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arxiv: 2501.11878 · v3 · submitted 2025-01-21 · ❄️ cond-mat.mtrl-sci

Development of an uncertainty-aware equation of state for gold

Lin H. Yang , James A. Gaffney This is my paper

Pith reviewed 2026-05-23 05:42 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords equation of stateGaussian processeserror-in-variablesdensity functional theorygolduncertainty quantificationwarm dense matterthermodynamic properties
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The pith

Gaussian processes with error-in-variables integration build uncertainty-aware equation of state tables for gold from DFT data.

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

The paper presents a framework that uses Gaussian processes to generate high-fidelity equation of state tables while systematically accounting for uncertainties in both the input variables such as temperature and density and the output properties such as pressure. It demonstrates the approach on first-principles density functional theory calculations for gold spanning densities up to 100 g/cc and temperatures up to 300 eV in the warm dense matter regime. The integration of error-in-variables allows the model to handle noise and limited data without assuming perfect inputs. A reader would care because reliable material models with quantified uncertainty support more trustworthy simulations of extreme conditions. The work further checks how well the uncertainties propagate through the final tables under data scarcity and experimental noise.

Core claim

By integrating Error-in-Variables into the Gaussian Process model, the framework navigates uncertainties in both input parameters like temperature and density and output variables including pressure and other thermodynamic properties. The methodology is demonstrated using first-principles density functional theory data for gold over maximum density compression up to 100 g/cc and extreme temperatures within the warm dense matter region reaching 300 eV. The resilience of uncertainty propagation in the resulting EOS tables is assessed under conditions including data scarcity and the intrinsic noise of experiments and simulations.

What carries the argument

Gaussian process regression with error-in-variables integration that treats uncertainties in both inputs and outputs when constructing equation of state tables.

If this is right

  • The EOS tables include quantified uncertainties that can be propagated into downstream hydrodynamic or material response calculations.
  • The model maintains performance when training data is sparse or noisy.
  • Systematic uncertainty handling applies to both simulation outputs and experimental inputs.
  • Tables cover the full warm dense matter regime for gold without separate ad-hoc corrections for uncertainty.

Where Pith is reading between the lines

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

  • The same error-in-variables Gaussian process approach could be retrained on data for other metals or compounds to produce comparable uncertainty-aware tables.
  • Coupling the output tables to existing simulation codes would allow direct sampling of uncertainty distributions in large-scale runs.
  • Extending the input space to include additional variables such as composition or magnetic field might reveal new sensitivity patterns without new first-principles runs.

Load-bearing premise

The first-principles DFT data for gold provides a sufficient and representative training set for the Gaussian process model to generalize across the full range of densities up to 100 g/cc and temperatures up to 300 eV without significant extrapolation errors.

What would settle it

Direct comparison of the generated EOS table pressures, energies, or other thermodynamic quantities against independent experimental shock data or separate high-accuracy simulations at densities and temperatures inside the claimed range but withheld from the training set.

Figures

Figures reproduced from arXiv: 2501.11878 by James A. Gaffney, Lin H. Yang.

Figure 1
Figure 1. Figure 1: FIG. 1: Schematic flow chart outlining the general process fo [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The Au EOS is derived by inputting DFT data into EIV-GP [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: EIV-GPs present a sophisticated method for correlat [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Comparison of cold EOS’s for fcc Au using different den [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: (a) Au isotherm data from DFT calculations (solid lin [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Debye temperature of gold as a function of density fro [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: The total free energy with predicted uncertainty is c [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Helmholtz free energy comparison between U790 and Y7 [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: (a) Comparing the internal energy of U790 and Y790, a n [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Comparison of shock Hugoniot (pressure verse densi [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Comparison of shock Hugoniot Us-up relation from th [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
read the original abstract

This study introduces a framework that employs Gaussian Processes (GPs) to develop high-fidelity equation of state (EOS) tables, essential for modeling material properties across varying temperatures and pressures. GPs offer a robust predictive modeling approach and are especially adept at handling uncertainties systematically. By integrating Error-in-Variables (EIV) into the GP model, we adeptly navigate uncertainties in both input parameters (like temperature and density) and output variables (including pressure and other thermodynamic properties). Our methodology is demonstrated using first-principles density functional theory (DFT) data for gold, observing its properties over maximum density compression (up to 100 g/cc) and extreme temperatures within the warm dense matter region (reaching 300 eV). Furthermore, we assess the resilience of our uncertainty propagation within the resultant EOS tables under various conditions, including data scarcity and the intrinsic noise of experiments and simulations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces a Gaussian Process framework augmented with Error-in-Variables (EIV) to construct uncertainty-aware equation of state (EOS) tables for gold, trained on first-principles DFT data spanning densities up to 100 g/cc and temperatures up to 300 eV in the warm dense matter regime, with additional assessment of uncertainty propagation under data scarcity and noise.

