Transformers have demonstrated effectiveness in solving partial differential equations (PDEs). However, extending them to solve PDEs on complex geometries remains a challenge. In this work, we propose SpiderSolver, a geometry-aware transformer that introduces spiderweb tokenization for handling complex domain geometry and irregularly discretized points. Our method partitions the irregular spatial domain into spiderweb-like patches, guided by the domain boundary geometry. SpiderSolver leverages a coarse-grained attention mechanism to capture global interactions across spiderweb tokens and a fine-grained attention mechanism to refine feature interactions between the domain boundary and its neighboring interior points. We evaluate SpiderSolver on PDEs with diverse domain geometries across seven datasets, including cars, airfoils, blood flow in the human thoracic aorta, as well as canonical cases governed by the Navier-Stokes, Darcy flow, elasticity, and plasticity equations. Experimental results demonstrate that SpiderSolver consistently achieves state-of-the-art performance across different datasets and metrics, with better generalization ability in the OOD setting. The code is available at https://github.com/Kai-Qi/SpiderSolver.

Periventricular white matter injury (PWMI) is the most frequent magnetic resonance imaging (MRI) finding in infants with Cerebral Palsy (CP). We aim to detect CP and identify subtle, sparse PWMI lesions in infants under two years of age with immature brain structures. Based on the characteristic that the responsible lesions are located within five target regions, we first construct a multi-modal dataset including 243 cases with the mask annotations of five target regions for delineating anatomical structures on T1-Weighted Imaging (T1WI) images, masks for lesions on T2-Weighted Imaging (T2WI) images, and categories (CP or Non-CP). Furthermore, we develop a bidirectional projection-based multi-modal fusion transformer (BiP-MFT), incorporating a Bidirectional Projection Fusion Module (BPFM) for integrating the features between five target regions on T1WI images and lesions on T2WI images. Our BiP-MFT achieves subject-level classification accuracy of 0.90, specificity of 0.87, and sensitivity of 0.94. It surpasses the best results of nine comparative methods, with 0.10, 0.08, and 0.09 improvements in classification accuracy, specificity and sensitivity respectively. Our BPFM outperforms eight compared feature fusion strategies using Transformer and U-Net backbones on our dataset. Ablation studies on the dataset annotations and model components justify the effectiveness of our annotation method and the model rationality. The proposed dataset and codes are available at https://github.com/Kai-Qi/BiP-MFT.

Recently, there are lots of literature on improving the robustness of SVM by constructing nonconvex functions, but they seldom theoretically study the robust property of the constructed functions. In this paper, based on our recent work, we present a novel capped asymmetric elastic net (CaEN) loss and equip it with the SVM as CaENSVM. We derive the influence function of the estimators of the CaENSVM to theoretically explain the robustness of the proposed method. Our results can be easily extended to other similar nonconvex loss functions. We further show that the influence function of the CaENSVM is bounded, so that the robustness of the CaENSVM can be theoretically explained. Other theoretical analysis demonstrates that the CaENSVM satisfies the Bayes rule and the corresponding generalization error bound based on Rademacher complexity guarantees its good generalization capability. Since CaEN loss is concave, we implement an efficient DC procedure based on the stochastic gradient descent algorithm (Pegasos) to solve the optimization problem. A host of experiments are conducted to verify the effectiveness of our proposed CaENSVM model.

Twin support vector machine (TWSVM), which constructs two nonparallel classifying hyperplanes, is widely applied to various fields. However, TWSVM solves two quadratic programming problems (QPPs) separately such that the final classifiers lack consistency and enough prediction accuracy. Moreover, by reason of only considering the 1-norm penalty for slack variables, TWSVM is not well defined in the geometrical view. In this article, we propose a novel elastic net nonparallel hyperplane support vector machine (ENNHSVM), which adopts elastic net penalty for slack variables and constructs two nonparallel separating hyperplanes simultaneously. We further discuss the properties of ENNHSVM theoretically and derive the violation tolerance upper bound to better demonstrate the relative violations of training samples in the same class. In particular, we design a safe screening rule for ENNHSVM to speed up the calculations. We finally compare the performance of ENNHSVM on both synthetic datasets and benchmark datasets with the Lagrangian SVM, the twin parametric-margin SVM, the elastic net SVM, the TWSVM, and the nonparallel hyperplane SVM.