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

• URL: http://arxiv.org/abs/2505.11622v1
• Date: Fri, 16 May 2025 18:38:50 GMT
• Title: The Stochastic Occupation Kernel (SOCK) Method for Learning Stochastic Differential Equations
• Authors: Michael L. Wells, Kamel Lahouel, Bruno Jedynak,
• Abstract summary: We present a novel kernel-based method for learning multivariate differential equations (SDEs)<n>We first estimate the drift term function, then the (matrix-valued) diffusion function given the drift.<n>We propose a simple learning procedure that retains strong predictive accuracy while using Fenchel duality to promote efficiency.
• Score: 0.786519149320184
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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