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Published February 2023 | public
Journal Article

One-shot learning of stochastic differential equations with data adapted kernels


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 [1] 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 → X_(t+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.

Additional Information

© 2022 Elsevier. MD, BH, HO acknowledge partial support by the Air Force Office of Scientific Research, USA under MURI award number FA9550-20-1-0358 (Machine Learning and Physics-Based Modeling and Simulation). MD, PT and HO acknowledge support from Beyond Limits (Learning Optimal Models) through CAST (The Caltech Center for Autonomous Systems and Technologies).

Additional details

August 22, 2023
October 25, 2023