p. 060001

    2017

  28. [28]

    Raissi, P

    M. Raissi, P. Perdikaris, G. E. Karniadakis, Machine learning of linear differential equations using gaussian processes, Journal of Computational Physics 348 (2017) 683–693

    2017

  29. [29]

    L. P. Swiler, M. Gulian, A. L. Frankel, C. Safta, J. D. Jakeman, A survey of constrained gaussian process regression: Approaches and implementation challenges, Journal of Machine Learning for Modeling and Computing 1 (2020)

    2020

  30. [30]

    A. P. Valentine, M. Sambridge, Gaussian process models—i. a framework for probabilistic continuous inverse theory, Geophysical Journal International 220 (2020) 1632–1647. 24

    2020

  31. [31]

    Dalton, A

    D. Dalton, A. Lazarus, H. Gao, D. Husmeier, Boundary constrained gaussian processes for robust physics-informed machine learning of linear partial differential equations, Journal of Machine Learning Research 25 (2024) 1–61

    2024

  32. [32]

    M.Pförtner, I.Steinwart, P.Hennig, J.Wenger, Physics-informedgaussianprocessregressiongeneralizes linear pde solvers, arXiv preprint arXiv:2212.12474 (2022)

    work page arXiv 2022

  33. [33]

    X. Li, M. Lange-Hegermann, B. Raitˇ a, Gaussian process regression for inverse problems in linear pdes, IFAC-PapersOnLine 59 (2025) 196–201

    2025

  34. [34]

    E. J. Parish, K. Duraisamy, A paradigm for data-driven predictive modeling using field inversion and machine learning, Journal of computational physics 305 (2016) 758–774

    2016

  35. [35]

    M. Wang, Q. Wang, T. A. Zaki, Discrete adjoint of fractional-step incompressible navier-stokes solver in curvilinear coordinates and application to data assimilation, Journal of Computational Physics 396 (2019) 427–450

    2019

  36. [36]

    M. R. Hestenes, E. Stiefel, Methods of conjugate gradients for solving linear systems, Journal of Research of the National Bureau of Standards 49 (1952) 409–436. doi:10.6028/jres.049.044

    work page doi:10.6028/jres.049.044 1952

  37. [37]

    Y. Li, E. Perlman, M. Wan, Y. Yang, C. Meneveau, R. Burns, S. Chen, A. Szalay, G. Eyink, A public turbulence database cluster and applications to study lagrangian evolution of velocity increments in turbulence, Journal of Turbulence (2008) N31

    2008

  38. [38]

    Perlman, R

    E. Perlman, R. Burns, Y. Li, C. Meneveau, Data exploration of turbulence simulations using a database cluster, in: Proceedings of the 2007 ACM/IEEE Conference on Supercomputing, 2007, pp. 1–11

    2007

  39. [39]

    Aronszajn, Theory of reproducing kernels, Transactions of the American Mathematical Society 68 (1950) 337–404

    N. Aronszajn, Theory of reproducing kernels, Transactions of the American Mathematical Society 68 (1950) 337–404

    1950

  40. [40]

    H. Taha, C. Gonzalez, M. Shorbagy, A minimization principle for incompressible fluid mechanics, Physics of Fluids 35 (2023)

    2023

  41. [41]

    Strang, A proposal for Toeplitz matrix calculations, Studies in Applied Mathematics 74 (1986) 171–176

    G. Strang, A proposal for Toeplitz matrix calculations, Studies in Applied Mathematics 74 (1986) 171–176

    1986

  42. [42]

    Greengard, J

    L. Greengard, J. Strain, The fast Gauss transform, SIAM Journal on Scientific and Statistical Com- puting 12 (1991) 79–94

    1991

  43. [43]

    J. Wang, C. Zhang, J. Katz, Gpu-based, parallel-line, omni-directional integration of measured pressure gradient field to obtain the 3d pressure distribution, Experiments in Fluids 60 (2019) 58

    2019

  44. [44]

    Bochner, Harmonic analysis and the theory of probability, Courier Corporation, 2005

    S. Bochner, Harmonic analysis and the theory of probability, Courier Corporation, 2005

    2005

  45. [45]

    M. L. Stein, Interpolation of spatial data: some theory for kriging, Springer Science & Business Media, 1999

    1999

  46. [46]

    Q. Wang, M. Wang, T. A. Zaki, What is observable from wall data in turbulent channel flow?, Journal of Fluid Mechanics 941 (2022) A48. doi:10.1017/jfm.2022.295

    work page doi:10.1017/jfm.2022.295 2022

  47. [47]

    Q. Wang, Y. Hasegawa, T. A. Zaki, Spatial reconstruction of steady scalar sources from remote mea- surements in turbulent flow, Journal of Fluid Mechanics 870 (2019) 316–352

    2019

  48. [48]

    E. C. Howell, J. D. Hanson, Development of a non-parametric gaussian process model in the three- dimensional equilibrium reconstruction code v3fit, Journal of Plasma Physics 86 (2020) 905860102. doi:10.1017/S0022377819000813

    work page doi:10.1017/s0022377819000813 2020

  49. [49]

    Y. Yang, Y. Xue, W. Zhao, S. Yao, C. Li, C. Wu, Fast flow field prediction of three-dimensional hypersonic vehicles using an improved gaussian process regression algorithm, Physics of Fluids 36 (2024). 25

    2024