Sponsor
The work at Portland State University was partly funded by the National Institute of Health RO1AG021155, R01EY032284, R01AG027161, the National Science Foundation #2136228, and the Google Research Award "Kernel PDE 2025".
Published In
arXiv preprint
Document Type
Pre-Print
Publication Date
2025
Abstract
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 (AD) subjects.
Rights
Copyright (c) 2025 The Authors
Persistent Identifier
https://archives.pdx.edu/ds/psu/44297
Citation Details
Wells, M. L.; Lahouel, Kamel; and Jedynak, Bruno, "The Stochastic Occupation Kernel (SOCK) Method for Learning Stochastic Differential Equations" (2025). Mathematics and Statistics Faculty Publications and Presentations. 441.
https://archives.pdx.edu/ds/psu/44297
Description
This is the author’s version of a work that was accepted for publication. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published.