# The Stochastic Occupation Kernel (SOCK) Method for Learning Stochastic Differential Equations

arXiv: https://arxiv.org/abs/2505.11622 · HTML: https://mlsanity.com/paper/2505.11622/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 44 references that are in the map.
- First posted on arXiv 16 May 2025.

## Builds on (most cited first, 17 of 17)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [A Rewriting System for Convex Optimization Problems](https://mlsanity.com/paper/1709.04494/) (2017)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Non-parametric Models for Non-negative Functions](https://mlsanity.com/paper/2007.03926/) (2020)
- [Stable Neural Stochastic Differential Equations in Analyzing Irregular Time Series Data](https://mlsanity.com/paper/2402.14989/) (2024)
- [A Kernel Two-Sample Test for Functional Data](https://mlsanity.com/paper/2008.11095/) (2020)
- [Learning Stochastic Differential Equations With Gaussian Processes Without Gradient Matching](https://mlsanity.com/paper/1807.05748/) (2018)
- [Latent SDEs on Homogeneous Spaces](https://mlsanity.com/paper/2306.16248/) (2023)
- [One-Shot Learning of Stochastic Differential Equations with Data Adapted Kernels](https://mlsanity.com/paper/2209.12086/) (2022)
- [The Occupation Kernel Method for Nonlinear System Identification](https://mlsanity.com/paper/1909.11792/) (2019)
- [DynGMA: a robust approach for learning stochastic differential equations from data](https://mlsanity.com/paper/2402.14475/) (2024)
- [Learning PSD-valued functions using kernel sums-of-squares](https://mlsanity.com/paper/2111.11306/) (2021)
- [Learning nonparametric ordinary differential equations from noisy data](https://mlsanity.com/paper/2206.15215/) (2022)
- [MOCK: an Algorithm for Learning Nonparametric Differential Equations via Multivariate Occupation Kernel Functions](https://mlsanity.com/paper/2306.10189/) (2023)
- [ROCK: A variational formulation for occupation kernel methods in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2503.13791/) (2025)

## Abstract (arXiv)

We present a novel kernel-based method for learning multivariate stochastic differential equations (SDEs). The method follows a two-step procedure: we first estimate the drift term function, then the (matrix-valued) diffusion function given the drift. Occupation kernels are integral functionals on a reproducing kernel Hilbert space (RKHS) that aggregate information over a trajectory. Our approach leverages vector-valued occupation kernels for estimating the drift component of the stochastic process. For diffusion estimation, we extend this framework by introducing operator-valued occupation kernels, enabling the estimation of an auxiliary matrix-valued function as a positive semi-definite operator, from which we readily derive the diffusion estimate. This enables us to avoid common challenges in SDE learning, such as intractable likelihoods, by optimizing a reconstruction-error-based objective. We propose a simple learning procedure that retains strong predictive accuracy while using Fenchel duality to promote efficiency. We validate the method on simulated benchmarks and a real-world dataset of Amyloid imaging in healthy and Alzheimer's disease subjects.