  Fourier-based wavefront sensors, such as the Pyramid Wavefront Sensor (PWFS),
are the current preference for high contrast imaging due to their high
sensitivity. However, these wavefront sensors have intrinsic nonlinearities
that constrain the range where conventional linear reconstruction methods can
be used to accurately estimate the incoming wavefront aberrations. We propose
to use Convolutional Neural Networks (CNNs) for the nonlinear reconstruction of
the wavefront sensor measurements. It is demonstrated that a CNN can be used to
accurately reconstruct the nonlinearities in both simulations and a lab
implementation. We show that solely using a CNN for the reconstruction leads to
suboptimal closed loop performance under simulated atmospheric turbulence.
However, it is demonstrated that using a CNN to estimate the nonlinear error
term on top of a linear model results in an improved effective dynamic range of
a simulated adaptive optics system. The larger effective dynamic range results
in a higher Strehl ratio under conditions where the nonlinear error is
relevant. This will allow the current and future generation of large
astronomical telescopes to work in a wider range of atmospheric conditions and
therefore reduce costly downtime of such facilities.
