# Non-parametric Estimation of Stochastic Differential Equations with Sparse Gaussian Processes

arXiv: https://arxiv.org/abs/1704.04375 · HTML: https://mlsanity.com/paper/1704.04375/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Apr 2017.

## Built on by (most cited first, 12 of 12)

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- [Learning the temporal evolution of multivariate densities via normalizing flows](https://mlsanity.com/paper/2107.13735/) (2021)
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- [Extracting stochastic dynamical systems with α-stable L'evy noise from data](https://mlsanity.com/paper/2109.14881/) (2021)
- [An end-to-end deep learning approach for extracting stochastic dynamical systems with α-stable L'evy noise](https://mlsanity.com/paper/2201.13114/) (2022)
- [In-Context Learning of Stochastic Differential Equations with Foundation Inference Models](https://mlsanity.com/paper/2502.19049/) (2025)
- [Extracting Stochastic Governing Laws by Nonlocal Kramers-Moyal Formulas](https://mlsanity.com/paper/2108.12570/) (2021)
- [Error Bounds of the Invariant Statistics in Machine Learning of Ergodic Ito Diffusions](https://mlsanity.com/paper/2105.10102/) (2021)
- [Weak Collocation Regression for Inferring Stochastic Dynamics with L'evy Noise](https://mlsanity.com/paper/2403.08292/) (2024)
- [Linear Noise Approximation Assisted Bayesian Inference on Mechanistic Model of Partially Observed Stochastic Reaction Network](https://mlsanity.com/paper/2405.02783/) (2024)
- [Time-dependent density estimation using binary classifiers](https://mlsanity.com/paper/2506.15505/) (2025)
- [A machine learning framework for uncovering stochastic nonlinear dynamics from noisy data](https://mlsanity.com/paper/2604.06081/) (2026)

## Abstract (arXiv)

The application of Stochastic Differential Equations (SDEs) to the analysis of temporal data has attracted increasing attention, due to their ability to describe complex dynamics with physically interpretable equations. In this paper, we introduce a non-parametric method for estimating the drift and diffusion terms of SDEs from a densely observed discrete time series. The use of Gaussian processes as priors permits working directly in a function-space view and thus the inference takes place directly in this space. To cope with the computational complexity that requires the use of Gaussian processes, a sparse Gaussian process approximation is provided. This approximation permits the efficient computation of predictions for the drift and diffusion terms by using a distribution over a small subset of pseudo-samples. The proposed method has been validated using both simulated data and real data from economy and paleoclimatology. The application of the method to real data demonstrates its ability to capture the behaviour of complex systems.