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

arXiv: https://arxiv.org/abs/2202.12927 · HTML: https://mlsanity.com/paper/2202.12927/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 86 references that are in the map.
- First posted on arXiv 25 Feb 2022.

## Builds on (most cited first, 18 of 18)

- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Stein Variational Gradient Descent as Gradient Flow](https://mlsanity.com/paper/1704.07520/) (2017)
- [Trainability and Accuracy of Neural Networks: An Interacting Particle System Approach](https://mlsanity.com/paper/1805.00915/) (2018)
- [SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergence](https://mlsanity.com/paper/2006.02509/) (2020)
- [A Non-Asymptotic Analysis for Stein Variational Gradient Descent](https://mlsanity.com/paper/2006.09797/) (2020)
- [Machine Learning from a Continuous Viewpoint](https://mlsanity.com/paper/1912.12777/) (2019)
- [On the Convergence of Gradient Descent Training for Two-layer ReLU-networks in the Mean Field Regime](https://mlsanity.com/paper/2005.13530/) (2020)
- [Scaling limit of the Stein variational gradient descent: the mean field regime](https://mlsanity.com/paper/1805.04035/) (2018)
- [Analysis of a Two-Layer Neural Network via Displacement Convexity](https://mlsanity.com/paper/1901.01375/) (2019)
- [An Introduction to Hamiltonian Monte Carlo Method for Sampling](https://mlsanity.com/paper/2108.12107/) (2021)
- [Velocity Field Generation for Density Control of Swarms using Heat Equation and Smoothing Kernels](https://mlsanity.com/paper/1710.00299/) (2017)
- [Coverage and Field Estimation on Bounded Domains by Diffusive Swarms](https://mlsanity.com/paper/1609.07695/) (2016)
- [A gradient flow perspective on the quantization problem](https://mlsanity.com/paper/1711.01652/) (2017)
- [Quantitative Assessment of Robotic Swarm Coverage](https://mlsanity.com/paper/1806.02488/) (2018)
- [Quantifying Robotic Swarm Coverage](https://mlsanity.com/paper/1905.07655/) (2019)
- [Asymptotic optimality of the triangular lattice for a class of optimal location problems](https://mlsanity.com/paper/2012.12129/) (2020)

## Built on by (most cited first, 6 of 6)

- [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022)
- [Theoretical Guarantees for Variational Inference with Fixed-Variance Mixture of Gaussians](https://mlsanity.com/paper/2406.04012/) (2024)
- [Interaction-Force Transport Gradient Flows](https://mlsanity.com/paper/2405.17075/) (2024)
- [Kernel Approximation of Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2410.20622/) (2024)
- [Inclusive KL Gradient Flows: Otto-Wasserstein, Fisher-Rao-Gaussian, and Local-Estimator Dynamics](https://mlsanity.com/paper/2411.00214/) (2024)
- [Variational Inference via Entropic Transport Descent](https://mlsanity.com/paper/2606.25265/) (2026)

## Abstract (arXiv)

As a counterpoint to classical stochastic particle methods for linear diffusion equations, we develop a deterministic particle method for the weighted porous medium equation (WPME) 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. That our method succeeds in capturing the long time behavior of WPME is significant from the perspective of related problems in quantization. Just as the Fokker-Planck equation provides a way to quantize a probability measure barρ by evolving an empirical measure according to stochastic Langevin dynamics so that the empirical measure flows toward barρ, our particle method provides a way to quantize barρ according to deterministic particle dynamics approximating WMPE. In this way, our method has natural applications to multi-agent coverage algorithms and sampling probability measures. A specific case of our method corresponds exactly to confined mean-field dynamics of training a two-layer neural network for a radial basis function 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 WPME. 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.