Title: DEEP ACCURATE SOLVER FOR THE GEODESIC PROBLEM. Abstract: A high order accurate deep learning method for computing geodesic distances on surfaces is introduced. The proposed method exploits a dynamic programming principle which lends itself to a scheme with quasi-linear computational complexity. We consider two main components for computing distances on surfaces; A numerical solver that locally approximates the distance function and an efficient causal ordering scheme by which points are updated. The quality of the distance approximation is determined by the local solver and is the main focus of this paper. A common approach to compute distances on continuous surfaces is by considering a discretized polygonal mesh approximating the surface, and estimating distances on the polygon. With such an approximation, the exact geodesic distances restricted to the polygon are at most second order accurate with respect to the distances on the corresponding continuous surface. Here, by order of accuracy we refer to the rate of convergence as a function of the average distance between sampled points. To improve the rate of convergence, we consider a neural network based local solver which implicitly approximates the structure of the continuous surface. The proposed solver circumvents the polyhedral representation, by directly consuming sampled mesh vertices for approximation of distances on the sampled continuous surfaces. We provide numerical evidence that the proposed update scheme, with appropriate local numerical support, provides better accuracy compared to the best possible polyhedral approximations and previous learning based methods. We introduce a trained solver which is third order accurate, with quasi-linear complexity in the number of sampled points.