Pith. sign in

REVIEW 3 minor 70 references

A Quadratic Order Reduction -- Gaussian Process Ordinary Differential Equation framework for the inference of Large Continuous Dynamical Systems

T0 review · 0 major / 3 minor · reviewed 2026-06-27 · grok-4.3

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

Pith's one-line read A Gaussian process ordinary differential equation model with quadratic order reduction forecasts large dynamical systems more accurately or cheaply than standard reduced-order methods while proving convergence in the smooth case.

desk verdict The paper combines GP-ODE with quadratic reduction and sphere projection for high-dimensional dynamical systems, with a stated convergence result for the base model and reported wins over several ROM methods. read the letter →

arxiv 2606.13063 v1 pith:DFUHDKDR submitted 2026-06-11 math.NA cs.NAstat.ML

Guglielmo Padula , Michele Girfoglio , Gianluigi Rozza This is my paper · ORCID
classification math.NAcs.NAstat.ML
keywords Gaussianprocessesordinarydifferentialequationsreduced-ordermodelingdynamicalsystemsforecastinguncertaintyquantificationquadraticreductionautonomous
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

This paper develops a framework to forecast the evolution of complex dynamical systems by modeling them as autonomous ordinary differential equations learned via Gaussian processes. It augments the base model with quadratic order reduction and sphere projection to manage high-dimensional data efficiently and keep the learned dynamics stable. The base GP-ODE is shown to converge to the true underlying equation under smoothness assumptions. Numerical tests indicate the complete approach beats several existing reduced-order forecasting techniques on accuracy or runtime while supplying uncertainty estimates. The method targets applications where reliable short-term predictions of large nonlinear systems are needed.

What carries the argument

The Gaussian Process Ordinary Differential Equation (GP-ODE) integrated with quadratic order model reduction and sphere projection, which enables learning stable latent dynamics from data.

What would settle it

A direct numerical test on a known smooth autonomous system where increasing data density fails to make the learned GP-ODE approach the true equation would disprove the convergence claim.

Watch

Extended reading notes

Core claim

The authors establish a Gaussian Process Ordinary Differential Equation model that provably converges to the true autonomous dynamical system in the smooth case. By incorporating quadratic order reduced-order modelling and sphere projection, the framework learns latent dynamics stably and efficiently, leading to better performance than Extended Dynamic Mode Decomposition, Bagging Optimised Dynamic Mode Decomposition, and Linear and Nonlinear Disambiguation Optimisation in accuracy or computational costs for forecasting complex systems.

Load-bearing premise

The dynamical system is autonomous and the observed data suffice to learn stable latent dynamics via quadratic reduction and sphere projection.

Editorial extensions

If this is right

  • The full model outperforms Extended Dynamic Mode Decomposition, Bagging Optimised Dynamic Mode Decomposition and Linear and Nonlinear Disambiguation Optimisation in accuracy or computational costs.
  • The base GP-ODE provably converges to the real autonomous equation in the smooth case.
  • Forecasts include rigorous uncertainty quantification.
  • Sphere projection preserves stability while quadratic reduction lowers the cost of learning latent dynamics in large systems.

Reading between the lines

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

  • The convergence result could be leveraged to obtain explicit error bounds when data contain moderate noise.
  • Adding explicit time dependence might allow the same structure to handle non-autonomous or forced systems.
  • The quadratic reduction step could be combined with other kernel choices to test whether similar stability and cost gains appear in related inference tasks.
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

0 major / 3 minor

Summary. The manuscript proposes a QOR-GP-ODE framework that integrates a base Gaussian Process Ordinary Differential Equation model with quadratic-order reduced-order modeling and sphere projection. The base GP-ODE is claimed to enable short-term forecasting with uncertainty quantification and to provably converge to the true autonomous ODE under smoothness assumptions; the full model is asserted to learn stable latent dynamics efficiently and to outperform EDMD, BODMD, and LNDO on accuracy or computational cost in numerical experiments on complex dynamical systems.