Significance. If the central claims hold after validation, the EIV-GP approach would provide a systematic method for propagating uncertainties in both input (density, temperature) and output (pressure, thermodynamic properties) variables when building EOS tables from simulation data, addressing a practical need in high-energy-density physics and material modeling under extreme conditions.

major comments (2)
  1. [Abstract] Abstract: The claim that the EIV-augmented GP produces a high-fidelity EOS across the full stated range is unsupported by any quantitative validation metrics (RMSE, uncertainty calibration, or held-out test errors), comparisons to existing EOS tables, or assessment of how well uncertainties are calibrated.
  2. [Abstract] Abstract: The generalization claim to 100 g/cc and 300 eV rests on the assumption that the DFT training set provides sufficient coverage to avoid extrapolation, yet no description of the data grid (number of points, spacing, or distribution) or figures demonstrating that these extremes lie within the convex hull of the training data is supplied; the EIV formalism propagates input/output noise but does not mitigate model-form error outside the data support.
minor comments (1)
  1. [Abstract] The abstract mentions assessment of resilience under data scarcity and noise but provides no concrete examples or quantitative results from those assessments.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough and constructive review of our manuscript. We address each major comment below and will revise the manuscript to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that the EIV-augmented GP produces a high-fidelity EOS across the full stated range is unsupported by any quantitative validation metrics (RMSE, uncertainty calibration, or held-out test errors), comparisons to existing EOS tables, or assessment of how well uncertainties are calibrated.

    Authors: We agree that the abstract's claim requires stronger quantitative support. In the revised manuscript we will add explicit validation results, including RMSE on held-out DFT data, uncertainty calibration diagnostics, and direct comparisons against standard gold EOS tables such as SESAME. These metrics will be presented in a new or expanded results subsection and referenced from the abstract. revision: yes

  2. Referee: [Abstract] Abstract: The generalization claim to 100 g/cc and 300 eV rests on the assumption that the DFT training set provides sufficient coverage to avoid extrapolation, yet no description of the data grid (number of points, spacing, or distribution) or figures demonstrating that these extremes lie within the convex hull of the training data is supplied; the EIV formalism propagates input/output noise but does not mitigate model-form error outside the data support.

    Authors: We accept the point that data coverage must be documented explicitly. The revised manuscript will include a detailed description of the DFT training grid (number of points, density and temperature spacing, and distribution) together with a figure that overlays the training data convex hull on the target prediction domain. We will also clarify in the text that the EIV-GP framework addresses input/output noise for interpolation within the trained domain and does not claim to correct model-form errors outside that support. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses independent DFT data as external input

full rationale

The paper trains a GP-EIV model on first-principles DFT data for gold to produce EOS tables. No load-bearing step reduces by construction to a fitted parameter or self-citation chain; the DFT points are treated as given external observations, and the model outputs predictions plus uncertainties without re-deriving the training values. The derivation chain is self-contained against the supplied data.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard assumptions of Gaussian process regression and the representativeness of the DFT dataset; no new entities are introduced.

free parameters (1)
  • GP kernel hyperparameters
    Length scales, variances, and noise terms in the Gaussian process are typically fitted to the training data.
axioms (1)
  • domain assumption The thermodynamic properties of gold can be modeled as a Gaussian process with additive noise in both inputs and outputs.
    Invoked when applying the EIV-GP model to the DFT data for EOS construction.

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Works this paper leans on

53 extracted references · 53 canonical work pages

  1. [1]

    R. M. More, K. H. Warren, D. A. Young, and G. B. Zimmerman, T he Physics of Fluids 31, 3059 (1988), ISSN 0031-9171, https://pubs.aip.org/aip/p fl/article-pdf/31/10/3059/12484353/3059 1 online.pdf, URL https://doi.org/10.1063/1.866963

    work page doi:10.1063/1.866963 1988

  2. [2]

    D. A. Young and E. M. Corey, Journal of Applied Physics 78, 3748 (1995), ISSN 0021-8979, https://pubs.aip.org/aip/jap/article-pdf/ 78/6/3748/8045523/3748 1 online.pdf, URL https://doi.org/10.1063/1.359955

    work page doi:10.1063/1.359955 1995

  3. [3]

