# One-Shot Learning of Stochastic Differential Equations with Data Adapted Kernels

arXiv: https://arxiv.org/abs/2209.12086 · HTML: https://mlsanity.com/paper/2209.12086/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 55 references that are in the map.
- First posted on arXiv 24 Sep 2022.

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

- [Gradient-based Hyperparameter Optimization through Reversible Learning](https://mlsanity.com/paper/1502.03492/) (2015)
- [Forward and Reverse Gradient-Based Hyperparameter Optimization](https://mlsanity.com/paper/1703.01785/) (2017)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Kernel Flows: from learning kernels from data into the abyss](https://mlsanity.com/paper/1808.04475/) (2018)
- [Kernel-based approximation of the Koopman generator and Schr"odinger operator](https://mlsanity.com/paper/2005.13231/) (2020)
- [Computational Graph Completion](https://mlsanity.com/paper/2110.10323/) (2021)
- [Learning Stochastic Differential Equations With Gaussian Processes Without Gradient Matching](https://mlsanity.com/paper/1807.05748/) (2018)
- [Learning dynamical systems from data: a simple cross-validation perspective](https://mlsanity.com/paper/2007.05074/) (2020)
- [Consistency of Empirical Bayes And Kernel Flow For Hierarchical Parameter Estimation](https://mlsanity.com/paper/2005.11375/) (2020)
- [Data-driven geophysical forecasting: Simple, low-cost, and accurate baselines with kernel methods](https://mlsanity.com/paper/2103.10935/) (2021)
- [Learning dynamical systems from data: A simple cross-validation perspective, part III: Irregularly-Sampled Time Series](https://mlsanity.com/paper/2111.13037/) (2021)
- [Kernel methods for center manifold approximation and a data-based version of the Center Manifold Theorem](https://mlsanity.com/paper/2012.00338/) (2020)
- [Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifolds](https://mlsanity.com/paper/1904.08622/) (2019)
- [A Note on Kernel Methods for Multiscale Systems with Critical Transitions](https://mlsanity.com/paper/1804.09415/) (2018)
- [Learning stochastic dynamical systems with neural networks mimicking the Euler-Maruyama scheme](https://mlsanity.com/paper/2105.08449/) (2021)
- [Data-driven prediction of a multi-scale Lorenz 96 chaotic system using deep learning methods: Reservoir computing, ANN, and RNN-LSTM](https://mlsanity.com/paper/1906.08829/) (2019)

## Built on by (most cited first, 15 of 15)

- [Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes](https://mlsanity.com/paper/2304.01294/) (2023)
- [Learning Stochastic Dynamical System via Flow Map Operator](https://mlsanity.com/paper/2305.03874/) (2023)
- [Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs](https://mlsanity.com/paper/2305.04962/) (2023)
- [A Training-Free Conditional Diffusion Model for Learning Stochastic Dynamical Systems](https://mlsanity.com/paper/2410.03108/) (2024)
- [Codiscovering graphical structure and functional relationships within data: A Gaussian Process framework for connecting the dots](https://mlsanity.com/paper/2311.17007/) (2023)
- [Modeling Unknown Stochastic Dynamical System Subject to External Excitation](https://mlsanity.com/paper/2406.15747/) (2024)
- [Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysis](https://mlsanity.com/paper/2503.01036/) (2025)
- [Data-driven Effective Modeling of Multiscale Stochastic Dynamical Systems](https://mlsanity.com/paper/2408.14821/) (2024)
- [Kernel Sum of Squares for Data Adapted Kernel Learning of Dynamical Systems from Data: A global optimization approach](https://mlsanity.com/paper/2408.06465/) (2024)
- [Recurrent Deep Kernel Learning of Dynamical Systems](https://mlsanity.com/paper/2405.19785/) (2024)
- [A Robust Model-Based Approach for Continuous-Time Policy Evaluation with Unknown L'evy Process Dynamics](https://mlsanity.com/paper/2504.01482/) (2025)
- [Identifying Unknown Stochastic Dynamics via Finite expression methods](https://mlsanity.com/paper/2504.07085/) (2025)
- [The Stochastic Occupation Kernel (SOCK) Method for Learning Stochastic Differential Equations](https://mlsanity.com/paper/2505.11622/) (2025)
- [Data-driven Effective Modeling of Stochastic Chemical Reaction Networks](https://mlsanity.com/paper/2608.25421/) (2026)
- [Neural means and kernel corrections for operator learning](https://mlsanity.com/paper/2609.00389/) (2026)

## Abstract (arXiv)

We consider the problem of learning Stochastic Differential Equations of the form dXt = f(Xt)dt+σ(Xt)dWt  from one sample trajectory. This problem is more challenging than learning deterministic dynamical systems because one sample trajectory only provides indirect information on the unknown functions f, σ, and stochastic process dWt representing the drift, the diffusion, and the stochastic forcing terms, respectively. We propose a method that combines Computational Graph Completion and data adapted kernels learned via a new variant of cross validation. Our approach can be decomposed as follows: (1) Represent the time-increment map Xt rightarrow Xt+dt as a Computational Graph in which f, σ and dWt appear as unknown functions and random variables. (2) Complete the graph (approximate unknown functions and random variables) via Maximum a Posteriori Estimation (given the data) with Gaussian Process (GP) priors on the unknown functions. (3) Learn the covariance functions (kernels) of the GP priors from data with randomized cross-validation. Numerical experiments illustrate the efficacy, robustness, and scope of our method.