Significance. If the convergence argument is rigorous and the experiments fairly compare against the cited ROM baselines under matched conditions, the work could supply a practically useful route to stable, uncertainty-aware forecasting of high-dimensional autonomous systems that improves on existing trade-offs between accuracy, stability, and interpretability.

minor comments (3)
  1. [Abstract] The abstract states that the base model 'provably converges' but does not indicate the section or theorem number containing the proof; adding an explicit forward reference would help readers locate the argument.
  2. [Abstract] Numerical experiments are summarized only at the level of 'outperforms ... in terms of accuracy or computational costs'; a table or figure reporting concrete error metrics, wall-clock times, and the precise test systems (including dimension and smoothness) would strengthen the comparison claims.
  3. [Abstract] The acronyms EDMD, BODMD, and LNDO are introduced without citations; supplying the original references for these baselines would allow readers to verify the comparison setup.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. The provided overview correctly reflects the core elements of the QOR-GP-ODE framework, including the base GP-ODE convergence property and the integration with quadratic reduced-order modeling.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The abstract and described claims present a GP-ODE base model with a stated convergence result under explicit smoothness and autonomy assumptions, plus empirical comparisons to other ROM methods. No load-bearing step reduces by construction to a fitted parameter renamed as prediction, a self-citation chain, or a self-definitional equivalence. The central claims remain independent of the paper's own fitted outputs or prior self-citations in the supplied text.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Ledger constructed from abstract only; full paper likely contains additional parameters and assumptions not visible here.

assumptions (1)
  • domain assumption The dynamical system is autonomous and smooth.
    Required for the stated convergence of the GP-ODE model to the true equation.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A Quadratic Order Reduction -- Gaussian Process Ordinary Differential Equation framework for the inference of Large Continuous Dynamical Systems." pith.science (2026). https://pith.science/paper/DFUHDKDR

@misc{pith2026260613063,
  author       = {Pith},
  title        = {Pith review of: A Quadratic Order Reduction -- Gaussian Process Ordinary Differential Equation framework for the inference of Large Continuous Dynamical Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DFUHDKDR}},
  note         = {Machine review of arXiv:2606.13063}
}
read the original abstract

Forecasting the evolution of complex dynamical systems remains a fundamentally challenging task, primarily due to pronounced nonlinear interactions, high-dimensional state spaces, and the concomitant requirement for rigorous and reliable uncertainty quantification. Contemporary reduced-order modelling (ROM) frameworks frequently exhibit inherent trade-offs among predictive accuracy, numerical stability, and interpretability, and thus often fail to achieve an optimal balance among these competing objectives. To address these limitations, we propose a framework for forecasting complex dynamical systems via a kernel autonomous ordinary differential equation approach based on Gaussian Processes and Quadratic Order Model Reduction. Our base method, the Gaussian Process Ordinary Differential Equations model, allows accurate short-term forecasting with uncertainty quantification, and it provably converges to the real autonomous equation in the smooth case. We integrate it with quadratic order reduced-order modelling and sphere projection for learning the latent dynamics efficiently while preserving stability. Numerical experiments demonstrate that our full model outperforms ROM forecasting methods such as Extended Dynamic Mode Decomposition, Bagging Optimised Dynamic Mode Decomposition and Linear and Nonlinear Disambiguation Optimisation in terms of accuracy or computational costs. These results demonstrate the potential of the framework as a robust and stable tool for forecasting complex dynamical systems with rigorous uncertainty quantification.

Figures

Figures reproduced from arXiv: 2606.13063 by the authors.

Figure 1
Figure 1. Top: theoretical error bound and numerical error for [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. The workflow of the QGPRODE algorithm. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Trajectory of the Lorentz system given by Eq. 40. [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Simulation of a Lorenz system. The error is computed with respect to the mean of [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]
Figure 5
Figure 5. Figure 5: Simulation of Eq. 43. The error is computed with respect to the mean of the [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]
Figure 6
Figure 6. Figure 6: BV-α test case. Prediction at time 100 of the QGPRODE with 95% confidence bounds and the true value. The temporal snapshots are saved each second, while the simulation is performed with ∆t = 2.5 · 10−4 . The spatial discretisation is 16 × 32; the first 80 samples are u…
Figure 7
Figure 7. Figure 7: BV-α test case. Dynamics of the QGPRODE and of the true model, and the L1 error, averaged through time, with uncertainty bands. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]
Figure 8
Figure 8. Figure 8: BV-α test case. Space-Averaged Absolute Value Difference between the QGPRODE dynamics and the True dynamics. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_8.png]
Figure 9
Figure 9. Figure 9: ERA5 temperature test case. Prediction at time step 99 of the QGPRODE with [PITH_FULL_IMAGE:figures/full_fig_p027_9.png]
Figure 10
Figure 10. Figure 10: ERA5 temperature test case. Dynamics of the QGPRODE and of the true model, [PITH_FULL_IMAGE:figures/full_fig_p027_10.png]
Figure 11
Figure 11. Figure 11: ERA5 temperature test case. Space-Averaged Absolute Value Difference between [PITH_FULL_IMAGE:figures/full_fig_p028_11.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

