偏微分方程保积分数值方法的一般框架

一种用于保守型偏微分方程的新型半解析多不变量保持积分器

A novel semi-analytical multiple invariants-preserving integrator for conservative PDEs

本文针对具有多个不变量泛函的保守型偏微分方程，提出基于离散变分导数和投影技术的新型半解析多不变量保持积分器，并在KdV方程上推导出质量-动量-能量保持格式。

四阶偏微分方程双曲逼近的一种保结构隐式-显式方法

A structure-preserving implicit-explicit method for a hyperbolic approximation of fourth-order PDEs

提出一种用于四阶PDE双曲逼近的保结构IMEX方法，证明其能量一致、保正且渐近稳定，并通过薄膜方程算例验证有效性。

基于局部拉回流映射的几何PDE任意高阶保结构参数有限元方法

Arbitrarily High-Order Structure-Preserving Parametric Finite Element Methods for Geometric PDEs via Local Pullback Flow Maps

本文提出基于局部拉回流映射的任意高阶保结构参数有限元方法，用于表面扩散和保体积平均曲率流等几何PDE，通过共形和等分布两种表述实现面积/体积精确保持及周长/表面积耗散。

一类非线性随机偏微分方程的保正性格式

Positivity-preserving schemes for some nonlinear stochastic PDEs

本文为纯时间相关 Brown 运动驱动的一类非线性随机热方程构造保正性数值格式，并通过数值实验与收敛结果说明其性质。

Global energy preserving model reduction for multi-symplectic PDEs

Many Hamiltonian systems can be recast in multi-symplectic form. We develop a reduced-order model (ROM) for multi-symplectic Hamiltonian partial differential equations (PDEs) that preserves the global energy. The full-order solutions are obtained by finite difference discretization in space and the global energy preserving average vector field (AVF) method. The ROM is constructed in the same way as the full-order model (FOM) applying proper orthogonal decomposition (POD) with the Galerkin projection. The reduced-order system has the same structure as the FOM, and preserves the discrete reduced…

哈密顿偏微分方程的高阶线性隐式能量保持算法

Arbitrary high-order linearly implicit energy-preserving algorithms for Hamiltonian PDEs

本文提出基于指数标量变量（ESAV）的任意高阶线性隐式能量保持格式，用于哈密顿偏微分方程，通过辛Runge-Kutta和傅里叶伪谱方法实现高效守恒数值求解。

为热力学一致的可逆-不可逆偏微分方程推导保结构降阶模型的一般数值框架

General Numerical Framework to Derive Structure Preserving Reduced Order Models for Thermodynamically Consistent Reversible-Irreversible PDEs

本文提出基于能量二次化（EQ）和广义Onsager原理的一般数值框架，为热力学一致的可逆-不可逆偏微分方程系统推导保结构降阶模型，并自动生成线性、易实现且唯一可解的保热力学定律数值算法。

反对称性、伪谱方法与守恒型偏微分方程

Antisymmetry, pseudospectral methods, and conservative PDEs

本文提出“对偶复合”方法，用于构造守恒型偏微分方程的能量保持离散化，推广了求和分部分方法，并探讨了与伪谱、伽辽金、抗混叠及哈密顿方法的关联。

Curl constraint-preserving reconstruction and the guidance it gives for mimetic scheme design

Several important PDE systems, like magnetohydrodynamics and computational electrodynamics, are known to support involutions where the divergence of a vector field evolves in divergence-free or divergence constraint-preserving fashion. Recently, new classes of PDE systems have emerged for hyperelasticity, compressible multiphase flows, so-called first order reductions of the Einstein field equations, or a novel first order hyperbolic reformulation of Schrödinger's equation, to name a few, where the involution in the PDE supports curl-free or curl constraint-preserving evolution of a vector fie…

真正多维的守恒有限体积格式用于非线性双曲PDEs

Genuinely multi-dimensional stationarity preserving Finite Volume formulation for nonlinear hyperbolic PDEs

本文提出了一种多维守恒有限体积方法，用于非线性守恒/平衡定律，通过多维守恒求积策略实现对稳态的保持，优于现有方法。

Von Neumann Stability Analysis of DG-like and PNPM-like Schemes for PDEs that have Globally Curl-Preserving Evolution of Vector Fields

