Partial Differential Equations (PDEs) are fundamental to the mathematical modeling of a wide range of physical, biological, and engineering systems. These systems span diverse fields, such as fluid dynamics, heat transfer, structural mechanics, and population dynamics. Traditionally, numerical methods such as Finite Difference Method (FDM), Finite Volume Method (FVM), Finite Element Method (FEM) [Smith, 1985, Zienkiewicz et al., 2005], and spectral methods have been employed to obtain approximate solutions to PDEs. Although these methods are highly effective, they ultimately reduce to solving linear systems of equations. This often leads to several challenges, such as when solving high-dimensional problems or performing adaptive solutions for complex domains.

Integrating invariance into data representations is a principled design in intelligent systems and web applications. Representations play a fundamental role, where systems and applications are both built on meaningful representations of digital inputs (rather than the raw data). In fact, the proper design/learning of such representations relies on priors w.r.t. the task of interest. Here, the concept of symmetry from the Erlangen Program may be the most fruitful prior -- informally, a symmetry of a system is a transformation that leaves a certain property of the system invariant. Symmetry priors are ubiquitous, e.g., translation as a symmetry of the object classification, where object category is invariant under translation. The quest for invariance is as old as pattern recognition and data mining itself. Invariant design has been the cornerstone of various representations in the era before deep learning, such as the SIFT. As we enter the early era of deep learning, the invariance principle is largely ignored and replaced by a data-driven paradigm, such as the CNN. However, this neglect did not last long before they encountered bottlenecks regarding robustness, interpretability, efficiency, and so on. The invariance principle has returned in the era of rethinking deep learning, forming a new field known as Geometric Deep Learning (GDL). In this tutorial, we will give a historical perspective of the invariance in data representations. More importantly, we will identify those research dilemmas, promising works, future directions, and web applications.

Diffusion models have revolutionized image generation, and their extension to video generation has shown promise. However, current video diffusion models~(VDMs) rely on a scalar timestep variable applied at the clip level, which limits their ability to model complex temporal dependencies needed for various tasks like image-to-video generation. To address this limitation, we propose a frame-aware video diffusion model~(FVDM), which introduces a novel vectorized timestep variable~(VTV). Unlike conventional VDMs, our approach allows each frame to follow an independent noise schedule, enhancing the model's capacity to capture fine-grained temporal dependencies. FVDM's flexibility is demonstrated across multiple tasks, including standard video generation, image-to-video generation, video interpolation, and long video synthesis. Through a diverse set of VTV configurations, we achieve superior quality in generated videos, overcoming challenges such as catastrophic forgetting during fine-tuning and limited generalizability in zero-shot methods.Our empirical evaluations show that FVDM outperforms state-of-the-art methods in video generation quality, while also excelling in extended tasks. By addressing fundamental shortcomings in existing VDMs, FVDM sets a new paradigm in video synthesis, offering a robust framework with significant implications for generative modeling and multimedia applications.

In this paper, we propose a new super-expressive activation function called the Parametric Elementary Universal Activation Function (PEUAF). We demonstrate the effectiveness of PEUAF through systematic and comprehensive experiments on various industrial and image datasets, including CIFAR10, Tiny-ImageNet, and ImageNet. Moreover, we significantly generalize the family of super-expressive activation functions, whose existence has been demonstrated in several recent works by showing that any continuous function can be approximated to any desired accuracy by a fixed-size network with a specific super-expressive activation function. Specifically, our work addresses two major bottlenecks in impeding the development of super-expressive activation functions: the limited identification of super-expressive functions, which raises doubts about their broad applicability, and their often peculiar forms, which lead to skepticism regarding their scalability and practicality in real-world applications.

The main objective of the Multiple Kernel k-Means (MKKM) algorithm is to extract non-linear information and achieve optimal clustering by optimizing base kernel matrices. Current methods enhance information diversity and reduce redundancy by exploiting interdependencies among multiple kernels based on correlations or dissimilarities. Nevertheless, relying solely on a single metric, such as correlation or dissimilarity, to define kernel relationships introduces bias and incomplete characterization. Consequently, this limitation hinders efficient information extraction, ultimately compromising clustering performance. To tackle this challenge, we introduce a novel method that systematically integrates both kernel correlation and dissimilarity. Our approach comprehensively captures kernel relationships, facilitating more efficient classification information extraction and improving clustering performance. By emphasizing the coherence between kernel correlation and dissimilarity, our method offers a more objective and transparent strategy for extracting non-linear information and significantly improving clustering precision, supported by theoretical rationale. We assess the performance of our algorithm on 13 challenging benchmark datasets, demonstrating its superiority over contemporary state-of-the-art MKKM techniques.