70 extracted references · 5 canonical work pages

  1. [1]

    Digital Twin in manufacturing: A categorical literature review and classification

    Werner Kritzinger, Matthias Karner, Georg Traar, Jan Henjes, and Wilfried Sihn. Digital Twin in manufacturing: A categorical literature review and classification. IFAC-PapersOnLine, 51(11):1016–1022, 2018

    2018

  2. [2]

    Brotzge, Don Berchoff, DaNa L

    Jerald A. Brotzge, Don Berchoff, DaNa L. Carlis, Frederick H. Carr, Rachel Hogan Carr, Jordan J. Gerth, Brian D. Gross, Thomas M. Hamill, Sue Ellen Haupt, Neil Jacobs, Amy McGovern, David J. Stensrud, Gary Szatkowski, Istvan Szunyogh, and Xuguang Wang. Challenges and Opportunities in Numerical Weather Prediction. Bulletin of the American Meteorological So...

    2023

  3. [3]

    Bresch, Daniela I

    Bruno Merz, Christian Kuhlicke, Michael Kunz, Massimiliano Pittore, Andrey Babeyko, David N. Bresch, Daniela I. V. Domeisen, Frauke Feser, Inga Koszalka, Heidi Kreibich, Florian Pantillon, Stefano Parolai, Joaquim G. Pinto, Heinz J¨ urgen Punge, Eleonora Ri- valta, Kai Schr¨ oter, Karen Strehlow, Ralf Weisse, and Andreas Wurpts. Impact Fore- casting to Su...

    2020

  4. [4]

    Sparse Identifi- cation for bifurcating phenomena in Computational Fluid Dynamics

    Lorenzo Tomada, Moaad Khamlich, Federico Pichi, and Gianluigi Rozza. Sparse Identifi- cation for bifurcating phenomena in Computational Fluid Dynamics. Computers & Fluids, 302:106841, November 2025

    2025

  5. [5]

    Machine Learning-based quadratic closures for non-intrusive Reduced Order Models

    Gabriele Codega, Anna Ivagnes, Nicola Demo, and Gianluigi Rozza. Machine Learning-based quadratic closures for non-intrusive Reduced Order Models, June 2025. arXiv:2506.09830 [math]

    work page Pith review arXiv 2025

  6. [6]

    Time Extrapolation with Graph Convolutional Autoencoder and Tensor Train Decomposition, 2025

    Yuanhong Chen, Federico Pichi, Zhen Gao, and Gianluigi Rozza. Time Extrapolation with Graph Convolutional Autoencoder and Tensor Train Decomposition, 2025. Version Number: 1

    2025

  7. [7]

    Koopman-Equivariant Gaussian Processes, February 2025

    Petar Bevanda, Max Beier, Armin Lederer, Alexandre Capone, Stefan Sosnowski, and San- dra Hirche. Koopman-Equivariant Gaussian Processes, February 2025. arXiv:2502.06645 [cs]

    work page arXiv 2025

  8. [8]

    Comparison and combination of reduced-order modelling techniques in 3D parametrized heat transfer problems

    Fabrizio Gelsomino and Gianluigi Rozza. Comparison and combination of reduced-order modelling techniques in 3D parametrized heat transfer problems. Mathematical and Computer Modelling of Dynamical Systems, 17(4):371–394, August 2011

    2011

Show all 70 references
  1. [9]

    Optimal transport-based displacement interpolation with data augmentation for reduced order modeling of nonlinear dynamical systems

    Moaad Khamlich, Federico Pichi, Michele Girfoglio, Annalisa Quaini, and Gianluigi Rozza. Optimal transport-based displacement interpolation with data augmentation for reduced order modeling of nonlinear dynamical systems. Journal of Computational Physics, 531:113938, June 2025

    2025

  2. [10]

    Carlos J. G. Rojas, Andreas Dengel, and Mateus Dias Ribeiro. Reduced-order Model for Fluid Flows via Neural Ordinary Differential Equations, 2021. Version Number: 2