This paper examines a class of PDEs where some part of the PDE system evolves a vector field whose curl remains zero or grows in proportion to specified source terms. Such PDEs are referred to as curl-free or curl-preserving respectively. In this paper we catalogue a class of DG-like schemes for such PDEs. To retain the globally curl-free or curl-preserving constraints, the components of the vector field, as well as their higher moments, have to be collocated at the edges of the mesh. They are updated by using potentials that are collocated at the vertices of the mesh. The resulting schemes : …

高维半线性非局部偏微分方程及带跳正倒向随机微分方程的深度前向-后向动态规划格式

Deep Forward-Backward Dynamic Programming Schemes for High-Dimensional Semilinear Nonlocal PDEs and FBSDE with Jumps

提出一种新的深度学习算法，通过深度神经网络同时逼近解和积分核，并采用数值微分逼近梯度，用于求解高维半线性非局部偏微分方程及带跳正倒向随机微分方程，数值实验验证了其有效性。

PDE参数辨识中无监督学习方法的数值分析

Numerical Analysis of Unsupervised Learning Approaches for Parameter Identification in PDEs

本文从经典数值分析视角系统综述了基于无监督深度学习的PDE扩散系数辨识方法，并提出统一框架以推导Galerkin有限元、混合方法及深度神经网络离散近似的严格误差界，突出条件稳定性估计的核心作用。

一种用于椭圆型偏微分方程的保正性单网格方法

A positivity-preserving unigrid method for elliptic PDEs

针对带正性约束的离散椭圆型PDE，本文提出基于多重网格单网格变体的迭代算法，使近似求解全过程保持非负，并通过一维、二维数值实验验证其有效性与计算权衡。

面向多辛偏微分方程的结构保持学习

Structure-preserving learning for multi-symplectic PDEs

针对辛投影需完全离散算子的局限，提出仅用数据、不依赖离散算子的非侵入式灰箱保能量机器学习方法，可保持多辛守恒律，在三类PDE上验证了有效性与超训练区间的泛化性。

显式强稳定性保持多阶段二导数时间步进方案

Explicit Strong Stability Preserving Multistage Two-Derivative Time-Stepping Schemes

本文提出显式多阶段二导数SSP时间步进方案，突破传统Runge-Kutta方法的四阶限制，设计出五阶方法并验证其收敛性与稳定性。

Random Sampling and Efficient Algorithms for Multiscale PDEs

We describe a numerical framework that uses random sampling to efficiently capture low-rank local solution spaces of multiscale PDE problems arising in domain decomposition. In contrast to existing techniques, our method does not rely on detailed analytical understanding of specific multiscale PDEs, in particular, their asymptotic limits. We present the application of the framework on two examples --- a linear kinetic equation and an elliptic equation with rough media. On these two examples, this framework achieves the asymptotic preserving property for the kinetic equations and numerical homo…

高阶拉格朗日乘子方案用于一般哈密顿PDEs

High-order Lagrange multiplier schemes for general Hamiltonian PDEs

本文提出了一种高阶拉格朗日乘子方案，用于构造任意阶的线性隐式能量守恒方案，以保持哈密顿PDEs的结构特性。

混合边界条件下演化偏微分方程的保结构量子方法线

Structure-Preserving Quantum Method of Lines for Evolutionary PDEs with Mixed Boundary Conditions

针对混合边界条件的二阶线性演化偏微分方程，提出基于Coons插值和边界感知离散化的保结构量子方法线，确保半离散系统稳定并兼容高效量子ODE求解器，通过哈密顿模拟实现最优复杂度。

Structure-preserving Nonlinear Filtering for Continuous and Discontinuous Galerkin Spectral/hp Element Methods

Finite element simulations have been used to solve various partial differential equations (PDEs) that model physical, chemical, and biological phenomena. The resulting discretized solutions to PDEs often do not satisfy requisite physical properties, such as positivity or monotonicity. Such invalid solutions pose both modeling challenges, since the physical interpretation of simulation results is not possible, and computational challenges, since such properties may be required to advance the scheme. We, therefore, consider the problem of computing solutions that preserve these structural soluti…

阐明用于哈密顿偏微分方程的几何自适应、保结构算子学习的 Brinkman 惩罚方法的共形结构

Elucidating the Conformal Structure of the Brinkman Penalisation Method for Geometry-Adapted, Structure-Preserving Operator Learning of Hamiltonian PDEs

本文阐明Brinkman惩罚方法在哈密顿PDE中的共形结构，提出保结构积分器与共形辛神经算子，实现复杂域上几何自适应的保结构算子学习，并通过数值实验验证了其正确能量行为。

