In this talk we introduce the semi-discrete optimal transport (OT) problem with entropic regularization. We characterize the solution using a governing, well-posed ordinary differential equation (ODE). This naturally yields an algorithm to solve the problem numerically, which turns out to be competitive for problems involving the squared Euclidean distance and exhibit superior performance when applied to various powers of the Euclidean distance. Finally, we note that the ODE approach yields an estimate on the rate of convergence of the solution as the regularization parameter vanishes, for a generic cost function.