    2021

  3. [11]

    Xuping Xie, Guannan Zhang, and Clayton G. Webster. Non-Intrusive Inference Reduced Order Model for Fluids Using Deep Multistep Neural Network. Mathematics, 7(8):757, August 2019

    2019

  4. [12]

    On latent dynamics learning in nonlinear reduced order modeling

    Nicola Farenga, Stefania Fresca, Simone Brivio, and Andrea Manzoni. On latent dynamics learning in nonlinear reduced order modeling. Neural Networks, 185:107146, May 2025

    2025

  5. [13]

    Model order re- duction assisted by deep neural networks (ROM-net)

    Thomas Daniel, Fabien Casenave, Nissrine Akkari, and David Ryckelynck. Model order re- duction assisted by deep neural networks (ROM-net). Advanced Modeling and Simulation in Engineering Sciences, 7(1):16, December 2020

    2020

  6. [14]

    An artificial neural network framework for reduced order modeling of transient flows

    Omer San, Romit Maulik, and Mansoor Ahmed. An artificial neural network framework for reduced order modeling of transient flows. Communications in Nonlinear Science and Numerical Simulation, 77:271–287, October 2019

    2019

  7. [15]

    Williams, Ioannis G

    Matthew O. Williams, Ioannis G. Kevrekidis, and Clarence W. Rowley. A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition. Journal of Nonlinear Science, 25(6):1307–1346, December 2015

    2015

  8. [16]

    Nathan Kutz

    Diya Sashidhar and J. Nathan Kutz. Bagging, optimized dynamic mode decompo- sition for robust, stable forecasting with spatial and temporal uncertainty quantifica- tion. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 380(222...

    2022

  9. [17]

    Peter J. Schmid. Dynamic mode decomposition of numerical and experimental data. Journal of Fluid Mechanics, 656:5–28, August 2010

    2010

  10. [18]

    A Dynamic Mode Decomposition Extension for the Forecasting of Parametric Dynamical Systems.SIAM Journal on Applied Dynamical Systems, 22(3):2432–2458, September 2023

    Francesco Andreuzzi, Nicola Demo, and Gianluigi Rozza. A Dynamic Mode Decomposition Extension for the Forecasting of Parametric Dynamical Systems.SIAM Journal on Applied Dynamical Systems, 22(3):2432–2458, September 2023

    2023

  11. [19]

    On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator

    Milan Korda and Igor Mezi´ c. On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator. Journal of Nonlinear Science, 28(2):687–710, April 2018

    2018

  12. [20]

    Brunton, Joshua L

    Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz. Discovering governing equa- tions from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113(15):3932–3937, April 2016

    2016

  13. [21]

    On the stability of projection- based model order reduction for convection-dominated laminar and turbulent flows

    Sebastian Grimberg, Charbel Farhat, and Noah Youkilis. On the stability of projection- based model order reduction for convection-dominated laminar and turbulent flows. Journal of Computational Physics, 419:109681, October 2020

    2020

  14. [22]

    Baddoo, Benjamin Herrmann, Beverley J

    Peter J. Baddoo, Benjamin Herrmann, Beverley J. McKeon, J. Nathan Kutz, and Steven L. Brunton. Physics-informed dynamic mode decomposition. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 479(2271):20220576, March 2023

    2023

  15. [23]

    Kaptanoglu, Jared L

    Alan A. Kaptanoglu, Jared L. Callaham, Aleksandr Aravkin, Christopher J. Hansen, and Steven L. Brunton. Promoting global stability in data-driven models of quadratic nonlinear dynamics. Physical Review Fluids, 6(9):094401, September 2021

    2021

  16. [24]

    Baddoo, Benjamin Herrmann, Beverley J

    Peter J. Baddoo, Benjamin Herrmann, Beverley J. McKeon, and Steven L. Brunton. Ker- nel learning for robust dynamic mode decomposition: linear and nonlinear disambiguation optimization. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 478(22...