随机多辛偏微分方程及其保结构数值方法

Stochastic multisymplectic PDEs and their structure-preserving numerical methods

本文针对多辛偏微分方程扩展构建随机多辛系统，推导其相关守恒律，提出保结构随机配点法并验证其在随机非线性薛定谔方程上的有效性。

数值特征方法在一类守恒系统中的稳定性分析。第一部分：周期边界条件

Stability analysis of the numerical Method of characteristics applied to a class of energy-preserving systems. Part I: Periodic boundary conditions

本文研究了特征方法在非耗散双曲偏微分方程周期边界条件下的数值稳定性，比较了三种求解器的稳定性差异。

A positivity- and monotonicity-preserving nonlinear operator splitting approach for approximating solutions to quenching-combustion semilinear partial differential equations

In recent years, there has been a large increase in interest in numerical algorithms which preserve various qualitative features of the original continuous problem. Herein, we propose and investigate a numerical algorithm which preserves qualitative features of so-called quenching combustion partial differential equations (PDEs). Such PDEs are often used to model solid-fuel ignition processes or enzymatic chemical reactions and are characterized by their singular nonlinear reaction terms and the exhibited positivity and monotonicity of their solutions on their time intervals of existence. In t…

基于泰勒级数条件的多阶段二导数时间推进方案的强稳定性保持分析

A Strong Stability Preserving Analysis for Multistage Two-Derivative Time-Stepping Schemes Based on Taylor Series Conditions

本文研究了基于泰勒级数条件的多阶段二导数时间推进方案的强稳定性保持性质，提出了SSP-TS方法，并证明其最大阶数为6，同时展示了该方法在求解偏微分方程中的有效性。

基于扰动物理信息神经网络的非线性偏微分方程单样本迁移学习

One-Shot Transfer Learning for Nonlinear PDEs with Perturbative PINNs

该研究提出结合扰动理论与单样本迁移学习的PINN框架，将非线性PDE分解为线性子问题，学习线性算子潜在表示后可闭式解新实例，在KPP-Fisher等方程上误差约1e-3且迁移快，扩展了迁移学习应用并明确有效范围。

约束满足的Krylov求解器用于保结构离散化

Constraint-satisfying Krylov solvers for structure-preserving discretisations

本文提出一种修改的FGMRES迭代法，在求解保结构离散化产生的大规模线性系统时强制执行约束，以保持守恒律和耗散系统的结构性质。

一种用神经网络求解抛物型偏微分方程的结构保持框架

A Structure-Preserving Framework for Solving Parabolic Partial Differential Equations with Neural Networks

提出Sidecar框架，通过引入辅助网络引导主神经网络求解器保持抛物型PDE的物理结构（如守恒与耗散），显著提升现有求解器的精度和结构保持能力。

利用使能量二次化的提升变换进行非线性保能模型约简

Nonlinear energy-preserving model reduction with lifting transformations that quadratize the energy

针对含一般非线性的保守PDE，提出通过能量二次化提升变换结合POD构建精确守恒二次能量的降阶模型，并在四类方程上验证其精度、在线效率与离线收益。

Multiscale Elliptic PDEs Upscaling and Function Approximation via Subsampled Data

There is an intimate connection between numerical upscaling of multiscale PDEs and scattered data approximation of heterogeneous functions: the coarse variables selected for deriving an upscaled equation (in the former) correspond to the sampled information used for approximation (in the latter). As such, both problems can be thought of as recovering a target function based on some coarse data that are either artificially chosen by an upscaling algorithm, or determined by some physical measurement process. The purpose of this paper is then to study that, under such a setup and for a specific e…

带不确定性的非线性偏微分方程双曲系统的新型高阶数值方法

New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties

本文提出一种结合随机空间五阶WENO插值与物理空间二阶重构的高阶有限体积方法，用于求解带不确定性的非线性双曲PDE系统，并在欧拉方程和浅水方程上验证了其无振荡、高阶精度、良好平衡性与保正性。

高阶强稳定保持二阶导数显式、隐式及IMEX方法综述

A review of high order strong stability preserving two-derivative explicit, implicit, and IMEX methods

本文综述了高阶强稳定保持（SSP）二阶导数显式、隐式及IMEX Runge--Kutta和一般线性方法，提出了保持空间离散强稳定性的充分条件，并设计了时间步长限制与刚性项无关的新型方法。

细胞层神经算子用于约束PDE的结构保持代理建模

Cellular Sheaf Neural Operators for Structure-Preserving Surrogate Modeling of Constrained PDEs

