In a variety of scientific and engineering applications, a fundamental computational task is to determine whether two objects (e.g. pebbles or sand grains) are in contact and, if so, to determine the point of contact and compute the contact forces (normal and shear). This task is clearly trivial when both objects are spheres (in 3D) or circles (in the plane). However, even for ellipses in the plane, it is a surprisingly mathematically rich and computationally delicate problem. Work by my supervisor Serge Prudhomme and his students and collaborators has resulted in an elegant mathematical analysis of the problem and a MATLAB implementation of a root-finding algorithm to produce datasets of large numbers of pairs of ellipses, in the hope of training a neural network. Neural networks, however, have not been particularly successful so far in fitting the data. We believe that this is because of certain inherent singularities in the geometry of the problem. In this talk, I will present some results from ongoing work in which I develop an approximate solution using an asymptotic analysis of the scaling around the problems singularities and then attempt to use equivariant machine learning techniques to estimate a correction term