Semi-supervised machine learning model for Lagrangian state estimation

In recent years, many studies have demonstrated the strength of supervised machine learning models for fluid state estimation. However, most of the studies assume that the sensors are fixed and that the high-resolution ground truth can be prepared. In practical situations, however, the sensors are not always fixed and may be floating. For example, in oceanography and river hydraulics, sensors are generally floating. Additionally, floating sensors make it more difficult to collect the high-resolution ground truth. We here propose a machine learning model for state estimation from such floating sensors without requiring high-resolution data for training. This model estimates velocity fields only from floating sensor measurements and is trained with a loss function using only sensor measurements and locations. We call this loss function as a "semi-supervised" loss function, since sensor measurements are used as the ground truth but high-resolution data of the entire velocity fields are not required. To demonstrate the model without high-resolution data for the training process, we consider two-dimensional decaying isotropic turbulence. Our results reveal that this model can estimate velocity fields with reasonable accuracy when the sensors are spatially dispersed to some extent in the domain. We also discuss the estimation accuracy dependence on machine learning methods, the number of sensors.