This WP will explore and develop the use of new PDE based algorithms for novel metric based training. Many applications of DL techniques, including imaging, rely on the comparisons between samples best described by probability distributions. Optimal Transport (OT) is a natural tool for this, e.g. using the (regularised) Wasserstein distance for the training loss. Currently, Sinkhorn type methods based on entropic regularisation are dominant. However, key questions remain in their use for large scale optimisation problems and mapping estimation. The principal challenges of this WP are to produce very fast and robust algorithms for training DL tasks, and then to apply these to challenging applications, to investigate the effect of using different cost functions and regularisations.