    C. J. Wu, D. A. Young, P. A. Sterne, and P. C. Myint, The Jour nal of Chemical Physics 151, 224505 (2019), ISSN 0021-9606, https://pubs.aip.org/aip/jcp/article- pdf/doi/10.1063/1.5128127/13332097/224505 1 online.pdf, URL https://doi.org/10.1063/1.5128127

    work page doi:10.1063/1.5128127/13332097/224505 2019

  4. [4]

    S. J. Ali, D. C. Swift, C. J. Wu, and R. G. Kraus, Journal of A pplied Physics 128, 185902 (2020), ISSN 0021-8979, https://pubs.aip.org/aip/jap/article- pdf/doi/10.1063/5.0029723/15258153/185902 1 online.pdf, URL https://doi.org/10.1063/5.0029723

    work page doi:10.1063/5.0029723/15258153/185902 2020

  5. [5]

    C. J. Wu, P. C. Myint, J. E. Pask, C. J. Prisbrey, A. A. Corre a, P. Suryanarayana, and J. B. Varley, The Journal of Physical Chemistry A 125, 1610 (2021), ISSN 1089-5639, doi: 10.1021/acs.jpca.0c09 809, URL https://doi.org/10.1021/acs.jpca.0c09809

    work page doi:10.1021/acs.jpca.0c09 2021

  6. [6]

    C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning (The MIT Press, 2005), ISBN 9780262256834, URL https://doi.org/10.7551/mitpress/3206.001.0001

    work page doi:10.7551/mitpress/3206.001.0001 2005

  7. [7]

    C. J. Wu, L. X. Benedict, P. C. Myint, S. Hamel, C. J. Prisbr ey, and J. R. Leek, Phys. Rev. B 108, 014102 (2023), URL https://link.aps.org/doi/10.1103/PhysRevB.108.014102

    work page doi:10.1103/physrevb.108.014102 2023

  8. [8]

    J. A. Gaffney, Private communication (2023)

    work page 2023

  9. [9]

    Girard and R

    A. Girard and R. Murray-Smith (2003)

    work page 2003

  10. [10]

    Murray-Smith and R

    R. Murray-Smith and R. Shorten, eds., Gaussian Processes: Prediction at a Noisy Input and Applica tion to Iterative 22 Multiple-Step Ahead Forecasting of Time-Series (Springer Berlin Heidelberg, Berlin, Heidelberg, 2005), p p. 158–184, ISBN 978-3-540-30560-6, URL https://doi.org/10.1007/978-3-540-30560-6_7

    work page doi:10.1007/978-3-540-30560-6_7 2005

  11. [11]

    McHutchon and C

    A. McHutchon and C. Rasmussen, Advances in neural infor mation processing systems 24 (2011)

    work page 2011

  12. [12]

    J. E. Johnson, V. Laparra, and G. Camps-Valls, IEEE Geos cience and Remote Sensing Letters 17, 391 (2020)

    work page 2020

  13. [13]

    Yokoo, N

    M. Yokoo, N. Kawai, K. G. Nakamura, and K.-i. Kondo, Appl ied Physics Letters 92, 051901 (2008), ISSN 0003-6951, https://pubs.aip.org/aip/apl/article- pdf/doi/10.1063/1.2840189/14660864/051901 1 online.pdf, URL https://doi.org/10.1063/1.2840189

    work page doi:10.1063/1.2840189/14660864/051901 2008

  14. [14]

    Briggs, F

    R. Briggs, F. Coppari, M. G. Gorman, R. F. Smith, S. J. Tra cy, A. L. Coleman, A. Fernandez-Pa˜ nella, M. Millot, J. H. Eggert, and D. E. Fratanduono, PHYSICAL REVIEW LETTERS 123 (2019)

    work page 2019

  15. [15]

    S. K. Han, R. F. Smith, D. Kim, J. K. Wicks, J. R. Rygg, A. La zicki, J. H. Eggert, and T. S. Duffy, PHYSICAL REVIEW B 103 (2021)

    work page 2021

  16. [16]

    W. Mao, H. Mao, P. Bell, A. Goncharov, V. Struzhkin, and R . Hemley, Physical Review B 66 (2002)

    work page 2002

  17. [17]

    H. K. Yuichi Akahama and A. K. Singh, Journal of Applied P hysics 92, 6 (2002)

    work page 2002

  18. [18]

    S.-H. Shim, T. S. Duffy, and K. Takemura, Earth and Planet ary Science Letters 203, 729 (2002), ISSN 0012-821X, URL https://www.sciencedirect.com/science/article/pii/S0012821X02009172

    work page 2002

  19. [19]

