Tao Tang

Dong Li, Zhonghua Qiao, Tao Tang

Recent results in the literature provide computational evidence that stabilized semi-implicit time-stepping method can efficiently simulate phase field problems involving fourth-order nonlinear dif- fusion, with typical examples like the Cahn-Hilliard equation and the thin film type equation. The up-to-date theoretical explanation of the numerical stability relies on the assumption that the deriva- tive of the nonlinear potential function satisfies a Lipschitz type condition, which in a rigorous sense, implies the boundedness of the numerical solution. In this work we remove the Lipschitz assumption on the nonlinearity and prove unconditional energy stability for the stabilized semi-implicit time-stepping methods. It is shown that the size of stabilization term depends on the initial energy and the perturba- tion parameter but is independent of the time step. The corresponding error analysis is also established under minimal nonlinearity and regularity assumptions.

Xiaodong Feng, Haojiong Shangguan, Tao Tang et al.

In this work, we present a hybrid numerical method for solving evolution partial differential equations (PDEs) by merging the time finite element method with deep neural networks. In contrast to the conventional deep learning-based formulation where the neural network is defined on a spatiotemporal domain, our methodology utilizes finite element basis functions in the time direction where the space-dependent coefficients are defined as the output of a neural network. We then apply the Galerkin or collocation projection in the time direction to obtain a system of PDEs for the space-dependent coefficients which is approximated in the framework of PINN. The advantages of such a hybrid formulation are twofold: statistical errors are avoided for the integral in the time direction, and the neural network's output can be regarded as a set of reduced spatial basis functions. To further alleviate the difficulties from high dimensionality and low regularity, we have developed an adaptive sampling strategy that refines the training set. More specifically, we use an explicit density model to approximate the distribution induced by the PDE residual and then augment the training set with new time-dependent random samples given by the learned density model. The effectiveness and efficiency of our proposed method have been demonstrated through a series of numerical experiments.

Jing Ren, Tao Tang, Hong Jia et al.

As data continues to grow in volume and complexity across domains such as finance, manufacturing, and healthcare, effective anomaly detection is essential for identifying irregular patterns that may signal critical issues. Recently, foundation models (FMs) have emerged as a powerful tool for advancing anomaly detection. They have demonstrated unprecedented capabilities in enhancing anomaly identification, generating detailed data descriptions, and providing visual explanations. This survey presents the first comprehensive review of recent advancements in FM-based anomaly detection. We propose a novel taxonomy that classifies FMs into three categories based on their roles in anomaly detection tasks, i.e., as encoders, detectors, or interpreters. We provide a systematic analysis of state-of-the-art methods and discuss key challenges in leveraging FMs for improved anomaly detection. We also outline future research directions in this rapidly evolving field.

Renqiang Luo, Tao Tang, Feng Xia et al.

Recent advancements in machine learning and deep learning have brought algorithmic fairness into sharp focus, illuminating concerns over discriminatory decision making that negatively impacts certain individuals or groups. These concerns have manifested in legal, ethical, and societal challenges, including the erosion of trust in intelligent systems. In response, this survey delves into the existing literature on algorithmic fairness, specifically highlighting its multifaceted social consequences. We introduce a novel taxonomy based on 'tolerance', a term we define as the degree to which variations in fairness outcomes are acceptable, providing a structured approach to understanding the subtleties of fairness within algorithmic decisions. Our systematic review covers diverse industries, revealing critical insights into the balance between algorithmic decision making and social equity. By synthesizing these insights, we outline a series of emerging challenges and propose strategic directions for future research and policy making, with the goal of advancing the field towards more equitable algorithmic systems.

Ling Guo, Hao Wu, Tao Zhou

We introduce in this work the normalizing field flows (NFF) for learning random fields from scattered measurements. More precisely, we construct a bijective transformation (a normalizing flow characterizing by neural networks) between a Gaussian random field with the Karhunen-Loève (KL) expansion structure and the target stochastic field, where the KL expansion coefficients and the invertible networks are trained by maximizing the sum of the log-likelihood on scattered measurements. This NFF model can be used to solve data-driven forward, inverse, and mixed forward/inverse stochastic partial differential equations in a unified framework. We demonstrate the capability of the proposed NFF model for learning Non Gaussian processes and different types of stochastic partial differential equations.