Accurate modeling and forecasting of physical processes necessitate more than knowing the appropriate model. Even with the exact evolution model, accurate prediction remains unfeasible without precise knowledge of the system`s state. For complex systems, it is often impossible to observe the entire system`s state. Instead, observations are limited to local measurements through a finite set of observers. This limitation motivates the art and science of data assimilation.
In this context, we have developed a novel nonlinear data assimilation algorithm that employs a feedback control penalty for solving partial differential equations. Our advancement on the existing AOT algorithm involves introducing a dynamic control process that enhances local feedback and achieves faster/predictable convergence rates. We have tested our algorithm on various one-dimensional and two-dimensional problems, demonstrating its efficacy in improving data assimilation for complex systems.