Learning Stochastic Dynamical Systems via Bridge Sampling

We develop algorithms to automate discovery of stochastic dynamical system models from noisy, vector-valued time series. By discovery, we mean learning both a nonlinear drift vector field and a diagonal diffusion matrix for an Itô stochastic differential equation in \(\mathbb {R}^d\). We parameterize the vector field using tensor products of Hermite polynomials, enabling the model to capture highly nonlinear and/or coupled dynamics. We solve the resulting estimation problem using expectation maximization (EM). This involves two steps. We augment the data via diffusion bridge sampling, with the goal of producing time series observed at a higher frequency than the original data. With this augmented data, the resulting expected log likelihood maximization problem reduces to a least squares problem. We provide an open-source implementation of this algorithm. Through experiments on systems with dimensions one through eight, we show that this EM approach enables accurate estimation for multiple time series with possibly irregular observation times. We study how the EM method performs as a function of the amount of data augmentation, as well as the volume and noisiness of the data.