    Dewaele, P

    A. Dewaele, P. Loubeyre, and M. Mezouar, Physical Revie w B 70, 094112 (2004)

    work page 2004

  20. [20]

    Dewaele, P

    A. Dewaele, P. Loubeyre, F. Occelli, O. Marie, and M. Mez ouar, Nature Communications 9, 2913 (2018), ISSN 2041-1723, URL https://doi.org/10.1038/s41467-018-05294-2

    work page doi:10.1038/s41467-018-05294-2 2018

  21. [21]

    Dewaele and P

    A. Dewaele and P. Loubeyre, High Pressure Research 27, 419 (2007), https://doi.org/10.1080/0895795070165962 7, URL https://doi.org/10.1080/08957950701659627

    work page doi:10.1080/0895795070165962 2007

  22. [22]

    Takemura and A

    K. Takemura and A. Dewael, Phys. Rev. B 78 (2008)

    work page 2008

  23. [23]

    S¨ oderlind, Phys

    P. S¨ oderlind, Phys. Rev. B 66 (2002)

    work page 2002

  24. [24]

    Dubrovinsky, N

    L. Dubrovinsky, N. Dubrovinskaia, W. A. Crichton, A. S. Mikhaylushkin, S. I. Simak, I. A. Abrikosov, J. S. De Almeida , R. Ahuja, W. Luo, and B. Johansson, Physical Review Letters 98, 045503 (2007), ISSN 0031-9007, 1079-7114

    work page 2007

  25. [25]

    Ishikawa, K

    T. Ishikawa, K. Kato, M. Nomura, N. Suzuki, H. Nagara, an d K. Shimizu, Physical Review B 88, 214110 (2013)

    work page 2013

  26. [26]

    Ahuja, S

    R. Ahuja, S. Rekhi, and B. Johansson, Physical Review B 63, 212101 (2001)

    work page 2001

  27. [27]

    J. C. Boettger, Physical Review B 67, 174107 (2003)

    work page 2003

  28. [28]

    X.-L. Z. Zhong-Li Liu, Ya-Ping Tao and L.-C. Cai, Comput ational Materials Science 114, 72 (2016)

    work page 2016

  29. [29]

    N. A. Smirnov, J. Phys.: Condens. Matter 29 (2017)

    work page 2017

  30. [30]

    C. E. Rasmussen, ArXiv (2006), URL https://ar5iv.org/html/2102.05497

    work page arXiv 2006

  31. [31]

    Vinet, J

    P. Vinet, J. Ferrante, J. H. Rose, and J. R. Smith, Journa l of Geophysical Research: Solid Earth 92, 9319 (1987), https://agupubs.onlinelibrary.wiley.com /doi/pdf/10.1029/JB092iB09p09319, URL 23 https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/JB092iB09p09319

    work page doi:10.1029/jb092ib09p09319 1987

  32. [32]

    Zhang, M

    S. Zhang, M. C. Marshall, L. H. Yang, P. A. Sterne, B. Mili tzer, M. D¨ ane, J. A. Gaffney, A. Shamp, T. Ogitsu, K. Caspersen, et al., Phys. Rev. E 102, 053203 (2020), URL https://link.aps.org/doi/10.1103/PhysRevE.102.053203

    work page doi:10.1103/physreve.102.053203 2020

  33. [33]

    Militzer, F

    B. Militzer, F. Gonz´ alez-Cataldo, S. Zhang, K. P. Driv er, and F. S. Soubiran, Phys. Rev. E 103, 013203 (2021), URL https://link.aps.org/doi/10.1103/PhysRevE.103.013203

    work page doi:10.1103/physreve.103.013203 2021

  34. [34]

    L. H. Yang, R. Q. Hood, J. E. Pask, and J. E. Klepeis, Journ al of Computer-Aided Materials Design 14, 337 (2007), ISSN 1573-4900, URL https://doi.org/10.1007/s10820-007-9053-1

    work page doi:10.1007/s10820-007-9053-1 2007

  35. [35]

    Giannozzi, S

    P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, et al., Journal of Physics: Condense d Matter 21, 395502 (2009), URL https://dx.doi.org/10.1088/0953-8984/21/39/395502

    work page doi:10.1088/0953-8984/21/39/395502 2009

  36. [36]

    Perdew and A

    J. Perdew and A. Zunger, Physical Review B 23, 5048 (1981), ISSN 0163-1829

    work page 1981

  37. [37]

    J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett . 77, 3865 (1996), URL https://link.aps.org/doi/10.1103/PhysRevLett.77.3865

    work page doi:10.1103/physrevlett.77.3865 1996

  38. [38]

