arXiv subjects

Jiang Yang

Publications and source records attributed to Jiang Yang.

At least 37 records · Page 2Linked to original sources

arXiv2026

Sinc Kolmogorov-Arnold network and its application for solving PDEs with singularities

In this paper, we propose to use Sinc interpolation in the context of Kolmogorov-Arnold Networks, neural networks with learnable activation functions, which recently gained attention as alternatives to Multilayer Perceptron. Many different function representations have already been tried, but we show that Sinc interpolation proposes a viable alternative, since it is known in numerical analysis to effectively represent both smooth functions and functions with singularities. This is important not only for function approximation but also for solving the partial differential equations with physics-informed neural networks. Through a series of experiments, we show that SincKANs provide better results in almost all of the examples we have considered.

cs.LG↗

arXiv2026

A second-order product-type implicit-explicit Runge-Kutta method preserving unit length and energy dissipation structures for gradient flows of vector fields

Gradient flows of unit vector fields arise in a wide range of physical models such as harmonic map heat flows, nematic liquid crystals, and magnetization dynamics. Designing numerical schemes that simultaneously preserve the unit length constraint and dissipate energy is essential for reliable simulations of such systems. Although projection methods can effectively enforce the unit length constraint, ensuring energy dissipation under projection, especially in high-order schemes, remains challenging. Unlike traditional implicit-explicit Runge-Kutta (IMEX-RK) methods, in this work we propose a general methodology for constructing product-type IMEX-RK schemes that offers greater adaptability to various models with the goal of designing structure-preserving numerical schemes. For gradient flows of unit vector fields with Dirichlet energy, we design a linear and second-order numerical scheme that simultaneously preserves energy dissipation and the unit length constraint by using product-type IMEX-RK methods and projection techniques. Numerical experiments verify the accuracy, stability, and structure-preserving properties of the scheme. According to our best knowledge, this is the first second-order linear scheme that can preserve both the unit length and the original Dirichlet energy for harmonic map heat flows.

math.NA↗

arXiv2026

Energy Dissipation Preserving Feature-based DNN Galerkin Methods for Gradient Flows

In recent years, deep learning methods, exemplified by Physics-Informed Neural Networks (PINNs), have been widely applied to the numerical solution of differential equations. However, these methods may suffer from limited accuracy, high training costs, and lack of robustness, particularly their inability to preserve the intrinsic physical structures of continuous PDE models, such as the energy dissipation property in gradient flow systems. To address these challenges, we propose a feature-based Deep Neural Network Galerkin (DNN-G) framework designed for structure-preserving simulations of gradient flows. Instead of treating neural networks merely as optimization-driven solvers, we employ them as adaptive feature generators that define nonlinear trial spaces within a Galerkin projection formulation.This formulation guarantees semi-discrete energy dissipation and can be naturally combined with energy stable time integration schemes. Several strategies for constructing neural basis functions are investigated, including random features, structured initialization, and problem-informed pre-training. Numerical experiments demonstrate that the proposed method preserves robust energy stability in high-dimensional settings and accurately captures complex topological transitions. With equivalent degrees of freedom, the DNN-G framework achieves higher accuracy than classical spectral methods, highlighting the effectiveness of neural feature representations for the numerical solution of partial differential equations.

math.NA↗

arXiv2026

Stability and convergence of multi-product expansion splitting methods with negative weights for semilinear parabolic equations

The operator splitting method has been widely used to solve differential equations by splitting the equation into more manageable parts. In this work, we resolves a long-standing problem -- how to establish the stability of multi-product expansion (MPE) splitting methods with negative weights. The difficulty occurs because negative weights in high-order MPE method cause the sum of the absolute values of weights larger than one, making standard stability proofs fail. In particular, we take the semilinear parabolic equation as a typical model and establish the stability of arbitrarily high-order MPE splitting methods with positive time steps but possibly negative weights. Rigorous convergence analysis is subsequently obtained from the stability result. Several numerical experiments validate the stability and accuracy of various high-order MPE splitting methods, highlighting their efficiency and robustness.

