Two-Step Bayesian Estimation of Sparse Dynamical Systems Using Data-Driven Closure Models

Music City Center 

Many problems in science and engineering involve observing sample trajectories from processes with partially known dynamics. Often the systems
are driven by an ODE with nonlinear dependence between elements of the state vector, and require solutions require expert knowledge to build mathematical
models. Recently, a growing literature has explored solving such systems using machine learning, with results suggesting data-driven modeling can
facilitate faster discovery. However, sparsity, missing data, and observation error limit existing methods.

We propose to address these challenges in two stages: First, we address sparsity and observation errors using data assimilation (DA) to in-fill trajectories of observed data. Second, we estimate dynamics using an MCMC-based model based on spline estimation and a sparse prior for the dynamics of the ODE. We also strengthen the sparse prior of the MCMC model with additional knowledge of the closure terms to improve model estimation. Considering the DA trajectories as additional priors, we average over DA ensembles to improve forecasting.

Spline estimation

Gaussian processes

Koopman operators

dynamical systems

missing data

uncertainty quantification 

Section on Physical and Engineering Sciences