    2022

  17. [25]

    Patsatzis, Gianluigi Rozza, Lucia Russo, and Con- stantinos Siettos

    Alessandro Della Pia, Dimitrios G. Patsatzis, Gianluigi Rozza, Lucia Russo, and Con- stantinos Siettos. Surrogate normal-forms for the numerical bifurcation and stability anal- ysis of navier-stokes flows via machine learning, September 2025. arXiv:2506.21275

    2025 arXiv

  18. [26]

    Ex- act Inference for Continuous-Time Gaussian Process Dynamics

    Katharina Ensinger, Nicholas Tagliapietra, Sebastian Ziesche, and Sebastian Trimpe. Ex- act Inference for Continuous-Time Gaussian Process Dynamics. Proceedings of the AAAI Conference on Artificial Intelligence, 38(11):11883–11891, March 2024

    2024

  19. [27]

    Learning unknown ODE models with Gaussian processes

    Markus Heinonen, Cagatay Yildiz, Henrik Mannerstr¨ om, Jukka Intosalmi, and Harri L¨ ahdesm¨ aki. Learning unknown ODE models with Gaussian processes. InProceedings of the 35th International Conference on Machine Learning, pages 1959–1968. PMLR, July 2018

    1959

  20. [28]

    Emilio Tanowe Maddalena, Paul Scharnhorst, and Colin N. Jones. Deterministic er- ror bounds for kernel-based learning techniques under bounded noise. Automatica, 134:109896, December 2021

    2021

  21. [29]

    Uniform Error and Posterior Vari- ance Bounds for Gaussian Process Regression with Application to Safe Control, 2021

    Armin Lederer, Jonas Umlauft, and Sandra Hirche. Uniform Error and Posterior Vari- ance Bounds for Gaussian Process Regression with Application to Safe Control, 2021. arXiv:2101.05328. 31

    2021

  22. [30]

    Gaussian Process Uniform Error Bounds with Unknown Hyperparameters for Safety-Critical Applications

    Alexandre Capone, Armin Lederer, and Sandra Hirche. Gaussian Process Uniform Error Bounds with Unknown Hyperparameters for Safety-Critical Applications. In Proceedings of the 39th International Conference on Machine Learning, pages 2609–2624. PMLR, June 2022

    2022

  23. [31]

    Consistency of support vector machines for forecasting the evolution of an unknown ergodic dynamical system from observations with unknown noise

    Ingo Steinwart and Marian Anghel. Consistency of support vector machines for forecasting the evolution of an unknown ergodic dynamical system from observations with unknown noise. The Annals of Statistics, 37(2), April 2009

    2009

  24. [32]

    Rui Tuo, Yan Wang, and C. F. Jeff Wu. On the Improved Rates of Convergence for Mat´ ern-Type Kernel Ridge Regression with Application to Calibration of Computer Mod- els. SIAM/ASA Journal on Uncertainty Quantification, 8(4):1522–1547, January 2020

    2020

  25. [33]

    Scattered Data Approximation

    Holger Wendland. Scattered Data Approximation. Cambridge University Press, 1 edition, December 2004

    2004

  26. [34]

    Carl Edward Rasmussen and Christopher K. I. Williams. Gaussian Processes for Machine Learning. The MIT Press, November 2005

    2005

  27. [35]

    Learning Stable Nonparametric Dynam- ical Systems with Gaussian Process Regression

    Wenxin Xiao, Armin Lederer, and Sandra Hirche. Learning Stable Nonparametric Dynam- ical Systems with Gaussian Process Regression. IFAC-PapersOnLine, 53(2):1194–1199, January 2020

    2020

  28. [36]

    Operator inference for non-intrusive model reduction with quadratic manifolds

    Rudy Geelen, Stephen Wright, and Karen Willcox. Operator inference for non-intrusive model reduction with quadratic manifolds. Computer Methods in Applied Mechanics and Engineering, 403:115717, January 2023

    2023

  29. [37]

    An analytic comparison of regularization methods for Gaussian Processes, May 2017

    Hossein Mohammadi, Rodolphe Le Riche, Nicolas Durrande, Eric Touboul, and Xavier Bay. An analytic comparison of regularization methods for Gaussian Processes, May 2017. arXiv:1602.00853

    2017 arXiv

  30. [38]

    Extension formulas and norm inequalities in Sobolev Hilbert spaces

    Saiboh Saturo. Extension formulas and norm inequalities in Sobolev Hilbert spaces. Mathematical Inequalities & Applications, 27(2), 2024

    2024

  31. [39]

    Pedregosa, G

    F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blon- del, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn: Machine Learning in Python. Journal of Machine Learni...

