Title: Shape-Tailored Deep Neural Networks Using PDEs for Segmentation. Abstract: We present Shape-Tailored Deep Neural Networks (ST-DNN). ST-DNN extend convolutional networks, which aggregate data from fixed shape (square) neighbor-hoods to compute descriptors, to be defined on arbitrarily shaped regions. This is useful for segmentation applications, where it is desired to have descriptors that aggregate data only within regions of segmentation to avoid mixing data from different regions, otherwise, the descriptors are difficult to group to a unique region.  We formulate these descriptors through partial differential equations (PDE) that naturally generalize convolution to arbitrary regions, and derive the methodology to jointly estimate the segmentation and ST-DNN descriptor. We also show that ST-DNN inherit covariance to translations and rotations from the PDE, a natural property of a segmentation method, which existing CNN based methods lack. ST-DNN are 3-4 order of magnitude smaller than typical CNN. We empirically show that they exceed segmentation performance compared to state-of-the-art CNN-based descriptors using 2-3 orders smaller training sets on the texture segmentation problem.