There is a large body of recent work on the approximation of diffusion equations by deterministic interacting particle systems. The analysis of these systems is typically carried out in Eulerian coordinates, despite the fact that the particle viewpoint is inherently Lagrangian. This is largely due to the fact that in the continuous setting, it may be extremely hard to solve diffusion equations in Lagrangian coordinates. In fact, the existence of Lagrangian solutions to the Porous Media Equation (PME) with general initial data was open until 2022. In this talk, I will discuss how to construct Lagrangian solutions to PME. I will then sketch how this analysis can be used to obtain convergence rates for certain deterministic versions of the score matching algorithm (a method for generating new samples from an unknown distribution given some data).