Deep learning has led to a dramatic leap on Single Image Super-Resolution (SISR) performances in recent years. %Despite the substantial advancement% While most existing work assumes a simple and fixed degradation model (e.g., bicubic downsampling), the research of Blind SR seeks to improve model generalization ability with unknown degradation. Recently, Kong et al pioneer the investigation of a more suitable training strategy for Blind SR using Dropout. Although such method indeed brings substantial generalization improvements via mitigating overfitting, we argue that Dropout simultaneously introduces undesirable side-effect that compromises model's capacity to faithfully reconstruct fine details. We show both the theoretical and experimental analyses in our paper, and furthermore, we present another easy yet effective training strategy that enhances the generalization ability of the model by simply modulating its first and second-order features statistics. Experimental results have shown that our method could serve as a model-agnostic regularization and outperforms Dropout on seven benchmark datasets including both synthetic and real-world scenarios.

Maxwell-Amp\`{e}re-Nernst-Planck (MANP) equations were recently proposed to model the dynamics of charged particles. In this study, we enhance a numerical algorithm of this system with deep learning tools. The proposed hybrid algorithm provides an automated means to determine a proper approximation for the dummy variables, which can otherwise only be obtained through massive numerical tests. In addition, the original method is validated for 2-dimensional problems. However, when the spatial dimension is one, the original curl-free relaxation component is inapplicable, and the approximation formula for dummy variables, which works well in a 2-dimensional scenario, fails to provide a reasonable output in the 1-dimensional case. The proposed method can be readily generalised to cases with one spatial dimension. Experiments show numerical stability and good convergence to the steady-state solution obtained from Poisson-Boltzmann type equations in the 1-dimensional case. The experiments conducted in the 2-dimensional case indicate that the proposed method preserves the conservation properties.

Deep learning has achieved remarkable success in the field of bearing fault diagnosis. However, this success comes with larger models and more complex computations, which cannot be transferred into industrial fields requiring models to be of high speed, strong portability, and low power consumption. In this paper, we propose a lightweight and deployable model for bearing fault diagnosis, referred to as BearingPGA-Net, to address these challenges. Firstly, aided by a well-trained large model, we train BearingPGA-Net via decoupled knowledge distillation. Despite its small size, our model demonstrates excellent fault diagnosis performance compared to other lightweight state-of-the-art methods. Secondly, we design an FPGA acceleration scheme for BearingPGA-Net using Verilog. This scheme involves the customized quantization and designing programmable logic gates for each layer of BearingPGA-Net on the FPGA, with an emphasis on parallel computing and module reuse to enhance the computational speed. To the best of our knowledge, this is the first instance of deploying a CNN-based bearing fault diagnosis model on an FPGA. Experimental results reveal that our deployment scheme achieves over 200 times faster diagnosis speed compared to CPU, while achieving a lower-than-0.4\% performance drop in terms of F1, Recall, and Precision score on our independently-collected bearing dataset. Our code is available at \url{https://github.com/asdvfghg/BearingPGA-Net}.

A ReLU network is a piecewise linear function over polytopes. Figuring out the properties of such polytopes is of fundamental importance for the research and development of neural networks. So far, either theoretical or empirical studies on polytopes only stay at the level of counting their number, which is far from a complete characterization of polytopes. To upgrade the characterization to a new level, here we propose to study the shapes of polytopes via the number of simplices obtained by triangulating the polytope. Then, by computing and analyzing the histogram of simplices across polytopes, we find that a ReLU network has relatively simple polytopes under both initialization and gradient descent, although these polytopes theoretically can be rather diverse and complicated. This finding can be appreciated as a novel implicit bias. Next, we use nontrivial combinatorial derivation to theoretically explain why adding depth does not create a more complicated polytope by bounding the average number of faces of polytopes with a function of the dimensionality. Our results concretely reveal what kind of simple functions a network learns and its space partition property. Also, by characterizing the shape of polytopes, the number of simplices be a leverage for other problems, \textit{e.g.}, serving as a generic functional complexity measure to explain the power of popular shortcut networks such as ResNet and analyzing the impact of different regularization strategies on a network's space partition.

The networks for point cloud tasks are expected to be invariant when the point clouds are affinely transformed such as rotation and reflection. So far, relative to the rotational invariance that has been attracting major research attention in the past years, the reflection invariance is little addressed. Notwithstanding, reflection symmetry can find itself in very common and important scenarios, e.g., static reflection symmetry of structured streets, dynamic reflection symmetry from bidirectional motion of moving objects (such as pedestrians), and left- and right-hand traffic practices in different countries. To the best of our knowledge, unfortunately, no reflection-invariant network has been reported in point cloud analysis till now. To fill this gap, we propose a framework by using quadratic neurons and PCA canonical representation, referred to as Cloud-RAIN, to endow point \underline{Cloud} models with \underline{R}eflection\underline{A}l \underline{IN}variance. We prove a theorem to explain why Cloud-RAIN can enjoy reflection symmetry. Furthermore, extensive experiments also corroborate the reflection property of the proposed Cloud-RAIN and show that Cloud-RAIN is superior to data augmentation. Our code is available at https://github.com/YimingCuiCuiCui/Cloud-RAIN.