Smoothing and parameter estimation by soft-adherence to governing equations

Highlights: • Smoothing and parameter estimation of nonlinear and high dimensional systems. • Comparison of the proposed method to the Ensemble Rauch-Tung-Striebel smoother. • Enabling parameter estimation without use of EM algorithm. • Testing the method on several canonical test problems for data-assimilation. • Encouraging reproducible and open research by publishing software to GitHub. -- Abstract: The analysis of high-dimensional dynamical systems generally requires the integration of simulation data with experimental measurements. Experimental data often has substantial amounts of measurement noise that compromises the ability to produce accurate dimensionality reduction, parameter estimation, reduced order models, and/or balanced models for control. Data assimilation attempts to overcome the deleterious effects of noise by producing a set of algorithms for state estimation from noisy and possibly incomplete measurements. Indeed, methods such as Kalman filtering and smoothing are vital tools for scientists in fields ranging from electronics to weather forecasting. In this work we develop a novel framework for smoothing data based on known or partially known nonlinear governing equations. The method yields superior results to current techniques when applied to problems with known deterministic dynamics. By exploiting the numerical time-stepping constraints of the deterministic system, an optimization formulation can readily extract the noise from the nonlinear dynamics in a principled manner. The superior performance is due in part to the fact that it optimizes global state estimates. We demonstrate the efficiency and efficacy of the method on a number of canonical examples, thus demonstrating its viability for the wide range of potential applications stated above.