  In recent years there has been great interest in using deep neural networks
(DNN) for super-resolution image reconstruction including for structured
illumination microscopy (SIM). While these methods have shown very promising
results, they all rely on data-driven, supervised training strategies that need
a large number of ground truth images, which is experimentally difficult to
realize. For SIM imaging, there exists a need for a flexible, general, and
open-source reconstruction method that can be readily adapted to different
forms of structured illumination. We demonstrate that we can combine a deep
neural network with the forward model of the structured illumination process to
reconstruct sub-diffraction images without training data. The resulting
physics-informed neural network (PINN) can be optimized on a single set of
diffraction limited sub-images and thus doesn't require any training set. We
show with simulated and experimental data that this PINN can be applied to a
wide variety of SIM methods by simply changing the known illumination patterns
used in the loss function and can achieve resolution improvements that match
well with theoretical expectations.
