Bayesian optimization of empirical model with state-dependent stochastic forcing

A method for optimal data simulation using random evolution operator is proposed. We consider a discrete data-driven model of the evolution operator that is a superposition of deterministic function and stochastic forcing, both parameterized with artificial neural networks (particularly, three-layer perceptrons). An important property of the model is its data-adaptive state-dependent (i.e. inhomogeneous over phase space) stochastic part. The Bayesian framework is applied to model construction and explained in detail. Particularly, the Bayesian criterion of model optimality is adopted to determine both the model dimension and the number of parameters (neurons) in the deterministic as well as in the stochastic parts on the base of statistical analysis of the data sample under consideration. On an example of data generated by the stochastic Lorenz-63 system we investigate this criterion and show that it allows to find a stochastic model which adequately reproduces invariant measure and other statistical properties of the system. Also, we demonstrate that the state-dependent stochastic part is optimal only for large enough duration of the time series.