Coupling Variational Data Assimilation and Operator Learning For Effective State Estimation on Complex Systems

We consider the problem of optimal recovery of the state of a complex system from a biased parametric model and noisy measurements. We propose a bias-aware Hybrid-AI approach for this recovery, by combining the Parameterized Background Data-Weak (PBDW) approach and machine learning in terms of the deep neural operator (Deeponet) architecture. The interest of PBDW is to merge a best-knowledge model and measurements in a weak form and estimate the state and the model's bias as a combination of anticipated (knowledge) and unanticipated (ignorance) uncertainties. In this framework, the anticipated uncertainty belongs to a background space built from a reduced model of the best-knowledge manifold, while the unanticipated uncertainty is here modeled using Deeponet for enhanced learning of the data-based correction. By integrating it in the PBDW sate estimate, Deeponet lies inside the kernel of the anticipated physics and thus strictly accommodates the deficient physics by locally learning the model bias. An optimal sensor selection strategy is added to the framework to capture relevant local information in an effective manner. The performance of the proposed approach, and its potential for solving complex physical systems, are shown on a 2D Helmholtz equation with various model biases coming from the source, boundary conditions, or both