A real-time variational data assimilation method with model bias identification and correction

Real-time monitoring on a physical system by means of a model-based digital twin may be difficult if occurring phenomena are multiphysics and multiscale. A main difficulty comes from the numerical complexity which is associated to an expensive computation hardly compatible with real-time. To overcome this issue, the high-fidelity parameterized physical model may be simplified which adds a model bias. Moreover, the parameter values can be inaccurate, and one part of the physics may be unknown. All those errors affect the effectiveness of numerical diagnostics and predictions and need to be corrected with assimilation techniques on observation data. Therefore, the monitoring of a process occurs in two stages: (1) the state estimation at the acquisition time which may be associated with an identification of the set of unknown parameters of the parametrized model and an update state which enriches the model; (2) a state prediction for future time steps with the updated model. The present study is mainly denoted to perform the state estimation using an extension, for time-dependent problems, of the Parameterized Background Data-Weak (PBDW) method introduced in [1]. This method is a non-intrusive, reduced basis and in-situ data assimilation framework for physical systems modeled by parametrized Partial Differential quations initially designed for steady-state problems. The key idea of the formulation is to seek an approximation to the true field employing projection-by-data, with a first contribution from a background estimate computed from a reduced-order method (ROM) enhanced on-the-fly, and a second contribution from an update state informed by the experimental observations (correction of model bias). Further research works [2,3] developed an extension to deal with noisy data and a nonlinear framework. Moreover, a priori error analysis was conducted by providing a bound on the state error and identifying individual contributions. In the present work, the state prediction for future time steps is also performed from an evaluation of the updated model and an extrapolation of the time function from the tensor-based decomposition (SVD) on prior updates. Numerical experiments are conducted on a thermal conduction problem in the context of heating on a Printed Circuit Board (PCB) with different cases of model bias: a bias on heat source, a biased boundary condition and an error on the constitutive equation. These numerical experiments show that the method significantly reduces the online computational time while providing relevant state evaluations and predictions. We thus illustrate the considerable improvement in prediction provided by the hybrid integration of a best-knowledge model and experimental observations