# Non-Parametric Learning of Stochastic Differential Equations with Non-asymptotic Fast Rates of Convergence

arXiv: https://arxiv.org/abs/2305.15557 · HTML: https://mlsanity.com/paper/2305.15557/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 62 references that are in the map.
- First posted on arXiv 24 May 2023.

## Builds on (most cited first, 16 of 16)

- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Non-parametric Models for Non-negative Functions](https://mlsanity.com/paper/2007.03926/) (2020)
- [Infinitely Deep Bayesian Neural Networks with Stochastic Differential Equations](https://mlsanity.com/paper/2102.06559/) (2021)
- [Towards a mathematical theory of trajectory inference](https://mlsanity.com/paper/2102.09204/) (2021)
- [Smoothing parameter and model selection for general smooth models](https://mlsanity.com/paper/1511.03864/) (2015)
- [Finding Global Minima via Kernel Approximations](https://mlsanity.com/paper/2012.11978/) (2020)
- [PSD Representations for Effective Probability Models](https://mlsanity.com/paper/2106.16116/) (2021)
- [Online Maximum Likelihood Estimation of the Parameters of Partially Observed Diffusion Processes](https://mlsanity.com/paper/1611.00170/) (2016)
- [Nonlinear Uncertainty Control with Iterative Covariance Steering](https://mlsanity.com/paper/1903.10919/) (2019)
- [Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions](https://mlsanity.com/paper/1510.05526/) (2015)
- [Learning PSD-valued functions using kernel sums-of-squares](https://mlsanity.com/paper/2111.11306/) (2021)
- [Sequential Convex Programming For Non-Linear Stochastic Optimal Control](https://mlsanity.com/paper/2009.05182/) (2020)
- [Joint Online Parameter Estimation and Optimal Sensor Placement for the Partially Observed Stochastic Advection-Diffusion Equation](https://mlsanity.com/paper/2009.08693/) (2020)
- [First-Order Pontryagin Maximum Principle for Risk-Averse Stochastic Optimal Control Problems](https://mlsanity.com/paper/2204.03036/) (2022)
- [Control Strategies for the Fokker-Planck Equation](https://mlsanity.com/paper/1707.07510/) (2017)
- [Parametric estimation of stochastic differential equations via online gradient descent](https://mlsanity.com/paper/2210.08800/) (2022)

## Built on by (most cited first, 4 of 4)

- [Learning diffusion at lightspeed](https://mlsanity.com/paper/2406.12616/) (2024)
- [Drift Estimation for Stochastic Differential Equations with Denoising Diffusion Models](https://mlsanity.com/paper/2602.17830/) (2026)
- [Using Linearized Optimal Transport to Predict the Evolution of Stochastic Particle Systems](https://mlsanity.com/paper/2408.01857/) (2024)
- [Safely Learning Controlled Stochastic Dynamics](https://mlsanity.com/paper/2506.02754/) (2025)

## Abstract (arXiv)

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.