math.NA↗

arXiv2026

Stability and convergence analysis of unconditionally original energy dissipative implicit-explicit Runge--Kutta methods for the phase field crystal models without Lipschitz assumptions

The phase field crystal (PFC) method is an efficient technique for simulating the evolution of crystalline microstructures at atomistic length scales and diffusive time scales. Due to the high-order derivatives (sixth-order) and the strongly nonlinear term (locally Lipschitz), developing high-order stable schemes and establishing corresponding error estimates is particularly challenging. In this study, we first establish a general framework for high-order implicit-explicit (IMEX) Runge--Kutta methods that preserves the original energy dissipation for auxiliary models with globally Lipschitz truncations on the nonlinear term. By employing the Sobolev embedding theorem and Cauchy's interlace theorem, we demonstrate that the solutions of the auxiliary models are identical to the solutions of the original models without the globally Lipschitz property, provided that the free energy of the initial value is well-defined. Furthermore, we rigorously prove the uniform boundedness of the solution in the L-infinity norm and unconditional global-in-time stability. This allows for a straightforward framework to derive optimal arbitrarily high-order L-infinity error estimate without relying on the Lipschitz assumption. In particular, compared to existing literature, the argument for error estimation is presented in a much more simplified and elegant manner, without imposing any constraints on time-step size or mesh grid size. In fact, the reported framework, built upon the truncated auxiliary problem for the original model, can be directly extended to a wide range of gradient flows, including Allen--Cahn equations, nonlocal PFC models, and epitaxial thin film growth equations, providing unconditional energy dissipation without enforcing Lipschitz continuity. Finally, we present numerical examples to validate our analytical results and demonstrate the effectiveness of capturing long-time dynamics.

math.NA↗

arXiv2025

Numerical Analysis of Simultaneous Reconstruction of Initial Condition and Potential in Subdiffusion

This paper investigates the simultaneous identification of a spatially dependent potential and the initial condition in a subdiffusion model based on two terminal observations. The existence, uniqueness, and conditional stability of the inverse problem are established under weak regularity assumptions through a constructive fixed-point iteration approach. The theoretical analysis further inspires the development of an easy-to-implement iterative algorithm. A fully discrete scheme is then proposed, combining the finite element method for spatial discretization, convolution quadrature for temporal discretization, and the quasi-boundary value method to handle the ill-posedness of recovering the initial condition. Inspired by the conditional stability estimate, we demonstrate the linear convergence of the iterative algorithm and provide a detailed error analysis for the reconstructed initial condition and potential. The derived \textsl{a priori} error estimate offers a practical guide for selecting regularization parameters and discretization mesh sizes based on the noise level. Numerical experiments are provided to illustrate and support our theoretical findings.

math.NA↗

arXiv2025

Table2LaTeX-RL: High-Fidelity LaTeX Code Generation from Table Images via Reinforced Multimodal Language Models

In this work, we address the task of table image to LaTeX code generation, with the goal of automating the reconstruction of high-quality, publication-ready tables from visual inputs. A central challenge of this task lies in accurately handling complex tables -- those with large sizes, deeply nested structures, and semantically rich or irregular cell content -- where existing methods often fail. We begin with a comprehensive analysis, identifying key challenges and highlighting the limitations of current evaluation protocols. To overcome these issues, we propose a reinforced multimodal large language model (MLLM) framework, where a pre-trained MLLM is fine-tuned on a large-scale table-to-LaTeX dataset. To further improve generation quality, we introduce a dual-reward reinforcement learning strategy based on Group Relative Policy Optimization (GRPO). Unlike standard approaches that optimize purely over text outputs, our method incorporates both a structure-level reward on LaTeX code and a visual fidelity reward computed from rendered outputs, enabling direct optimization of the visual output quality. We adopt a hybrid evaluation protocol combining TEDS-Structure and CW-SSIM, and show that our method achieves state-of-the-art performance, particularly on structurally complex tables, demonstrating the effectiveness and robustness of our approach.

cs.AI↗

arXiv2025

GePUP-ES: High-order Energy-stable Projection Methods for the Incompressible Navier-Stokes Equations with No-slip Conditions

