  In this work, we present and study Continuous Generative Neural Networks
(CGNNs), namely, generative models in the continuous setting: the output of a
CGNN belongs to an infinite-dimensional function space. The architecture is
inspired by DCGAN, with one fully connected layer, several convolutional layers
and nonlinear activation functions. In the continuous $L^2$ setting, the
dimensions of the spaces of each layer are replaced by the scales of a
multiresolution analysis of a compactly supported wavelet. We present
conditions on the convolutional filters and on the nonlinearity that guarantee
that a CGNN is injective. This theory finds applications to inverse problems,
and allows for deriving Lipschitz stability estimates for (possibly nonlinear)
infinite-dimensional inverse problems with unknowns belonging to the manifold
generated by a CGNN. Several numerical simulations, including signal
deblurring, illustrate and validate this approach.
