We consider the acoustic wave equation in very large and heterogeneous media. Its solution is notoriously costly, and the popularity of finite differences is well established, as it enables solving quickly and with good accuracy providing the medium does not include strong topography variations. However, when many wavelengths are propagated, the solution can be polluted by numerical dispersion.This problem affects both finite differences and finite elements methods, with the latter being practical to track topography and complex geometries. One way to reduce pollution is to increase the number of points per wavelength. This generates very high computational costs. The aim of this work is to establish whether it is possible to use learning methods to reduce pollution while keeping low computational costs. FD schemes with optimized coefficients have been proposed in the literature (He, Z., Zhang, J. and Yao, Z., 2019) to minimize numerical dispersion and we compare their performance on different test cases, with the perspective of reducing as much as possible the number of points per wavelength. To go further, we proposed a new method, based on Machine Learning while keeping a small stencil for approximating the differential operators. It amounts to computing on a coarse grid with a small number of points per wavelength, leading to fast simulations but with numerical dispersion, which we correct with Machine learning. This correction is applied during the simulation at selected time steps. This approach is however unstable for long-time simulations, generating unphysical artifacts. We consider another approach, which consists in modifying globally finite element approximations by enriching the standard polynomial approximation space with a prior approximation of the solution obtained with Physics-Informed-Neural-Networks.