    J. P. Perdew, K. Burke, and M. Ernzerhof, Physical Revie w Letters 78, 1396 (1997)

    work page 1997

  39. [39]

    A. V. Terentjev, L. A. Constantin, and J. M. Pitarke, Phy s. Rev. B 98, 214108 (2018), URL https://link.aps.org/doi/10.1103/PhysRevB.98.214108

    work page doi:10.1103/physrevb.98.214108 2018

  40. [40]

    Dal Corso, Physical Review B 82, 075116 (2010)

    A. Dal Corso, Physical Review B 82, 075116 (2010)

    work page 2010

  41. [41]

    Dal Corso, Physical Review B 86, 085135 (2012)

    A. Dal Corso, Physical Review B 86, 085135 (2012)

    work page 2012

  42. [42]

    Tadano and S

    T. Tadano and S. Tsuneyuki, Phys. Rev. B 92, 054301 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.92.054301

    work page doi:10.1103/physrevb.92.054301 2015

  43. [43]

    D. R. Hamann, Phys. Rev. B 88, 085117 (2013), URL https://link.aps.org/doi/10.1103/PhysRevB.88.085117

    work page doi:10.1103/physrevb.88.085117 2013

  44. [44]

    Pask, private communication (2023)

    J. Pask, private communication (2023)

    work page 2023

  45. [45]

    J. R. Neighbours and G. A. Alers, Phys. Rev. 111, 707 (1958), URL https://link.aps.org/doi/10.1103/PhysRev.111.707

    work page doi:10.1103/physrev.111.707 1958

  46. [46]

    L. H. Yang, P. S¨ oderlind, D. C. Swift, and J. A. Gaffney, u npublished (2024)

    work page 2024

  47. [47]

    F. Zhou, W. Nielson, Y. Xia, and V. Ozoli¸ nˇ s, Phys. Rev. B 100, 184308 (2019), URL https://link.aps.org/doi/10.1103/PhysRevB.100.184308

    work page doi:10.1103/physrevb.100.184308 2019

  48. [48]

    Kusaba and T

    K. Kusaba and T. Kikegawa, Solid State Communications 149, 371 (2009), ISSN 0038-1098, URL https://www.sciencedirect.com/science/article/pii/S0038109808007187

    work page 2009

  49. [49]

    Nagai, R

    R. Nagai, R. Akashi, and O. Sugino, npj Computational Ma terials 6, 43 (2020), ISSN 2057-3960, URL https://doi.org/10.1038/s41524-020-0310-0

    work page doi:10.1038/s41524-020-0310-0 2020

  50. [50]

    Christensen, T

    R. Christensen, T. Bligaard, and K. W. Jacobsen, in Uncertainty Quantification in Multiscale Materials Modeli ng, edited by Y. Wang and D. L. McDowell (Woodhead Publishing, 2020), El sevier Series in Mechanics of Advanced Materials, pp. 77–91, ISBN 978-0-08-102941-1, URL https://www.sciencedirect.com/science/article/pii/B9780081029411000031. 24

    work page 2020

  51. [51]

    Lejaeghere, in Uncertainty Quantification in Multiscale Materials Modeli ng, edited by Y

    K. Lejaeghere, in Uncertainty Quantification in Multiscale Materials Modeli ng, edited by Y. Wang and D. L. McDowell (Woodhead Publishing, 2020), Elsevier Series in Mechanics of Advanced Materials, pp. 41–76, ISBN 978-0-08-102941-1, URL https://www.sciencedirect.com/science/article/pii/B978008102941100002X

    work page 2020

  52. [52]

    Gygi, Journal of Chemical Theory and Computation 19, 1300 (2023), ISSN 1549-9618, doi: 10.1021/acs.jctc.2c01 191, URL https://doi.org/10.1021/acs.jctc.2c01191

    F. Gygi, Journal of Chemical Theory and Computation 19, 1300 (2023), ISSN 1549-9618, doi: 10.1021/acs.jctc.2c01 191, URL https://doi.org/10.1021/acs.jctc.2c01191

    work page doi:10.1021/acs.jctc.2c01 2023

  53. [53]

    S¨ oderlind, private communication (2023)

    P. S¨ oderlind, private communication (2023). IX. APPENDIX:HANDLING INPUT AND OUTPUT NOISE WITH SQUARED E XPONENTIAL KERNELS At their core, GPs define a prior over functions and, given data, inf er a posterior function space that captures the underlying data-generating process [6, 30]. The kernel or cov ariance function characterizes the smoothness and var...

    work page 2023