  Euler's elastica constitute an appealing variational image inpainting model.
It minimises an energy that involves the total variation as well as the level
line curvature. These components are transparent and make it attractive for
shape completion tasks. However, its gradient flow is a singular, anisotropic,
and nonlinear PDE of fourth order, which is numerically challenging: It is
difficult to find efficient algorithms that offer sharp edges and good rotation
invariance. As a remedy, we design the first neural algorithm that simulates
inpainting with Euler's Elastica. We use the deep energy concept which employs
the variational energy as neural network loss. Furthermore, we pair it with a
deep image prior where the network architecture itself acts as a prior. This
yields better inpaintings by steering the optimisation trajectory closer to the
desired solution. Our results are qualitatively on par with state-of-the-art
algorithms on elastica-based shape completion. They combine good rotation
invariance with sharp edges. Moreover, we benefit from the high efficiency and
effortless parallelisation within a neural framework. Our neural elastica
approach only requires 3x3 central difference stencils. It is thus much simpler
than other well-performing algorithms for elastica inpainting. Last but not
least, it is unsupervised as it requires no ground truth training data.
