  Deep operator networks (DeepONets) are trained to predict the linear
amplification of instability waves in high-speed boundary layers and to perform
data assimilation. In contrast to traditional networks that approximate
functions, DeepONets are designed to approximate operators. Using this
framework, we train a DeepONet to take as inputs an upstream disturbance and a
downstream location of interest, and to provide as output the perturbation
field downstream in the boundary layer. DeepONet thus approximates the
linearized and parabolized Navier-Stokes operator for this flow. Once trained,
the network can perform predictions of the downstream flow for a wide variety
of inflow conditions, without the need to calculate the whole trajectory of the
perturbations, and at a very small computational cost compared to
discretization of the original equations. In addition, we show that DeepONets
can solve the inverse problem, where downstream wall measurements are adopted
as input and a trained network can predict the upstream disturbances that led
to these observations. This capability, along with the forward predictions,
allows us to perform a full data assimilation cycle: starting from
wall-pressure data, we predict the upstream disturbance using the inverse
DeepONet and its evolution using the forward DeepONet.
