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Non-parametric Learning of Stochastic Differential Equations with Non-asymptotic Fast Rates of Convergence.

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Title: Non-parametric Learning of Stochastic Differential Equations with Non-asymptotic Fast Rates of Convergence.
Authors: Bonalli, Riccardo1 (AUTHOR) riccardo.bonalli@cnrs.fr, Rudi, Alessandro2 (AUTHOR) alessandro.rudi@sdabocconi.it
Source: Foundations of Computational Mathematics. Jun2026, Vol. 26 Issue 3, p1497-1552. 56p.
Subjects: Stochastic differential equations, Nonparametric estimation, Reproducing kernel (Mathematics), Numerical analysis, Fokker-Planck equation, Drift diffusion models, Kernel functions
Abstract: We propose a novel non-parametric learning paradigm for the identification of drift and diffusion coefficients of multi-dimensional non-linear stochastic differential equations, which relies upon discrete-time observations of the state. The key idea essentially consists of fitting a RKHS-based approximation of the corresponding Fokker–Planck equation to such observations, yielding theoretical estimates of non-asymptotic learning rates which, unlike previous works, become increasingly tighter when the regularity of the unknown drift and diffusion coefficients becomes higher. Our method being kernel-based, offline pre-processing may be profitably leveraged to enable efficient numerical implementation, offering excellent balance between precision and computational complexity. [ABSTRACT FROM AUTHOR]
Copyright of Foundations of Computational Mathematics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Label: Abstract
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  Data: We propose a novel non-parametric learning paradigm for the identification of drift and diffusion coefficients of multi-dimensional non-linear stochastic differential equations, which relies upon discrete-time observations of the state. The key idea essentially consists of fitting a RKHS-based approximation of the corresponding Fokker–Planck equation to such observations, yielding theoretical estimates of non-asymptotic learning rates which, unlike previous works, become increasingly tighter when the regularity of the unknown drift and diffusion coefficients becomes higher. Our method being kernel-based, offline pre-processing may be profitably leveraged to enable efficient numerical implementation, offering excellent balance between precision and computational complexity. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Foundations of Computational Mathematics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s10208-025-09705-x
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      – Code: eng
        Text: English
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        PageCount: 56
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    Subjects:
      – SubjectFull: Stochastic differential equations
        Type: general
      – SubjectFull: Nonparametric estimation
        Type: general
      – SubjectFull: Reproducing kernel (Mathematics)
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      – SubjectFull: Numerical analysis
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      – SubjectFull: Fokker-Planck equation
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      – SubjectFull: Drift diffusion models
        Type: general
      – SubjectFull: Kernel functions
        Type: general
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      – TitleFull: Non-parametric Learning of Stochastic Differential Equations with Non-asymptotic Fast Rates of Convergence.
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              Text: Jun2026
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