提出细胞层神经算子，通过面向细胞复形的消息传递和Hodge拉普拉斯算子，在代理模型中保持PDE的几何约束和物理结构，在湍流MHD和聚变平衡任务中提升了结构敏感诊断指标。

用于非线性双曲型偏微分方程不确定性量化的基于样条的随机配置方法

Spline-Based Stochastic Collocation Methods for Uncertainty Quantification in Nonlinear Hyperbolic PDEs

针对基于广义多项式混沌的随机配置方法难以捕捉非线性双曲PDE不连续解、易产生吉布斯振荡的问题，提出采用保形样条插值的替代方法，通过数值算例验证其在准确量化不确定性同时抑制振荡的优势。

通过松弛和投影技术在时间演化的自然梯度方法中保持守恒定律

Preserving Conservation Laws in the Time-Evolving Natural Gradient Method via Relaxation and Projection Techniques

本文改进了时间演化自然梯度方法，通过松弛算法和投影技术确保解保持守恒律，提升PDE求解的准确性和物理意义。

时空FLAVORS：多尺度PDE的有限差分、多辛和伪谱积分器

Space-time FLAVORS: finite difference, multisymlectic, and pseudospectral integrators for multiscale PDEs

本文提出一类新的刚性问题偏微分方程积分器，基于流平均方法，在空间和时间上使用介观步长，具有多尺度、通用、非侵入、双尺度收敛和保结构等性质。

Numerical approximation of port-Hamiltonian systems for hyperbolic or parabolic PDEs with boundary control

We consider the design of structure-preserving discretization methods for the solution of systems of boundary controlled Partial Differential Equations (PDEs) thanks to the port-Hamiltonian formalism. We first provide a novel general structure of infinite-dimensional port-Hamiltonian systems (pHs) for which the Partitioned Finite Element Method (PFEM) straightforwardly applies. The proposed strategy is applied to abstract multidimensional linear hyperbolic and parabolic systems of PDEs. Then we show that instructional model problems based on the wave equation, Mindlin equation and heat equatio…

用于BBM方程的能量守恒移动网格方法

Energy preserving moving mesh methods applied to the BBM equation

本文提出了一种基于分区 unity 方法的能量守恒数值方法，并将其扩展到移动网格方法中，用于精确保持BBM方程的哈密顿量近似。

一种混合 SIAC——用于数值 PDE 解中不连续性的数据驱动后处理滤波器

A hybrid SIAC -- data-driven post-processing filter for discontinuities in solutions to numerical PDEs

提出仅用于最终时刻的混合 SIAC-CNN 后处理滤波器，结合矩约束与数据驱动修正，在保持光滑区精度的同时降低激波附近误差。

面向时间相关偏微分方程的物理信息神经求解器的预测性移动样本方法

Predictive Moving Sample Method for Physics-Informed Neural Solvers of Time-Dependent PDEs

提出预测性移动样本方法（PMSM）及其窗口重置变体（WR-PMSM），通过渐进时间步进和简化速度场损失，利用残差动态移动配点，高效求解时间相关偏微分方程。

一维KWC系统的保结构格式

Structure-preserving scheme for 1D KWC system

本文针对一维KWC相场模型，提出保结构数值格式，确保值域保持与能量耗散，并给出收敛条件与误差估计。

任意阶数的稳定性保持稳定有限元：多维非线性双曲问题的应用。应用于欧拉方程与重力

Arbitrary order stationarity preserving stabilized finite elements for multidimensional nonlinear hyperbolic problems. Application to the Euler equations with gravity

本文提出任意高阶稳定性保持稳定有限元方法，用于多维非线性双曲守恒律问题，通过局部预处理物理通量，实现高精度稳态近似，并应用于欧拉方程与重力问题，提升鲁棒性和精度。

三次哈密顿系统的线性隐式全局能量保持降阶模型

Linearly Implicit Global Energy Preserving Reduced-order Models for Cubic Hamiltonian Systems

针对具有三次不变量的大规模多辛偏微分方程，提出一种线性隐式全局能量保持方法构建降阶模型，证明其保持全局与局部能量，并通过三个数值例子验证了方法效率。

A positivity-preserving and energy stable scheme for a quantum diffusion equation

