Haomin Zhou

Haoran Sun, Hanjun Dai, Bo Dai et al.

It is known that gradient-based MCMC samplers for continuous spaces, such as Langevin Monte Carlo (LMC), can be derived as particle versions of a gradient flow that minimizes KL divergence on a Wasserstein manifold. The superior efficiency of such samplers has motivated several recent attempts to generalize LMC to discrete spaces. However, a fully principled extension of Langevin dynamics to discrete spaces has yet to be achieved, due to the lack of well-defined gradients in the sample space. In this work, we show how the Wasserstein gradient flow can be generalized naturally to discrete spaces. Given the proposed formulation, we demonstrate how a discrete analogue of Langevin dynamics can subsequently be developed. With this new understanding, we reveal how recent gradient-based samplers in discrete spaces can be obtained as special cases by choosing particular discretizations. More importantly, the framework also allows for the derivation of novel algorithms, one of which, \textit{Discrete Langevin Monte Carlo} (DLMC), is obtained by a factorized estimate of the transition matrix. The DLMC method admits a convenient parallel implementation and time-uniform sampling that achieves larger jump distances. We demonstrate the advantages of DLMC on various binary and categorical distributions.

Nathan Gaby, Xiaojing Ye, Haomin Zhou

We develop a novel computational framework to approximate solution operators of evolution partial differential equations (PDEs). By employing a general nonlinear reduced-order model, such as a deep neural network, to approximate the solution of a given PDE, we realize that the evolution of the model parameter is a control problem in the parameter space. Based on this observation, we propose to approximate the solution operator of the PDE by learning the control vector field in the parameter space. From any initial value, this control field can steer the parameter to generate a trajectory such that the corresponding reduced-order model solves the PDE. This allows for substantially reduced computational cost to solve the evolution PDE with arbitrary initial conditions. We also develop comprehensive error analysis for the proposed method when solving a large class of semilinear parabolic PDEs. Numerical experiments on different high-dimensional evolution PDEs with various initial conditions demonstrate the promising results of the proposed method.

Qujiangxue Chen, Jianbo Cui, Luca Dieci et al.

In this paper, we study dynamical optimal transport on a connected graph from the perspective of the Benamou-Brenier formulation, where densities are assigned to vertices and velocities to edges. However, directly using Newton's method on the resulting nonlinear systems encounters two potential difficulties: (i) if the graph contains cycles, edge variables are not unique, and (ii) there is no guarantee that the density variables remain positive. To address these challenges, we introduce a finite-difference-type Newton method that eliminates cycle-induced redundancies through a spanning-tree gauge, resulting in a reduced set of independent variables and a well-posed, sparse linear system. For the lattice graph arising from the continuous optimal transport problem, density positivity can also be guaranteed by using an upwind discretization subject to a CFL-type condition. We further demonstrate the versatility of the proposed scheme by applying it to a range of problems, including optimal transport on lattices and random graphs, inverse optimal transport problems, and social network analysis.