Physics-Aware Tuning of Unscented Kalman Filter:Statistical Framework for Solving Inverse Problems Involving Nonlinear Dynamical Systems and Missing Data

Abstract The performance of the Unscented Kalman Filter (UKF) strongly depends on the proper choice of measurement and process uncertainty matrices, as well as on the scaling parameter of the unscented transform. To avoid cumbersome trial-and-error manual settings in finding the optimal hyperparameters, we introduce a hands-off meta-optimization framework, which incorporates a nonlinear mesh adaptive direct search optimization algorithm in an offline outer loop, paired with a physics-aware loss function. The novelty of this approach is twofold. First, a physics-aware loss function is used to optimize the UKF hyperparameters. It minimizes the physical discrepancy induced by the data-driven correction of the prior states during filtering. Notably, sensor data are not directly incorporated into the calculation of the loss function, which expedites the tuning of the filter in weakly informative data scenarios, especially when the underlying physics is well understood. Second, UKF relaxation is embedded in the optimization to make the measurement and process noise covariance matrices adaptive, which greatly reduces the dimension of the optimization space while remaining very general with respect to the structure of the matrices. We demonstrate the effectiveness of the proposed framework, as the cornerstone of a future digital twin technology combining data- and physics-based models, through various classical problems in solid mechanics, rotating machinery, civil engineering, and fluid-solid interaction.