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One of the oldest and most studied subject in scientific computing is algorithms for solving partial differential equations (PDEs). A long list of numerical methods have been proposed and successfully used for various applications. In recent years, deep learning methods have shown their superiority for high-dimensional PDEs where traditional methods fail. However, for low dimensional problems, it remains unclear whether these methods have a real advantage over traditional algorithms as a direct solver. In this work, we propose the random feature method (RFM) for solving PDEs, a natural bridge between traditional and machine learning-based algorithms. RFM is based on a combination of well-known ideas: 1. representation of the approximate solution using random feature functions; 2. collocation method to take care of the PDE; 3. the penalty method to treat the boundary conditions, which allows us to treat the boundary condition and the PDE in the same footing. We find it crucial to add several additional components including multi-scale representation and rescaling the weights in the loss function. We demonstrate that the method exhibits spectral accuracy and can compete with traditional solvers in terms of both accuracy and efficiency. In addition, we find that RFM is particularly suited for complex problems with complex geometry, where both traditional and machine learning-based algorithms encounter difficulties.

Multi-Instance Learning(MIL) aims to learn the mapping between a bag of instances and the bag-level label. Therefore, the relationships among instances are very important for learning the mapping. In this paper, we propose an MIL algorithm based on a graph built by structural relationship among instances within a bag. Then, Graph Convolutional Network(GCN) and the graph-attention mechanism are used to learn bag-embedding. In the task of medical image classification, our GCN-based MIL algorithm makes full use of the structural relationships among patches(instances) in an original image space domain, and experimental results verify that our method is more suitable for handling medical high-resolution images. We also verify experimentally that the proposed method achieves better results than previous methods on five bechmark MIL datasets and four medical image datasets., Comment: 22 pages,5 figures

In this paper, we present a method for automatically generating support structures for fabricating 3D objects with fusion deposition modeling (FDM) printers. First, we search for the parts of the 3D object model that cannot be printed normally (called unprintable parts). Then, we add some necessary support structures to all those parts. These support structures connect those unprintable parts to the object itself, or to the bottom plane. In addition, the support structure is corrected to increase stability while printing the object. For models of different sizes, we can adjust the density and width of the support structures in order to ensure stability and reliability in the printing process. Compared with existing methods, our method can reduce material cost and computation time.