One-Shot Learning of Stochastic Differential Equations with Data Adapted Kernels

Abstract

We consider the problem of learning Stochastic Differential Equations of the form dX_t = f(X_t)dt+(X_t)dW_t 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, , and stochastic process dW_t representing the drift, the diffusion, and the stochastic forcing terms, respectively. We propose a method that combines Computational Graph Completion 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 X_t X_t+dt as a Computational Graph in which f, and dW_t 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.

Tasks

Reproductions