Inspired by the unconstrained PPE (UPPE) formulation [Liu, Liu, & Pego 2007 Comm. Pure Appl. Math., 60 pp. 1443], we previously proposed the GePUP formulation [Zhang 2016 J. Sci. Comput., 67 pp. 1134] for numerically solving the incompressible Navier-Stokes equations (INSE) on no-slip domains. In this paper, we propose GePUP-E and GePUP-ES, variants of GePUP that feature (a) electric boundary conditions with no explicit enforcement of the no-penetration condition, (b) equivalence to the no-slip INSE, (c) exponential decay of the divergence of an initially non-solenoidal velocity, and (d) monotonic decrease of the kinetic energy. Different from UPPE, the GePUP-E and GePUP-ES formulations are of strong forms and are designed for finite volume/difference methods under the framework of method of lines. Furthermore, we develop semi-discrete algorithms that preserve (c) and (d) and fully discrete algorithms that are fourth-order accurate for velocity both in time and in space. These algorithms employ algebraically stable time integrators in a black-box manner and only consist of solving a sequence of linear equations in each time step. Results of numerical tests confirm our analysis.

math.NA↗

arXiv2025

$ε$-rank and the Staircase Phenomenon: New Insights into Neural Network Training Dynamics

Understanding the training dynamics of deep neural networks (DNNs), particularly how they evolve low-dimensional features from high-dimensional data, remains a central challenge in deep learning theory. In this work, we introduce the concept of $ε$-rank, a novel metric quantifying the effective feature of neuron functions in the terminal hidden layer. Through extensive experiments across diverse tasks, we observe a universal staircase phenomenon: during training process implemented by the standard stochastic gradient descent methods, the decline of the loss function is accompanied by an increase in the $ε$-rank and exhibits a staircase pattern. Theoretically, we rigorously prove a negative correlation between the loss lower bound and $ε$-rank, demonstrating that a high $ε$-rank is essential for significant loss reduction. Moreover, numerical evidences show that within the same deep neural network, the $ε$-rank of the subsequent hidden layer is higher than that of the previous hidden layer. Based on these observations, to eliminate the staircase phenomenon, we propose a novel pre-training strategy on the initial hidden layer that elevates the $ε$-rank of the terminal hidden layer. Numerical experiments validate its effectiveness in reducing training time and improving accuracy across various tasks. Therefore, the newly introduced concept of $ε$-rank is a computable quantity that serves as an intrinsic effective metric characteristic for deep neural networks, providing a novel perspective for understanding the training dynamics of neural networks and offering a theoretical foundation for designing efficient training strategies in practical applications.

cs.LG↗

arXiv2025

Structured First-Layer Initialization Pre-Training Techniques to Accelerate Training Process Based on $\varepsilon$-Rank

Training deep neural networks for scientific computing remains computationally expensive due to the slow formation of diverse feature representations in early training stages. Recent studies identify a staircase phenomenon in training dynamics, where loss decreases are closely correlated with increases in $\varepsilon$-rank, reflecting the effective number of linearly independent neuron functions. Motivated by this observation, this work proposes a structured first-layer initialization (SFLI) pre-training method to enhance the diversity of neural features at initialization by constructing $\varepsilon$-linearly independent neurons in the input layer. We present systematic initialization schemes compatible with various activation functions and integrate the strategy into multiple neural architectures, including modified multi-layer perceptrons and physics-informed residual adaptive networks. Extensive numerical experiments on function approximation and PDE benchmarks, demonstrate that SFLI significantly improves the initial $\varepsilon$-rank, accelerates convergence, mitigates spectral bias, and enhances prediction accuracy. With the help of SILP, we only need to add one line of code to conventional existing algorithms.

math.NA↗

arXiv2025

Numerical Reconstruction and Analysis of Backward Semilinear Subdiffusion Problems

