Yun Liu

Yun Liu, Chen Cui, Shi Shu et al.

The numerical solution of partial differential equations (PDEs) is fundamental to scientific and engineering computing. In the presence of strong anisotropy, material heterogeneity, and complex geometries, however, classical iterative solvers often suffer from reduced efficiency and require substantial problem-dependent tuning. The Fourier neural solver (FNS) is a learning-based hybrid iterative solver for such problems without extensive manual parameter tuning, but its original design is primarily effective for scalar PDEs on structured meshes and is difficult to extend directly to unstructured meshes or strongly coupled PDE systems. Building on the FNS framework, we introduce block smoothing operators and graph neural networks to construct a solver for unstructured systems, termed the graph Fourier neural solver (G-FNS). We further incorporate a coordinate transformation network to develop the adaptive graph Fourier neural solver (AG-FNS), and then extend this formulation to a frequency-domain multilevel variant, ML-AG-FNS. Rigorous analysis shows that, under suitable mathematical assumptions, the proposed method achieves mesh-independent convergence rate. Error-spectrum visualizations further indicate that AG-FNS can capture complex multiscale error modes. Extensive experiments on two-dimensional anisotropic diffusion and on two- and three-dimensional isotropic/anisotropic linear elasticity problems over unstructured meshes demonstrate strong robustness and efficiency. The proposed framework can be used either as a solver or as a preconditioner for Krylov subspace methods. Overall, it substantially extends the original FNS methodology and broadens the applicability of this class of neural solvers.

Guolei Sun, Yun Liu, Jingyun Liang et al.

Due to the fact that fully supervised semantic segmentation methods require sufficient fully-labeled data to work well and can not generalize to unseen classes, few-shot segmentation has attracted lots of research attention. Previous arts extract features from support and query images, which are processed jointly before making predictions on query images. The whole process is based on convolutional neural networks (CNN), leading to the problem that only local information is used. In this paper, we propose a TRansformer-based Few-shot Semantic segmentation method (TRFS). Specifically, our model consists of two modules: Global Enhancement Module (GEM) and Local Enhancement Module (LEM). GEM adopts transformer blocks to exploit global information, while LEM utilizes conventional convolutions to exploit local information, across query and support features. Both GEM and LEM are complementary, helping to learn better feature representations for segmenting query images. Extensive experiments on PASCAL-5i and COCO datasets show that our approach achieves new state-of-the-art performance, demonstrating its effectiveness.