    2011

  32. [40]

    PyDMD: Python Dynamic Mode Decomposition

    Nicola Demo, Marco Tezzele, and Gianluigi Rozza. PyDMD: Python Dynamic Mode Decomposition. The Journal of Open Source Software, 3(22):530, February 2018

    2018

  33. [41]

    Ichinaga, Francesco Andreuzzi, Nicola Demo, Marco Tezzele, Karl Lapo, Gianluigi Rozza, Steven L

    Sara M. Ichinaga, Francesco Andreuzzi, Nicola Demo, Marco Tezzele, Karl Lapo, Gianluigi Rozza, Steven L. Brunton, and J. Nathan Kutz. PyDMD: A Python Package for Robust Dynamic Mode Decomposition. Journal of Machine Learning Research, 25(417):1–9, 2024

    2024

  34. [42]

    A novel Large Eddy Simula- tion model for the Quasi-Geostrophic equations in a Finite Volume setting

    Michele Girfoglio, Annalisa Quaini, and Gianluigi Rozza. A novel Large Eddy Simula- tion model for the Quasi-Geostrophic equations in a Finite Volume setting. Journal of Computational and Applied Mathematics, 418:114656, January 2023

    2023

  35. [43]

    ERA5 hourly data on single levels from 1940 to present, 2018

    Copernicus Climate Change Service, Climate Data Store. ERA5 hourly data on single levels from 1940 to present, 2018. 32

    1940

  36. [44]

    Consistency of Gaussian Process Regression in Metric Spaces

    Peter Koepernik and Florian Pfaff. Consistency of Gaussian Process Regression in Metric Spaces. Journal of Machine Learning Research, 22(244):1–27, 2021

    2021

  37. [45]

    Buhmann, and Gerlind Plonka-Hoch

    Stefano De Marchi, Martin D. Buhmann, and Gerlind Plonka-Hoch. Kernel Functions and Meshless Methods. Dolomites Research Notes on Approximation, 4(DRNA Volume 4.2):1–63, September 2011

    2011

  38. [46]

    H. Inci, T. Kappeler, and P. Topalov. On the regularity of the composition of diffeomor- phisms. Memoirs of the American Mathematical Society, 226(1062):1, 2013

    2013

  39. [47]

    Behzadan and M

    A. Behzadan and M. Holst. Multiplication in Sobolev spaces, revisited. Arkiv f¨ or Matematik, 59(2):275–306, 2021

    2021

  40. [48]

    Fixed Point Theorems and Applications, volume 116 of UNITEXT

    Vittorino Pata. Fixed Point Theorems and Applications, volume 116 of UNITEXT. Springer International Publishing, Cham, 2019

    2019

  41. [49]

    Micchelli, Yuesheng Xu, and Haizhang Zhang

    Charles A. Micchelli, Yuesheng Xu, and Haizhang Zhang. Universal Kernels. Journal of Machine Learning Research, 7:2651–2667, 2006

    2006

  42. [50]

    Fast algorithms for convex quadratic programming and mul- ticommodity flows

    S Kapoor and P M Vaidya. Fast algorithms for convex quadratic programming and mul- ticommodity flows. In Proceedings of the eighteenth annual ACM symposium on Theory of computing - STOC ’86, pages 147–159, Berkeley, California, United States, 1986. ACM Press

    1986

  43. [51]

    Morris W. Hirsch. Differential Topology, volume 33. Springer New York, 1976

    1976

  44. [52]

    John M. Lee. Introduction to Smooth Manifolds, volume 218 of Graduate Texts in Mathematics. Springer New York, New York, NY, 2012. A Theoretical Analysis of GPRODE In this section, we analyse the convergence properties of the GPRODE methodology. The primary objective of GPRODE...

