Young Ju Lee

Young Ju Lee, Hengguang Li

We study the regularity and finite element approximation of the axisymmetric Stokes problem on a polygonal domain $Ω$. In particular, taking into account the singular coefficients in the equation and non-smoothness of the domain, we establish the well-posedness and full regularity of the solution in new weighted Sobolev spaces $\maK^m_{μ, 1}(Ω)$. Using our a priori results, we give a specific construction of graded meshes on which the Taylor-Hood mixed method approximates singular solutions at the optimal convergence rate. Numerical tests are presented to confirm the theoretical results in the paper.

Minh Phu Vuong, Young-Ju Lee, Iván Ojeda-Ruiz et al.

Due to the growing concern about unsavory behaviors of machine learning models toward certain demographic groups, the notion of 'fairness' has recently drawn much attention from the community, thereby motivating the study of fairness in graph clustering. Fair graph clustering aims to partition the set of nodes in a graph into $k$ disjoint clusters such that the proportion of each protected group within each cluster is consistent with the proportion of that group in the entire dataset. It is, however, computationally challenging to incorporate fairness constraints into existing graph clustering algorithms, particularly for large graphs. To address this problem, we propose FairAD, a computationally efficient fair graph clustering method. It first constructs a new affinity matrix based on the notion of algebraic distance such that fairness constraints are imposed. A graph coarsening process is then performed on this affinity matrix to find representative nodes that correspond to $k$ clusters. Finally, a constrained minimization problem is solved to obtain the solution of fair clustering. Experiment results on the modified stochastic block model and six public datasets show that FairAD can achieve fair clustering while being up to 40 times faster compared to state-of-the-art fair graph clustering algorithms.

Ruitong Shan, Young Ju Lee, Jiwei Jia

In this paper, we propose a numerical method for fractional Laplace equations that combines finite difference discretization with shallow neural network approximation. The fractional Laplace operator is discretized using a directional representation of Riemann--Liouville type, which leads to a finite difference approximation of the nonlocal operator. In two dimensions, the angular integral is approximated by a quadrature rule, and auxiliary points are introduced along each direction to facilitate the evaluation of the operator. Based on the resulting discrete system, the solution is then represented by a shallow neural network constructed through the orthogonal greedy algorithm (OGA).

Ye Lin, Young Ju Lee, Jiwei Jia

GreenLearning networks (GL) directly learn Green's function in physical space, making them an interpretable model for capturing unknown solution operators of partial differential equations (PDEs). For many PDEs, the corresponding Green's function exhibits asymptotic smoothness. In this paper, we propose a framework named Green Multigrid networks (GreenMGNet), an operator learning algorithm designed for a class of asymptotically smooth Green's functions. Compared with the pioneering GL, the new framework presents itself with better accuracy and efficiency, thereby achieving a significant improvement. GreenMGNet is composed of two technical novelties. First, Green's function is modeled as a piecewise function to take into account its singular behavior in some parts of the hyperplane. Such piecewise function is then approximated by a neural network with augmented output(AugNN) so that it can capture singularity accurately. Second, the asymptotic smoothness property of Green's function is used to leverage the Multi-Level Multi-Integration (MLMI) algorithm for both the training and inference stages. Several test cases of operator learning are presented to demonstrate the accuracy and effectiveness of the proposed method. On average, GreenMGNet achieves $3.8\%$ to $39.15\%$ accuracy improvement. To match the accuracy level of GL, GreenMGNet requires only about $10\%$ of the full grid data, resulting in a $55.9\%$ and $92.5\%$ reduction in training time and GPU memory cost for one-dimensional test problems, and a $37.7\%$ and $62.5\%$ reduction for two-dimensional test problems.

Ye Lin, Jiwei Jia, Young Ju Lee et al.

Greedy algorithms, particularly the orthogonal greedy algorithm (OGA), have proven effective in training shallow neural networks for fitting functions and solving partial differential equations (PDEs). In this paper, we extend the application of OGA to the tasks of linear operator learning, which is equivalent to learning the kernel function through integral transforms. Firstly, a novel greedy algorithm is developed for kernel estimation rate in a new semi-inner product, which can be utilized to approximate the Green's function of linear PDEs from data. Secondly, we introduce the OGA for point-wise kernel estimation to further improve the approximation rate, achieving orders of accuracy improvement across various tasks and baseline models. In addition, we provide a theoretical analysis on the kernel estimation problem and the optimal approximation rates for both algorithms, establishing their efficacy and potential for future applications in PDEs and operator learning tasks.