This paper aims to develop and analyze a numerical scheme for solving the backward problem of semilinear subdiffusion equations. We establish the existence, uniqueness, and conditional stability of the solution to the inverse problem by applying the smoothing and asymptotic properties of solution operators and constructing a fixed-point iteration. This derived conditional stability further inspires a numerical reconstruction scheme. To address the mildly ill-posed nature of the problem, we employ the quasi-boundary value method for regularization. A fully discrete scheme is proposed, utilizing the finite element method for spatial discretization and convolution quadrature for temporal discretization. A thorough error analysis of the resulting discrete system is provided for both smooth and nonsmooth data. This analysis relies on the smoothing properties of discrete solution operators, some nonstandard error estimates optimal with respect to data regularity in the direct problem, and the arguments used in stability analysis. The derived a priori error estimate offers guidance for selecting the regularization parameter and discretization parameters based on the noise level. Moreover, we propose an easy-to-implement iterative algorithm for solving the fully discrete scheme and prove its linear convergence. Numerical examples are provided to illustrate the theoretical estimates and demonstrate the necessity of the assumption required in the analysis.

math.NA↗

arXiv2024

Higher-Order Energy-Decreasing Exponential Time Differencing Runge-Kutta methods for Gradient Flows

In this paper, we develop a general framework for constructing higher-order, unconditionally energy-stable exponential time differencing Runge-Kutta methods applicable to a range of gradient flows. Specifically, we identify conditions sufficient for ETDRK schemes to maintain the original energy dissipation. Our analysis reveals that commonly used third-order and fourth-order ETDRK schemes fail to meet these conditions. To address this, we introduce new third-order ETDRK schemes, designed with appropriate stabilization, which satisfy these conditions and thus guarantee the unconditional energy decaying property. We conduct extensive numerical experiments with these new schemes to verify their accuracy, stability, behavior under large time steps, long-term evolution, and adaptive time stepping strategy across various gradient flows. This study is the first to examine the unconditional energy stability of high-order ETDRK methods, and we are optimistic that our framework will enable the development of ETDRK schemes beyond the third order that are unconditionally energy stable.

math.NA↗

arXiv2024

Computational and analytical studies of a new nonlocal phase-field crystal model in two dimensions

A nonlocal phase-field crystal (NPFC) model is presented as a nonlocal counterpart of the local phase-field crystal (LPFC) model and a special case of the structural PFC (XPFC) derived from classical field theory for crystal growth and phase transition. The NPFC incorporates a finite range of spatial nonlocal interactions that can account for both repulsive and attractive effects. The specific form is data-driven and determined by a fitting to the materials structure factor, which can be much more accurate than the LPFC and previously proposed fractional variant. In particular, it is able to match the experimental data of the structure factor up to the second peak, an achievement not possible with other PFC variants studied in the literature. Both LPFC and fractional PFC (FPFC) are also shown to be distinct scaling limits of the NPFC, which reflects the generality. The advantage of NPFC in retaining material properties suggests that it may be more suitable for characterizing liquid-solid transition systems. Moreover, we study numerical discretizations using Fourier spectral methods, which are shown to be convergent and asymptotically compatible, making them robust numerical discretizations across different parameter ranges. Numerical experiments are given in the two-dimensional case to demonstrate the effectiveness of the NPFC in simulating crystal structures and grain boundaries.

math.NA↗

arXiv2024

Energy diminishing implicit-explicit Runge--Kutta methods for gradient flows

This study focuses on the development and analysis of a group of high-order implicit-explicit (IMEX) Runge--Kutta (RK) methods that are suitable for discretizing gradient flows with nonlinearity that is Lipschitz continuous. We demonstrate that these IMEX-RK methods can preserve the original energy dissipation property without any restrictions on the time-step size, thanks to a stabilization technique. The stabilization constants are solely dependent on the minimal eigenvalues that result from the Butcher tables of the IMEX-RKs. Furthermore, we establish a simple framework that can determine whether an IMEX-RK scheme is capable of preserving the original energy dissipation property or not. We also present a heuristic convergence analysis based on the truncation errors. This is the first research to prove that a linear high-order single-step scheme can ensure the original energy stability unconditionally for general gradient flows. Additionally, we provide several high-order IMEX-RK schemes that satisfy the established framework. Notably, we discovered a new four-stage third-order IMEX-RK scheme that reduces energy. Finally, we provide numerical examples to demonstrate the stability and accuracy properties of the proposed methods.

