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As a counterpoint to classical stochastic particle methods for lineardiffusion equations, we develop a deterministic particle method for theweighted porous medium equation (WPME) and prove its convergence on boundedtime intervals. This generalizes related work on blob methods for unweightedporous medium equations. From a numerical analysis perspective, our method hasseveral advantages: it is meshfree, preserves the gradient flow structure ofthe underlying PDE, converges in arbitrary dimension, and captures the correctasymptotic behavior in simulations.  That our method succeeds in capturing the long time behavior of WPME issignificant from the perspective of related problems in quantization. Just asthe Fokker-Planck equation provides a way to quantize a probability measure$\bar{\rho}$ by evolving an empirical measure according to stochastic Langevindynamics so that the empirical measure flows toward $\bar{\rho}$, our particlemethod provides a way to quantize $\bar{\rho}$ according to deterministicparticle dynamics approximating WMPE. In this way, our method has naturalapplications to multi-agent coverage algorithms and sampling probabilitymeasures.  A specific case of our method corresponds exactly to confined mean-fielddynamics of training a two-layer neural network for a radial basis functionactivation function. From this perspective, our convergence result shows that,in the overparametrized regime and as the variance of the radial basisfunctions goes to zero, the continuum limit is given by WPME. This generalizesprevious results, which considered the case of a uniform data distribution, tothe more general inhomogeneous setting. As a consequence of our convergenceresult, we identify conditions on the target function and data distribution forwhich convexity of the energy landscape emerges in the continuum limit.

A blob method for inhomogeneous diffusion with applications to multi-agent control and sampling