Title: Total Variation by PINN
Primary area: applications to computer vision, audio, language, and other modalities
Keywords: total variation, physics-informed neural networks, unsupervised learning, image denoising, operator learning

Abstract: PINN approximation for solutions of non-linear equations is much more difficult than the linear analogue. This is due to the non-differentiability of the solution at some points and/or non-differentibility of the coefficients of the PDE. We develop a PINN approach for $\Gamma$-convergence inspired TV regularization \`a la Chambolle and Lions. We use smooth approximations of the TV functional that makes the problem differentiable at all points. Our coordinate-based neural network representation enables 
 gradient computation while maintaining the continuous PDE formulation. We train the network  through increasingly less smooth approximations, gradually approaching the original TV solution.
We next extend our results from a single image and a single PDE to a class of images. This is done via operator learning that maps any initial image to its TV solution % \cite{Lu2021_DeepONet}, 
where a single network learns the denoising operator across multiple images. Experiments on 2D and 3D data demonstrate that our method achieves competitive denoising quality with classical TV solvers.

TL;DR: We develop a PINN approach for $\Gamma$-convergence inspired TV regularization