math.NA↗

arXiv2023

Action Sensitivity Learning for Temporal Action Localization

Temporal action localization (TAL), which involves recognizing and locating action instances, is a challenging task in video understanding. Most existing approaches directly predict action classes and regress offsets to boundaries, while overlooking the discrepant importance of each frame. In this paper, we propose an Action Sensitivity Learning framework (ASL) to tackle this task, which aims to assess the value of each frame and then leverage the generated action sensitivity to recalibrate the training procedure. We first introduce a lightweight Action Sensitivity Evaluator to learn the action sensitivity at the class level and instance level, respectively. The outputs of the two branches are combined to reweight the gradient of the two sub-tasks. Moreover, based on the action sensitivity of each frame, we design an Action Sensitive Contrastive Loss to enhance features, where the action-aware frames are sampled as positive pairs to push away the action-irrelevant frames. The extensive studies on various action localization benchmarks (i.e., MultiThumos, Charades, Ego4D-Moment Queries v1.0, Epic-Kitchens 100, Thumos14 and ActivityNet1.3) show that ASL surpasses the state-of-the-art in terms of average-mAP under multiple types of scenarios, e.g., single-labeled, densely-labeled and egocentric.

cs.CV↗

arXiv2023

Life Regression based Patch Slimming for Vision Transformers

Vision transformers have achieved remarkable success in computer vision tasks by using multi-head self-attention modules to capture long-range dependencies within images. However, the high inference computation cost poses a new challenge. Several methods have been proposed to address this problem, mainly by slimming patches. In the inference stage, these methods classify patches into two classes, one to keep and the other to discard in multiple layers. This approach results in additional computation at every layer where patches are discarded, which hinders inference acceleration. In this study, we tackle the patch slimming problem from a different perspective by proposing a life regression module that determines the lifespan of each image patch in one go. During inference, the patch is discarded once the current layer index exceeds its life. Our proposed method avoids additional computation and parameters in multiple layers to enhance inference speed while maintaining competitive performance. Additionally, our approach requires fewer training epochs than other patch slimming methods.

cs.CV↗

arXiv2023

On profiniteness and Hausdorffness of topological residuated lattices

The aim of this paper is to study the profiniteness of compact topological residuated lattices and the existence of Hausdorff topological residuated lattices. Firstly, we study profinite residuated lattices and obtain sufficient and necessary conditions for profiniteness in compact topological residuated lattices. These conditions include topological and algebraic characterizations. Moreover, it order to study the existence of Hausdorf topological residuated lattices, we investigate finiteness conditions in residuated lattices. Finally, we investigate linear topological residuated lattices and give the class of residuated lattices that can be endowed with a non-trivial Hausdorff topology.

math.LO↗

arXiv2022

Long time $H^1$-stability of fast L2-1$_σ$ method on general nonuniform meshes for subdiffusion equations

In this work, the global-in-time $H^1$-stability of a fast L2-1$_σ$ method on general nonuniform meshes is studied for subdiffusion equations, where the convolution kernel in the Caputo fractional derivative is approximated by sum of exponentials. Under some mild restrictions on time stepsize, a bilinear form associated with the fast L2-1$_σ$ formula is proved to be positive semidefinite for all time. As a consequence, the uniform global-in-time $H^1$-stability of the fast L2-1$_σ$ schemes can be derived for both linear and semilinear subdiffusion equations, in the sense that the $H^1$-norm is uniformly bounded as the time tends to infinity. To the best of our knowledge, this appears to be the first work for the global-in-time $H^1$-stability of fast L2-1$_σ$ scheme on general nonuniform meshes for subdiffusion equations. Moreover, the sharp finite time $H^1$-error estimate for the fast L2-1$_σ$ schemes is reproved based on more delicate analysis of coefficients where the restriction on time step ratios is relaxed comparing to existing works.

math.NA↗