arXiv subjects

Xianyi Zeng

Publications and source records attributed to Xianyi Zeng.

15 recordsLinked to original sources

arXiv2025

On the Stability Barrier of Hermite Type Discretizations of Advection Equations

In this paper we establish a stability barrier of a class of high-order Hermite-type discretization of 1D advection equations underlying the hybrid-variable (HV) and active flux (AF) methods. These methods seek numerical approximations to both cell-averages and nodal solutions and evolves them in time simultaneously. It was shown in earlier work that the HV methods are supraconvergent, providing that the discretization uses more unknowns in the upwind direction than the downwind one, similar to the "upwind condition" of classical finite-difference schemes. Although it is well known that the stencil of finite-difference methods could not be too biased towards the upwind direction for stability consideration, known as "stability barrier", such a barrier has not been established for Hermite-type methods. In this work, we first show by numerical evidence that a similar barrier exists for HV methods and make a conjecture on the sharp bound on the stencil. Next, we prove the existence of stability barrier by showing that the semi-discretized HV methods are unstable given a stencil sufficiently biased towards the upwind direction. Tighter barriers are then proved using combinatorical tools, and finally we extend the analysis to studying other Hermite-type methods built on approximating nodal solutions and derivatives, such as those widely used in Hermite WENO methods.

math.NA↗

arXiv2024

Harnessing the Power of Neural Operators with Automatically Encoded Conservation Laws

Neural operators (NOs) have emerged as effective tools for modeling complex physical systems in scientific machine learning. In NOs, a central characteristic is to learn the governing physical laws directly from data. In contrast to other machine learning applications, partial knowledge is often known a priori about the physical system at hand whereby quantities such as mass, energy and momentum are exactly conserved. Currently, NOs have to learn these conservation laws from data and can only approximately satisfy them due to finite training data and random noise. In this work, we introduce conservation law-encoded neural operators (clawNOs), a suite of NOs that endow inference with automatic satisfaction of such conservation laws. ClawNOs are built with a divergence-free prediction of the solution field, with which the continuity equation is automatically guaranteed. As a consequence, clawNOs are compliant with the most fundamental and ubiquitous conservation laws essential for correct physical consistency. As demonstrations, we consider a wide variety of scientific applications ranging from constitutive modeling of material deformation, incompressible fluid dynamics, to atmospheric simulation. ClawNOs significantly outperform the state-of-the-art NOs in learning efficacy, especially in small-data regimes.

cs.LG↗

arXiv2024

Equivariant Hopf bifurcation arising in circular-distributed predator-prey interaction with taxis

In this paper, we study the Rosenzweig-MacArthur predator-prey model with predator-taxis and time delay defined on a disk. Theoretically, we studied the equivariant Hopf bifurcation around the positive constant steady-state solution. Standing and rotating waves have been investigated through the theory of isotropic subgroups and Lyapunov-Schmidt reduction. The existence conditions, the formula for the periodic direction and the periodic variation of bifurcation periodic solutions are obtained. Numerically, we select appropriate parameters and conduct numerical simulations to illustrate the theoretical results and reveal quite complicated dynamics on the disk. Different types of rotating and standing waves, as well as more complex spatiotemporal patterns with random initial values, are new dynamic phenomena that do not occur in one-dimensional intervals.

math.DS↗

arXiv2023

Spatiotemporal Patterns Induced by Turing-Hopf Interaction and Symmetry on a Disk

Turing bifurcation and Hopf bifurcation are two important kinds of transitions giving birth to inhomogeneous solutions, in spatial or temporal ways. On a disk, these two bifurcations may lead to equivariant Turing-Hopf bifurcations. In this paper, normal forms for three kinds of Turing-Hopf bifurcations are given and the breathing, standing wave-like, and rotating wave-like patterns are found in numerical examples.

math.DS↗

arXiv2023

An Explicit Fourth-Order Hybrid-Variable Method for Euler Equations with A Residual-Consistent Viscosity

In this paper we present a formally fourth-order accurate hybrid-variable method for the Euler equations in the context of method of lines. The hybrid-variable (HV) method seeks numerical approximations to both cell-averages and nodal solutions and evolves them in time simultaneously; and it is proved in previous work that these methods are inherent superconvergent. Taking advantage of the superconvergence, the method is built on a third-order discrete differential operator, which approximates the first spatial derivative at each grid point, only using the information in the two neighboring cells. Stability and accuracy analyses are conducted in the one-dimensional case for the linear advection equation; whereas extension to nonlinear systems including the Euler equations is achieved using characteristic decomposition and the incorporation of a residual-consistent viscosity to capture strong discontinuities. Extensive numerical tests are presented to assess the numerical performance of the method for both 1D and 2D problems.

math.NA↗

arXiv2023

Equivariant Hopf Bifurcation in a Class of Partial Functional Differential Equations on a Circular Domain

