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

2025/05/16 by Michael Wells, Wells, Michael L., Kamel Lahouel +3
Computer Science · Mathematics · Physics and Astronomy · #46E22 #46N10 #60H10 #62J07 #65C20 #FOS: Computer and information sciences #Generative Adversarial Networks and Image Synthesis #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Model Reduction and Neural Networks #Statistical Methods and Inference

paper · pdf · doi:10.48550/arxiv.2505.11622

openalex publication_date 2025/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 subjects.

Citations

• ROCK: A variational formulation for occupation kernel methods in Reproducing Kernel Hilbert Spaces
• DynGMA: a robust approach for learning stochastic differential equations from data
• Stable Neural Stochastic Differential Equations in Analyzing Irregular Time Series Data
• MOCK: an Algorithm for Learning Nonparametric Differential Equations via Multivariate Occupation Kernel Functions
• Learning nonparametric ordinary differential equations from noisy data
• Learning PSD-valued functions using kernel sums-of-squares
• Neural SDEs as Infinite-Dimensional GANs
• A Kernel Two-Sample Test for Functional Data
• Non-parametric Models for Non-negative Functions
• Learning Stochastic Differential Equations With Gaussian Processes\n Without Gradient Matching
• Neural Ordinary Differential Equations
• CVXPY: A Python-Embedded Modeling Language for Convex Optimization
• Discovering governing equations from data by sparse identification of nonlinear dynamical systems
• Undergraduate News: Inheritance Tax
• The Pricing of Options and Corporate Liabilities

Related

• The Stochastic Occupation Kernel Method for System Identification
• Non-Parametric Learning of Stochastic Differential Equations with Non-asymptotic Fast Rates of Convergence
• DynGMA: a robust approach for learning stochastic differential equations from data
• Learning Stochastic Differential Equations With Gaussian Processes\n Without Gradient Matching
• Nonparametric learning of stochastic differential equations from sparse and noisy data