26
25

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

Abstract

We consider the problem of learning Stochastic Differential Equations of the form dXt=f(Xt)dt+σ(Xt)dWtdX_t = f(X_t)dt+\sigma(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 ff, σ\sigma, and stochastic process dWtdW_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 XtXt+dtX_t \rightarrow X_{t+dt} as a Computational Graph in which ff, σ\sigma and dWtdW_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.

View on arXiv
Comments on this paper