  In this study, our goal is to integrate classical mathematical models with
deep neural networks by introducing two novel deep neural network models for
image segmentation known as Double-well Nets. Drawing inspirations from the
Potts model, our models leverage neural networks to represent a region force
functional. We extend the well-know MBO (Merriman-Bence-Osher) scheme to solve
the Potts model. The widely recognized Potts model is approximated using a
double-well potential and then solved by an operator-splitting method, which
turns out to be an extension of the well-known MBO scheme. Subsequently, we
replace the region force functional in the Potts model with a UNet-type
network, which is data-driven and is designed to capture multiscale features of
images, and also introduce control variables to enhance effectiveness. The
resulting algorithm is a neural network activated by a function that minimizes
the double-well potential. What sets our proposed Double-well Nets apart from
many existing deep learning methods for image segmentation is their strong
mathematical foundation. They are derived from the network approximation theory
and employ the MBO scheme to approximately solve the Potts model. By
incorporating mathematical principles, Double-well Nets bridge the MBO scheme
and neural networks, and offer an alternative perspective for designing
networks with mathematical backgrounds. Through comprehensive experiments, we
demonstrate the performance of Double-well Nets, showcasing their superior
accuracy and robustness compared to state-of-the-art neural networks. Overall,
our work represents a valuable contribution to the field of image segmentation
by combining the strengths of classical variational models and deep neural
networks. The Double-well Nets introduce an innovative approach that leverages
mathematical foundations to enhance segmentation performance.
