  Neural networks allow solving many ill-posed inverse problems with
unprecedented performance. Physics informed approaches already progressively
replace carefully hand-crafted reconstruction algorithms in real applications.
However, these networks suffer from a major defect: when trained on a given
forward operator, they do not generalize well to a different one. The aim of
this paper is twofold. First, we show through various applications that
training the network with a family of forward operators allows solving the
adaptivity problem without compromising the reconstruction quality
significantly.Second, we illustrate that this training procedure allows
tackling challenging blind inverse problems.Our experiments include partial
Fourier sampling problems arising in magnetic resonance imaging (MRI) with
sensitivity estimation and off-resonance effects, computerized tomography (CT)
with a tilted geometry and image deblurring with Fresnel diffraction kernels.