Circular domains frequently appear in the fields of ecology, biology and chemistry. In this paper, we investigate the equivariant Hopf bifurcation of partial functional differential equations with Neumann boundary condition on a two-dimensional disk. The properties of these bifurcations around equilibriums are analyzed rigorously by studying the equivariant normal forms. Two reaction-diffusion systems with discrete time delays are selected as numerical examples to verify the theoretical results, in which spatially inhomogeneous periodic solutions including standing waves and rotating waves, and spatially homogeneous periodic solutions are found near the bifurcation points.

math.DS↗

arXiv2022

Lightweight Transformer in Federated Setting for Human Activity Recognition

Human activity recognition (HAR) is a machine learning task with important applications in healthcare especially in the context of home care of patients and older adults. HAR is often based on data collected from smart sensors, particularly smart home IoT devices such as smartphones, wearables and other body sensors. Deep learning techniques like convolutional neural networks (CNNs) and recurrent neural networks (RNNs) have been used for HAR, both in centralized and federated settings. However, these techniques have certain limitations: RNNs cannot be easily parallelized, CNNs have the limitation of sequence length, and both are computationally expensive. Moreover, in home healthcare applications the centralized approach can raise serious privacy concerns since the sensors used by a HAR classifier collect a lot of highly personal and sensitive data about people in the home. In this paper, to address some of such challenges facing HAR, we propose a novel lightweight (one-patch) transformer, which can combine the advantages of RNNs and CNNs without their major limitations, and also TransFed, a more privacy-friendly, federated learning-based HAR classifier using our proposed lightweight transformer. We designed a testbed to construct a new HAR dataset from five recruited human participants, and used the new dataset to evaluate the performance of the proposed HAR classifier in both federated and centralized settings. Additionally, we use another public dataset to evaluate the performance of the proposed HAR classifier in centralized setting to compare it with existing HAR classifiers. The experimental results showed that our proposed new solution outperformed state-of-the-art HAR classifiers based on CNNs and RNNs, whiling being more computationally efficient.

cs.CV↗

arXiv2022

Embedded domain Reduced Basis Models for the shallow water hyperbolic equations with the Shifted Boundary Method

We consider fully discrete embedded finite element approximations for a shallow water hyperbolic problem and its reduced-order model. Our approach is based on a fixed background mesh and an embedded reduced basis. The Shifted Boundary Method for spatial discretization is combined with an explicit predictor/multi-corrector time integration to integrate in time the numerical solutions to the shallow water equations, both for the full and reduced-order model. In order to improve the approximation of the solution manifold also for geometries that are untested during the offline stage, the snapshots have been pre-processed by means of an interpolation procedure that precedes the reduced basis computation. The methodology is tested on geometrically parametrized shapes with varying size and position.

math.NA↗

arXiv2020

A Deep Reinforcement Learning Based Multi-Criteria Decision Support System for Textile Manufacturing Process Optimization

Textile manufacturing is a typical traditional industry involving high complexity in interconnected processes with limited capacity on the application of modern technologies. Decision-making in this domain generally takes multiple criteria into consideration, which usually arouses more complexity. To address this issue, the present paper proposes a decision support system that combines the intelligent data-based random forest (RF) models and a human knowledge based analytical hierarchical process (AHP) multi-criteria structure in accordance to the objective and the subjective factors of the textile manufacturing process. More importantly, the textile manufacturing process is described as the Markov decision process (MDP) paradigm, and a deep reinforcement learning scheme, the Deep Q-networks (DQN), is employed to optimize it. The effectiveness of this system has been validated in a case study of optimizing a textile ozonation process, showing that it can better master the challenging decision-making tasks in textile manufacturing processes.

cs.AI↗

arXiv2020

Multi-Objective Optimization of the Textile Manufacturing Process Using Deep-Q-Network Based Multi-Agent Reinforcement Learning

Multi-objective optimization of the textile manufacturing process is an increasing challenge because of the growing complexity involved in the development of the textile industry. The use of intelligent techniques has been often discussed in this domain, although a significant improvement from certain successful applications has been reported, the traditional methods failed to work with high-as well as human intervention. Upon which, this paper proposed a multi-agent reinforcement learning (MARL) framework to transform the optimization process into a stochastic game and introduced the deep Q-networks algorithm to train the multiple agents. A utilitarian selection mechanism was employed in the stochastic game, which (-greedy policy) in each state to avoid the interruption of multiple equilibria and achieve the correlated equilibrium optimal solutions of the optimizing process. The case study result reflects that the proposed MARL system is possible to achieve the optimal solutions for the textile ozonation process and it performs better than the traditional approaches.

cs.AI↗

arXiv2020

On the Stability of Explicit Finite Difference Methods for Advection-Diffusion Equations

