# Approximate Bayes learning of stochastic differential equations

arXiv: https://arxiv.org/abs/1702.05390 · HTML: https://mlsanity.com/paper/1702.05390/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Feb 2017.

## Built on by (most cited first, 11 of 11)

- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [Learning About Structural Errors in Models of Complex Dynamical Systems](https://mlsanity.com/paper/2401.00035/) (2023)
- [In-Context Learning of Stochastic Differential Equations with Foundation Inference Models](https://mlsanity.com/paper/2502.19049/) (2025)
- [Robust and Scalable SDE Learning: A Functional Perspective](https://mlsanity.com/paper/2110.05167/) (2021)
- [Federated Neural Nonparametric Point Processes](https://mlsanity.com/paper/2410.05637/) (2024)
- [State-space deep Gaussian processes with applications](https://mlsanity.com/paper/2111.12604/) (2021)
- [Geometric path augmentation for inference of sparsely observed stochastic nonlinear systems](https://mlsanity.com/paper/2301.08102/) (2023)
- [Geometric constraints improve inference of sparsely observed stochastic dynamics](https://mlsanity.com/paper/2304.00423/) (2023)
- [Towards Fast Coarse-graining and Equation Discovery with Foundation Inference Models](https://mlsanity.com/paper/2510.12618/) (2025)
- [Principled model selection for stochastic dynamics](https://mlsanity.com/paper/2501.10339/) (2025)
- [A machine learning framework for uncovering stochastic nonlinear dynamics from noisy data](https://mlsanity.com/paper/2604.06081/) (2026)

## Abstract (arXiv)

We introduce a nonparametric approach for estimating drift and diffusion functions in systems of stochastic differential equations from observations of the state vector. Gaussian processes are used as flexible models for these functions and estimates are calculated directly from dense data sets using Gaussian process regression. We also develop an approximate expectation maximization algorithm to deal with the unobserved, latent dynamics between sparse observations. The posterior over states is approximated by a piecewise linearized process of the Ornstein-Uhlenbeck type and the maximum a posteriori estimation of the drift is facilitated by a sparse Gaussian process approximation.