    2012

  45. [53]

    If assumption 2 holds, then if we defineˆσ N = supx∈B σN, we get lim N→+∞ ˆσN = 0 (55) uniformly

  46. [54]

    If assumption 1 holds, then sup x∈B ||µN(x)−f(x)|| 2 ≤2||f|| RKHS(U,R m)σN(x),∀x∈U.(56)

  47. [55]

    If assumptions 1,2 hold, thensup x∈B ||µN(x)−f(x)|| 2 →0. Proof.1. The inequality is found in [29, 44] if the inverse matrix is adopted of the pseu- doinverse, but the proof follows straightforwardly also for the pseudo-inverse case

  48. [56]

    Follows from the previous point. 37

  49. [57]

    Our case follows by applying the scalar version to every component and by computing theℓ 2 norm

    The scalar version with the inverse matrix instead of the pseudoinverse is found in [28, 45], but the proof follows straightforwardly also for the pseudo-inverse case. Our case follows by applying the scalar version to every component and by computing theℓ 2 norm

  50. [58]

    Theorem 1 establishes the core approximation guarantees for the Gaussian process model

    Follows from assumptions and from points 2 and 3. Theorem 1 establishes the core approximation guarantees for the Gaussian process model. The first part shows that the posterior varianceσ N(x) can be bounded in terms of the distance to the training points, emphasising how the ...

  51. [59]

    58, or 59, 57

    LetE N(t),V N the solutions of Eq. 58, or 59, 57. Then given the sequence of stochastic processY N(t+t N)with meanE N(t)−X(t+t N)and varianceV N(t)if holds that the quantitysup t∈[0,M] E[||YN(t+t N)−Y(t+t N)||2]converges whenN→+∞to0,∀t∈ [0, M]

  52. [60]

    LetY N the solution of Eq. 60. ThenE[||Y(t+t N)−Y N(t+t N)||2]converges in mean square to0∀t∈[0, M]. Proof.1. Follows from Theorem 1 and Proposition 1

  53. [61]

    Follows from Theorem 1 and Propositions 2,3

  54. [62]

    41 We have demonstrated that the GPRODE framework yields a mathematically consistent procedure for inferring dynamical systems from observational data

    Follows from Theorem 1 and Propositions 4. 41 We have demonstrated that the GPRODE framework yields a mathematically consistent procedure for inferring dynamical systems from observational data. In particular, the Gaussian process–based estimator converges uniformly to the tru...

  55. [63]

    Let YN(t)∼M N( ˜EN(t), VN(t)) (85) where ˜EN(t), ˜VN(t)are defined in Eq. 83. Thensup t∈[0,M] E[||YN(t+t N)−Y(t+t N)||2 2] converges to0

  56. [64]

    Let ˜YN(t)the solution of Eq. 84. Thesup t∈[0,M] E[||YN(t+t N)−Y(t+t N)||2 2]converges to0. 43 Proof.1. We have ||(˜µN −f(x))|| 2 ≤ ||(˜µN −Ψ N[f](x))|| 2 +||(f(x)−Ψ N[f](x))|| 2 ≤sup x∈B ||(˜µN −Ψ N(x))||2 +C ΨN (f)(∆tN)q. (86) As a consequence sup x∈B ||(˜µN −f)|| ≤2||Ψ N(x)...

  57. [65]

    Follows from point 1 and Proposition 1

  58. [66]

    Follows from point 1 and Proposition 2

  59. [67]

    We have shown convergence of the ODE with derivatives estimated using a finite-difference scheme to the true ODE

    Follows from point 1 and 4. We have shown convergence of the ODE with derivatives estimated using a finite-difference scheme to the true ODE. Definition 2.A functionalΦ(X, f, h)is an Initial Value Problem solver of orderpif given the sequenceX Φ,k∈Ngiven by ( XΦ,0,f =X, XΦ,k+1...

  60. [68]

    The operatorF:B→H k+m 2 (U,R m)given byF(y) =f(y(x))is well defined and further- more Lipschitz

  61. [69]

    it exists a timeh 0 such thatX(h, x)∈B,∀h≤h 0. Proof.1. Lety∈B. As||Dy−I|| ∞ ≤ ||y−id||<1 we have thatyis invertible andB⊂ D k+2+m 2 (U,R m) and a consequenceFis well defined [46] . Now||F(y)−F(z)|| H k+m 2 = || R 1 0 (∇f(y(x)+t(y(x)−z(x)))dt) T (y(x)−z(x))|| H k+m 2 , so asH ...

  62. [70]

    We are now ready to prove the convergence of the Forward difference method

    AsFis Lipschitz, the thesis follows from the local uniqueness theorem of ODE in Banach Spaces [48]. We are now ready to prove the convergence of the Forward difference method. Lets consider ||f(x)− X(h, x)−x h || H 3+m 2 (U) .(98) By applying the definition ofX(h, x) and by ch...

Pith tools

Reviewed June 27, 2026 · model on record in the stance chip above.