Deep learning-enhanced reduced-order ensemble Kalman filter for efficient Bayesian data assimilation of parametric PDEs

Bayesian data assimilation for systems governed by parametric partial differential equations (PDEs) is computationally demanding due to the need for multiple forward model evaluations. Reduced-order models (ROMs) have been widely used to reduce the computational burden. However, traditional ROM techniques rely on linear mode superposition, which frequently fails to capture nonlinear time-dependent dynamics efficiently and leads to biases in the assimilation results. To address these limitations, we introduce a new deep learning-enhanced reduced-order ensemble Kalman filter (DR-EnKF) method for Bayesian data assimilation. The proposed approach employs a two-tiered learning framework. First, the full-order model is reduced using operator inference, which finds the primary dynamics of the system through long-term simulations generated from coarse-grid data. Second, a model error network is trained with short-term simulation data from a fine grid to learn about the ROM-induced discrepancy. The learned network is then used online to correct the ROM-based EnKF, resulting in more accurate state updates during the assimilation process. The performance of the proposed method is evaluated on several benchmark problems, including the Burgers' equation, the FitzHugh-Nagumo model, and advection-diffusion-reaction systems. The results show considerable computational speedup without compromising accuracy, making this approach an effective tool for large-scale data assimilation tasks.