# Learning Stochastic Differential Equations With Gaussian Processes Without Gradient Matching

arXiv: https://arxiv.org/abs/1807.05748 · HTML: https://mlsanity.com/paper/1807.05748/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jul 2018.

## Built on by (most cited first, 23 of 23)

- [Learning effective stochastic differential equations from microscopic simulations: linking stochastic numerics to deep learning](https://mlsanity.com/paper/2106.09004/) (2021)
- [Learning Stochastic Dynamical System via Flow Map Operator](https://mlsanity.com/paper/2305.03874/) (2023)
- [One-Shot Learning of Stochastic Differential Equations with Data Adapted Kernels](https://mlsanity.com/paper/2209.12086/) (2022)
- [Scalable Inference in SDEs by Direct Matching of the Fokker-Planck-Kolmogorov Equation](https://mlsanity.com/paper/2110.15739/) (2021)
- [Variational Gaussian Process Diffusion Processes](https://mlsanity.com/paper/2306.02066/) (2023)
- [Learning stochastic dynamical systems with neural networks mimicking the Euler-Maruyama scheme](https://mlsanity.com/paper/2105.08449/) (2021)
- [Modeling Unknown Stochastic Dynamical System Subject to External Excitation](https://mlsanity.com/paper/2406.15747/) (2024)
- [Data-driven Effective Modeling of Multiscale Stochastic Dynamical Systems](https://mlsanity.com/paper/2408.14821/) (2024)
- [Fully Bayesian Differential Gaussian Processes through Stochastic Differential Equations](https://mlsanity.com/paper/2408.06069/) (2024)
- [Learning particle swarming models from data with Gaussian processes](https://mlsanity.com/paper/2106.02735/) (2021)
- [Robust and Scalable SDE Learning: A Functional Perspective](https://mlsanity.com/paper/2110.05167/) (2021)
- [Infinite-dimensional optimization and Bayesian nonparametric learning of stochastic differential equations](https://mlsanity.com/paper/2205.15368/) (2022)
- [Drift Estimation for Diffusion Processes Using Neural Networks Based on Discretely Observed Independent Paths](https://mlsanity.com/paper/2511.11161/) (2025)
- [Learning Collective Behaviors from Observation](https://mlsanity.com/paper/2311.00875/) (2023)
- [Data-Driven Model Selections of Second-Order Particle Dynamics via Integrating Gaussian Processes with Low-Dimensional Interacting Structures](https://mlsanity.com/paper/2311.00902/) (2023)
- [Geometric path augmentation for inference of sparsely observed stochastic nonlinear systems](https://mlsanity.com/paper/2301.08102/) (2023)
- [Nonparametric learning of stochastic differential equations from sparse and noisy data](https://mlsanity.com/paper/2508.11597/) (2025)
- [Traversing Time with Multi-Resolution Gaussian Process State-Space Models](https://mlsanity.com/paper/2112.03230/) (2021)
- [The Past as a Stochastic Process](https://mlsanity.com/paper/2112.05876/) (2021)
- [The Stochastic Occupation Kernel (SOCK) Method for Learning Stochastic Differential Equations](https://mlsanity.com/paper/2505.11622/) (2025)
- [Data-driven Learning of Interaction Laws in Multispecies Particle Systems with Gaussian Processes: Convergence Theory and Applications](https://mlsanity.com/paper/2511.02053/) (2025)
- [Data-driven Effective Modeling of Stochastic Chemical Reaction Networks](https://mlsanity.com/paper/2608.25421/) (2026)
- [Simulation-Free Learning of GP-SDEs from Irregular Observations](https://mlsanity.com/paper/2609.33112/) (2026)

## Abstract (arXiv)

We introduce a novel paradigm for learning non-parametric drift and diffusion functions for stochastic differential equation (SDE). The proposed model learns to simulate path distributions that match observations with non-uniform time increments and arbitrary sparseness, which is in contrast with gradient matching that does not optimize simulated responses. We formulate sensitivity equations for learning and demonstrate that our general stochastic distribution optimisation leads to robust and efficient learning of SDE systems.