  In many applications, it is important to reconstruct a fluid flow field, or
some other high-dimensional state, from limited measurements and limited data.
In this work, we propose a shallow neural network-based learning methodology
for such fluid flow reconstruction. Our approach learns an end-to-end mapping
between the sensor measurements and the high-dimensional fluid flow field,
without any heavy preprocessing on the raw data. No prior knowledge is assumed
to be available, and the estimation method is purely data-driven. We
demonstrate the performance on three examples in fluid mechanics and
oceanography, showing that this modern data-driven approach outperforms
traditional modal approximation techniques which are commonly used for flow
reconstruction. Not only does the proposed method show superior performance
characteristics, it can also produce a comparable level of performance with
traditional methods in the area, using significantly fewer sensors. Thus, the
mathematical architecture is ideal for emerging global monitoring technologies
where measurement data are often limited.
