As a counterpoint to classical stochastic particle methods for linear diffusion equations, such as Langevin dynamics for the Fokker-Planck equation, we develop a deterministic particle method for the weighted porous medium equation and prove its convergence on bounded time intervals. This generalizes related work on blob methods for unweighted porous medium equations. From a numerical analysis perspective, our method has several advantages: it is meshfree, preserves the gradient flow structure of the underlying PDE, converges in arbitrary dimension, and captures the correct asymptotic behavior in simulations.
The fact that our method succeeds in capturing the long time behavior of the weighted porous medium equation is significant from the perspective of related problems in quantization. Just as the Fokker-Planck equation provides a way to quantize a probability measure
ρ
¯
\bar {\rho }
by evolving an empirical measure
ρ
N
(
t
)
=
1
N
∑
i
=
1
N
δ
X
i
(
t
)
\rho ^N(t) = \frac {1}{N} \sum _{i=1}^N \delta _{X^i(t)}
according to stochastic Langevin dynamics so that
ρ
N
(
t
)
\rho ^N(t)
flows toward
ρ
¯
\bar {\rho }
, our particle method provides a way to quantize
ρ
¯
\bar {\rho }
according to deterministic particle dynamics approximating the weighted porous medium equation. In this way, our method has natural applications to multi-agent coverage algorithms and sampling probability measures.
A specific case of our method corresponds to confined mean-field dynamics of training a two-layer neural network for a radial basis activation function. From this perspective, our convergence result shows that, in the overparametrized regime and as the variance of the radial basis functions goes to zero, the continuum limit is given by the weighted porous medium equation. This generalizes previous results, which considered the case of a uniform data distribution, to the more general inhomogeneous setting. As a consequence of our convergence result, we identify conditions on the target function and data distribution for which convexity of the energy landscape emerges in the continuum limit.