  We introduce a new class of iterative image reconstruction algorithms for
radio interferometry, at the interface of convex optimization and deep
learning, inspired by plug-and-play methods. The approach consists in learning
a prior image model by training a deep neural network (DNN) as a denoiser, and
substituting it for the handcrafted proximal regularization operator of an
optimization algorithm. The proposed AIRI (``AI for Regularization in
radio-interferometric Imaging'') framework, for imaging complex intensity
structure with diffuse and faint emission from visibility data, inherits the
robustness and interpretability of optimization, and the learning power and
speed of networks. Our approach relies on three steps. Firstly, we design a low
dynamic range training database from optical intensity images. Secondly, we
train a DNN denoiser at a noise level inferred from the signal-to-noise ratio
of the data. We use training losses enhanced with a nonexpansiveness term
ensuring algorithm convergence, and including on-the-fly database dynamic range
enhancement via exponentiation. Thirdly, we plug the learned denoiser into the
forward-backward optimization algorithm, resulting in a simple iterative
structure alternating a denoising step with a gradient-descent data-fidelity
step. We have validated AIRI against CLEAN, optimization algorithms of the SARA
family, and a DNN trained to reconstruct the image directly from visibility
data. Simulation results show that AIRI is competitive in imaging quality with
SARA and its unconstrained forward-backward-based version uSARA, while
providing significant acceleration. CLEAN remains faster but offers lower
quality. The end-to-end DNN offers further acceleration, but with far lower
quality than AIRI.
