Hybrid twinning using PBDW and DeepONet for the effective state estimation and prediction on partially known systems

The accurate estimation of the state of complex uncertain physical systems requires reconciling theoretical models, with inherent imperfections, with noisy experimental data. In this work, we propose an effective hybrid approach that combines physics-based modeling with data-driven learning to enhance state estimation and further prediction. Our method builds upon the Parameterized Background Data-Weak (PBDW) framework, which naturally integrates a reduced-order representation of the best-available model with measurement data to account for both anticipated and unanticipated uncertainties. To address model discrepancies not captured by the reduced-order space, and learn the structure of model deviation, we incorporate a Deep Operator Network (DeepONet) constrained to be an orthogonal complement of the best-knowledge manifold. This ensures that the learned correction targets only the unknown components of model bias, preserving the interpretability and fidelity of the physical model. An optimal sensor placement strategy is also investigated to maximize information gained from measurements. We validate the proposed approach on a representative problem involving the Helmholtz equation under various sources of modeling error, including those arising from boundary conditions and source terms.