We propose a new fully-discretized finite difference scheme for a quantum diffusion equation, in both one and two dimensions. This is the first fully-discretized scheme with proven positivity-preserving and energy stable properties using only standard finite difference discretization. The difficulty in proving the positivity-preserving property lies in the lack of a maximum principle for fourth order PDEs. To overcome this difficulty, we reformulate the scheme as an optimization problem based on variational structure and use the singular nature of the energy functional near the boundary values…

Structure-preserving numerical methods for stochastic Poisson systems

We propose a class of numerical integration methods for stochastic Poisson systems (SPSs) of arbitrary dimensions. Based on the Darboux-Lie theorem, we transform the SPSs to their canonical form, the generalized stochastic Hamiltonian systems (SHSs), via canonical coordinate transformations found by solving certain PDEs defined by the Poisson brackets of the SPSs. An a-generating function approach with α\in [0,1] is then used to create symplectic discretizations of the SHSs, which are then transformed back by the inverse coordinate transformation to numerical integrators for the SPSs. These in…

DDC-PINNs：基于神经网络驱动的域分解和经典ODE求解器的预测-校正方法用于时变PDEs

DDC-PINNs: A Predictor-Corrector Approach Based on Neural Network-Driven Domain Decomposition and Classical ODE Solvers for Time-Dependent PDEs

本文提出DDC-PINNs方法，通过域分解和因果策略解决传统PINNs在求解时变PDEs时的时序因果性不足和表达能力限制问题，将PDEs简化为ODEs并利用经典ODE求解器提高精度和泛化能力。

高维偏微分方程的有限表达式逼近：无维数灾难

Finite Expression Approximation of High-Dimensional PDEs Without the Curse of Dimensionality

本文证明有限表达式符号表示类可克服高维PDE的维数灾难，通过多级皮卡德框架传播及构造确定性逼近，实现多项式复杂度保证，为有限表达式方法奠定理论基础。

强稳定性保持多步Runge-克莱茨方法

Strong Stability Preserving Multistep Runge-Kutta Methods

本文研究了高阶空间离散化方法的强稳定性特性，推导了显式多步Runge-Kutta方法的SSP系数上限，并通过数值优化得到高阶方法，测试了其在激波方程和Buckley-Leverett方程中的性能。

能量保持的降阶算子推断用于高效设计与控制

Energy-Preserving Reduced Operator Inference for Efficient Design and Control

针对能量保持二次算子的PDE，提出能量保持算子推断（EP-OpInf）方法，通过约束优化在降阶模型中保留能量结构，实现高效准确的多查询设计与控制。

Performance Assessment of Energy-preserving, Adaptive Time-step Variational Integrators

A fixed time-step variational integrator cannot preserve momentum, energy, and symplectic form simultaneously for nonintegrable systems. This barrier can be overcome by treating time as a discrete dynamic variable and deriving adaptive time-step variational integrators that conserve the energy in addition to being symplectic and momentum-preserving. Their utility, however, is still an open question due to the numerical difficulties associated with solving the discrete governing equations. In this work, we investigate the numerical performance of energy-preserving, adaptive time-step variationa…

从数据中学习变分偏微分方程的离散模型

Learning of discrete models of variational PDEs from data

提出一种保结构的机器学习架构，通过训练离散拉格朗日密度的神经网络模型从时空格点数据中学习变分偏微分方程的离散模型，并引入正则化技术提升数值鲁棒性，能识别训练数据中缺失的行波等简单解。

On the Spatial and Temporal Order of Convergence of Hyperbolic PDEs

In this work, we determine the full expression for the global truncation error of hyperbolic partial differential equations (PDEs). In particular, we use theoretical analysis and symbolic algebra to find exact expressions for the coefficients of the generic global truncation error. Our analysis is valid for any hyperbolic PDE, be it linear or non-linear, and employing finite difference, finite volume, or finite element discretization in space, and advanced in time with a predictor-corrector, multistep, or a deferred correction method, belonging to the Method of Lines. Furthermore, we discuss t…

一种局部梯度保持的水平集函数重新初始化方法

A locally gradient-preserving reinitialization for level set functions

本文提出一种局部梯度保持的水平集重新初始化方法，通过求解三个Hamilton-Jacobi型PDEs来保持界面梯度，适用于欧拉浸入边界法等应用。

通过广义核插值保持辛结构的哈密顿动力学预测

Symplecticity-Preserving Prediction of Hamiltonian Dynamics by Generalized Kernel Interpolation

本文提出了一种保持辛结构的核插值方法，用于高效预测高维离散PDEs的哈密顿动力学，通过结构保持降阶技术实现长时精度提升。