We propose a particle method for numerically solving the Landau equation, inspired by the score-based transport modeling (SBTM) method for the Fokker-Planck equation. This method can preserve some important physical properties of the Landau equation, such as the conservation of mass, momentum, and energy, and decay of estimated entropy. We prove that matching the gradient of the logarithm of the approximate solution is enough to recover the true solution to the Landau equation with Maxwellian molecules. Several numerical experiments in low and moderately high dimensions are performed, with particular emphasis on comparing the proposed method with the traditional particle or blob method.

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 $\bar{\rho}$ by evolving an empirical measure according to stochastic Langevin dynamics so that the empirical measure flows toward $\bar{\rho}$, our particle method provides a way to quantize $\bar{\rho}$ 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.

There has been recent interest in developing scalable Bayesian sampling methods for big-data analysis, such as stochastic gradient MCMC (SG-MCMC) and Stein variational gradient descent (SVGD). A standard SG-MCMC algorithm simulates samples from a discrete-time Markov chain to approximate a target distribution, thus samples could be highly correlated, an undesired property for SG-MCMC. In contrary, SVGD directly optimizes a set of particles to approximate a target distribution, and thus is able to obtain good approximate with relatively much fewer samples. In this paper, we propose a principle particle-optimization framework based on Wasserstein gradient flows to unify SG-MCMC and SVGD, and to allow new algorithms to be developed. Our framework interprets SG-MCMC as particle optimization, revealing strong connections between SG-MCMC and SVGD. The key component of our framework is several particle-approximate techniques to efficiently solve the original partial differential equations on the space of probability measures. Extensive experiments on both synthetic data and deep neural networks demonstrate the effectiveness and efficiency of our framework for scalable Bayesian sampling.