Particle filtering is used to compute nonlinear estimates of complex systems. It samples trajectories from a chosen distribution and computes the estimate as a weighted average of them. Easy-to-sample distributions often lead to degenerate samples where only one trajectory carries all the weight, negatively affecting the resulting performance of the estimate. While much research has been done on the design of appropriate sampling distributions that would lead to controlled degeneracy, in this paper our objective is to learn sampling distributions. Leveraging the framework of algorithm unrolling, we model the sampling distribution as a multivariate normal, and we use neural networks to learn both the mean and the covariance. We carry out unsupervised training of the model to minimize weight degeneracy, relying only on the observed measurements of the system. We show in simulations that the resulting particle filter yields good estimates in a wide range of scenarios.