# Infinite-dimensional optimization and Bayesian nonparametric learning of stochastic differential equations

arXiv: https://arxiv.org/abs/2205.15368 · HTML: https://mlsanity.com/paper/2205.15368/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 67 references that are in the map.
- First posted on arXiv 30 May 2022.

## Builds on (most cited first, 4 of 4)

- [Learning Stochastic Differential Equations With Gaussian Processes Without Gradient Matching](https://mlsanity.com/paper/1807.05748/) (2018)
- [Bayesian inference for diffusion driven mixed-effects models](https://mlsanity.com/paper/1507.06807/) (2015)
- [A variational approach to path estimation and parameter inference of hidden diffusion processes](https://mlsanity.com/paper/1508.00506/) (2015)
- [Bayesian Variable Selection for Multi-Outcome Models Through Shared Shrinkage](https://mlsanity.com/paper/1904.11594/) (2019)

## Built on by (most cited first, 3 of 3)

- [Nonparametric learning of stochastic differential equations from sparse and noisy data](https://mlsanity.com/paper/2508.11597/) (2025)
- [Nonparametric learning of covariate-based Markov jump processes using RKHS techniques](https://mlsanity.com/paper/2505.03119/) (2025)
- [Variational Smoothing and Inference for SDEs from Sparse Data with Dynamic Neural Flows](https://mlsanity.com/paper/2605.05606/) (2026)

## Abstract (arXiv)

The paper has two major themes. The first part of the paper establishes certain general results for infinite-dimensional optimization problems on Hilbert spaces. These results cover the classical representer theorem and many of its variants as special cases and offer a wider scope of applications. The second part of the paper then develops a systematic approach for learning the drift function of a stochastic differential equation by integrating the results of the first part with Bayesian hierarchical framework. Importantly, our Baysian approach incorporates low-cost sparse learning through proper use of shrinkage priors while allowing proper quantification of uncertainty through posterior distributions. Several examples at the end illustrate the accuracy of our learning scheme.