One-Shot Learning of Stochastic Differential Equations with Computational Graph Completion
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We consider the problem of learning Stochastic Differential Equations of the form dX_t = f(X_t)dt+σ(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, σ, and stochastic process dW_t representing the drift, the diffusion, and the stochastic forcing terms, respectively. We propose a simple kernel-based solution to this problem that can be decomposed as follows: (1) Represent the time-increment map X_t → X_t+dt as a Computational Graph in which f, σ 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.
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