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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 Rd . 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. 

In this paper, we develop methods to find the most sparse perturbation to a given Markov chain (either discrete- or continuous-time) such that the perturbed Markov chain achieves a desired equilibrium. 

We propose and analyze a block coordinate descent proximal algorithm (BCD-prox) for simultaneous filtering and parameter estimation of ODE models. As we show on ODE systems with up to d=40 dimensions, as compared to state-of-the-art methods, BCD-prox exhibits increased robustness (to noise, parameter initialization, and hyperparameters), decreased training times, and improved accuracy of both filtered states and estimated parameters. We show how BCD-prox can be used with multistep numerical discretizations, and we establish convergence of BCD-prox under hypotheses that include real systems of interest.