In this paper we study the stability of explicit finite difference discretizations of linear advection-diffusion equations (ADE) with arbitrary order of accuracy in the context of method of lines. The analysis first focuses on the stability of the system of ordinary differential equations (ODE) that is obtained by discretizing the ADE in space and then extends to fully discretized methods where explicit Runge-Kutta methods are used for integrating the ODE system. In particular, it is proved that all stable semi-discretization of the ADE gives rise to a conditionally stable fully discretized method if the time-integrator is at least first-order accurate, whereas high-order spatial discretization of the advection equation cannot yield a stable method if the temporal order is too low. In the second half of this paper, we extend the analysis to a partially dissipative wave system and obtain the stability results for both semi-discretized and fully-discretized methods. Finally, the major theoretical predictions are verified numerically.

math.NA↗

arXiv2020

A reinforcement learning based decision support system in textile manufacturing process

This paper introduced a reinforcement learning based decision support system in textile manufacturing process. A solution optimization problem of color fading ozonation is discussed and set up as a Markov Decision Process (MDP) in terms of tuple {S, A, P, R}. Q-learning is used to train an agent in the interaction with the setup environment by accumulating the reward R. According to the application result, it is found that the proposed MDP model has well expressed the optimization problem of textile manufacturing process discussed in this paper, therefore the use of reinforcement learning to support decision making in this sector is conducted and proven that is applicable with promising prospects.

cs.LG↗

arXiv2018

On Finite Volume Discretization of Infiltration Dynamics in Tumor Growth Models

We address numerical challenges in solving hyperbolic free boundary problems described by spherically symmetric conservation laws that arise in the modeling of tumor growth due to immune cell infiltrations. In this work, we normalize the radial coordinate to transform the free boundary problem to a fixed boundary one, and utilize finite volume methods to discretize the resulting equations. We show that the conventional finite volume methods fail to preserve constant solutions and the incompressibility condition, and they typically lead to inaccurate, if not wrong, solutions even for very simple tests. These issues are addressed in a new finite volume framework with segregated flux computations that satisfy sufficient conditions for ensuring the so-called totality conservation law and the geometric conservation law. Classical first-order and second-order finite volume methods are enhanced in this framework. Their performance is assessed by various benchmark tests to show that the enhanced methods are able to preserve the incompressibility constraint and produce much more accurate results than the conventional ones.

math.NA↗

arXiv2016

A hybrid FD-FV method for first-order hyperbolic conservation laws on Cartesian grids: The smooth problem case

We present a class of hybrid FD-FV (finite difference and finite volume) methods for solving general hyperbolic conservation laws written in first-order form. The presentation focuses on one- and two-dimensional Cartesian grids; however, the generalization to higher dimensions is straightforward. These methods use both cell-averaged values and nodal values as dependent variables to discretize the governing partial differential equation (PDE) in space, and they are combined with method of lines for integration in time. This framework is absent of any Riemann solvers while it achieves numerical conservation naturally. This paper focuses on the accuracy and linear stability of the proposed FD-FV methods, thus we suppose in addition that the solutions are sufficiently smooth. In particular, we prove that the spatial-order of the FD-FV method is typically one-order higher than that of the discrete differential operator, which is involved in the construction of the method. In addition, the methods are linearly stable subjected to a Courant-Friedrich-Lewy condition when appropriate time-integrators are used. The numerical performance of the methods is assessed by a number of benchmark problems in one and two dimensions. These examples include the linear advection equation, nonlinear Euler equations, the solid dynamics problem for linear elastic orthotropic materials, and the Buckley-Leverett equation.

math.NA↗

arXiv2014

A general approach to enhance slope limiters on non-uniform rectilinear grids

Most slope limiter functions in high-resolution finite volume methods to solve hyperbolic conservation laws are designed assuming one-dimensional uniform grids, and they are also used to compute slope limiters in computations on non-uniform rectilinear grids. However, this strategy may lead to either loss of total variation diminishing (TVD) stability for 1D linear problems or the loss of formal second-order accuracy if the grid is highly non-uniform. This is especially true when the limiter function is not piecewise linear. Numerical evidences are provided to support this argument for two popular finite volume strategies: MUSCL in space and method of lines in time (MUSCL-MOL), and capacity-form differencing. In order to deal with this issue, this paper presents a general approach to study and enhance the slope limiter functions for highly non-uniform grids in the MUSCL-MOL framework. This approach extends the classical reconstruct-evolve-project procedure to general grids, and it gives sufficient conditions for a slope limiter function leading to a TVD stable, formal second-order accuracy in space, and symmetry preserving numerical scheme on arbitrary grids. Several widely used limiter functions, including the smooth ones by van Leer and van Albada, are enhanced to satisfy these conditions. These properties are confirmed by solving various one-dimensional and two-dimensional benchmark problems using the enhanced limiters on highly non-uniform rectilinear grids.

math.NA↗