Systematic Physics Constrained Parameter Estimation of Stochastic Differential Equations

A systematic Bayesian framework is developed for physics constrained parameter inference ofstochastic differential equations (SDE) from partial observations. The physical constraints arederived for stochastic climate models but are applicable for many fluid systems. A condition isderived for global stability of stochastic climate models based on energy conservation. Stochasticclimate models are globally stable when a quadratic form, which is related to the cubic nonlinearoperator, is negative definite. A new algorithm for the efficient sampling of such negative definite matrices is developed and also for imputing unobserved data which improve the accuracy of theparameter estimates. The performance of this framework is evaluated on two conceptual climatemodels

Linear theory for filtering nonlinear multiscale systems with model error

We study filtering of multiscale dynamical systems with model error arising from unresolved smaller scale processes. The analysis assumes continuous-time noisy observations of all components of the slow variables alone. For a linear model with Gaussian noise, we prove existence of a unique choice of parameters in a linear reduced model for the slow variables. The linear theory extends to to a non-Gaussian, nonlinear test problem, where we assume we know the optimal stochastic parameterization and the correct observation model. We show that when the parameterization is inappropriate, parameters chosen for good filter performance may give poor equilibrium statistical estimates and vice versa. Given the correct parameterization, it is imperative to estimate the parameters simultaneously and to account for the nonlinear feedback of the stochastic parameters into the reduced filter estimates. In numerical experiments on the two-layer Lorenz-96 model, we find that parameters estimated online, as part of a filtering procedure, produce accurate filtering and equilibrium statistical prediction. In contrast, a linear regression based offline method, which fits the parameters to a given training data set independently from the filter, yields filter estimates which are worse than the observations or even divergent when the slow variables are not fully observed