Engineering Analysis with Boundary Elements最新文献_第4页

Mc-PINN for solving Conservative Allen–Cahn equations using Metropolis–Hasting based sampling Mc-PINN用于求解基于Metropolis-Hasting采样的保守Allen-Cahn方程

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-03-01 Epub Date: 2026-01-14 DOI: 10.1016/j.enganabound.2026.106643

Anjali Singh, Rajen Kumar Sinha

The conservative Allen–Cahn (CAC) equation is a second-order nonlinear partial differential equation that models phase separation in binary mixtures while preserving the total volume. Physics-informed neural networks (PINNs) have demonstrated considerable success in approximating solutions to various classes of partial differential equations; however, their application to CAC models remains challenging. These difficulties stem from the presence of a small interfacial parameter

ϵ

in the nonlinear term

F^{'} (u)

and highly nonlinear mass-correction terms

G_{i}

,

i = 1, 2, 3

, which significantly degrade the approximation accuracy and mass conservation properties of standard PINNs. In this work, we propose a novel hybrid mass-constrained physics-informed neural network (Mc-PINN) framework for efficiently and accurately solving three types of CAC models in convex polygonal domains. The proposed method integrates deep learning with operator-splitting strategies to decompose the original CAC equations into simpler subproblems. One subproblem admits an analytical solution, while the other is solved using the Mc-PINN scheme. To further enhance efficiency, a Metropolis–Hastings based adaptive sampling strategy is introduced. In addition, we derive error estimates for the proposed method applied to all three CAC models. Numerical experiments demonstrate the robustness, accuracy, and mass-conservation capability of the proposed framework.

保守的Allen-Cahn （CAC）方程是一个二阶非线性偏微分方程，它在保持总体积的情况下模拟二元混合物的相分离。物理信息神经网络（pinn）在逼近各种偏微分方程的解方面取得了相当大的成功；然而，它们在CAC模型中的应用仍然具有挑战性。这些困难源于非线性项F ' (u)中存在一个小的界面参数，以及高度非线性的质量校正项Gi， i=1,2,3，这显著降低了标准pin的近似精度和质量守恒特性。在这项工作中，我们提出了一种新的混合质量约束物理信息神经网络（Mc-PINN）框架，用于有效和准确地求解凸多边形域中的三种类型的CAC模型。该方法将深度学习与算子分裂策略相结合，将原始CAC方程分解为更简单的子问题。其中一个子问题允许解析解，而另一个子问题则使用Mc-PINN方案求解。为了进一步提高效率，引入了一种基于Metropolis-Hastings的自适应采样策略。此外，我们推导了适用于所有三种CAC模型的方法的误差估计。数值实验证明了该框架的鲁棒性、准确性和质量守恒性。

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引用次数: 0

Discontinuous isogeometric boundary elements for direct acoustic–structural coupling with Reissner–Mindlin shells Reissner-Mindlin壳直接声结构耦合的不连续等几何边界元

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-03-01 Epub Date: 2026-01-07 DOI: 10.1016/j.enganabound.2025.106629

Ahmed Mostafa Shaaban , Elena Atroshchenko , Steffen Marburg

A discontinuous isogeometric boundary element method is proposed for the acoustic model, which is based on the Helmholtz time-harmonic wave propagation equation, as part of acoustic–structural interaction problems. The discontinuous formulation is combined with an offset-based collocation scheme, which enhances accuracy over the continuous approach while simplifying integration, the computation of free terms, and the representation of highly distorted elements in pole-based models. The dynamic structural model, representing the second component of the interaction problem, is formulated using isogeometric Reissner–Mindlin shell theory, which is particularly effective for modeling thin curved structures. The acoustic and structural models are directly coupled through conforming numerical meshes on the interaction surface. Isogeometric analysis is applied for both the acoustic and structural formulations, as well as to the coupling scheme, due to its ability to represent exact geometries and its superior accuracy compared to conventional numerical approaches.

Numerical examples are presented to assess the performance of the proposed solution, with results compared against available analytical solutions and previously reported outcomes based on alternative coupling strategies.

作为声-结构相互作用问题的一部分，提出了一种基于Helmholtz时谐波传播方程的不连续等几何边界元方法。不连续公式与基于偏移量的配置方案相结合，提高了连续方法的精度，同时简化了基于极点模型的积分、自由项的计算和高度扭曲元素的表示。采用等几何Reissner-Mindlin壳理论建立了代表相互作用问题第二部分的动力结构模型，该模型对薄弯曲结构的建模特别有效。声学模型和结构模型在相互作用面上通过统一的数值网格直接耦合。等几何分析既适用于声学和结构公式，也适用于耦合方案，因为它能够表示精确的几何形状，并且与传统的数值方法相比具有更高的精度。给出了数值示例来评估所提出的解决方案的性能，并将结果与可用的解析解决方案和先前基于替代耦合策略的报告结果进行了比较。

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引用次数: 0

An inter-well interference quantitative evaluation approach of multiple multi-stage fractured horizontal well with non-uniform simulated reservoirs volume in tight gas reservoirs 致密气藏非均匀模拟储层体积多段多级压裂水平井井间干扰定量评价方法

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-03-01 Epub Date: 2026-01-09 DOI: 10.1016/j.enganabound.2025.106627

Youjie Xu , Rui Yong , Lianjin Zhang , Fei Zhang , Zhenglin Mao , Xixiang Liu

The multiple multi-stage fractured horizontal wells (MFHWs) interference has been identified in tight gas reservoirs, but current method and model cannot achieve quantitative inter-well interference evaluation of non-uniform fractured region. Therefore, MFHWs mathematical model is established with consideration of multiple sub-region permeability difference. Coupling method of boundary element and source function is employed to solved the mathematical model. Inter-well interference factor is defined and used to evaluate inter-well interference degree. Total interference factor minimum value is used to determine the optimal rate ratio. The result shows that larger production time and well distance leads to small inter-well interference factor, but they have no influence optimal rate ratio. The optimal rate ratio will increase with the increasing of number of adjacent well fractures. If permeability of center region is larger than that of other region, inter-well interference factor and total interference factor will decrease and optimal rate ratio will increase. The model and method can evaluate inter-well interference degree and optimal rate ratio quantitatively, which provides guidance for horizontal well fracturing and parameter optimization design.

致密气藏存在多级压裂水平井干扰，但现有的方法和模型无法实现非均匀压裂区的井间干扰定量评价。为此，建立了考虑多分区渗透率差的MFHWs数学模型。采用边界元与源函数耦合的方法求解数学模型。定义了井间干扰系数，并用其来评价井间干扰程度。采用总干扰因子最小值确定最优率比。结果表明，较大的生产时间和井距可以减小井间干扰系数，但不影响最佳产量比。最佳速率比随着邻井裂缝数量的增加而增大。中心区渗透率大于其他区域时，井间干扰系数和总干扰系数减小，最优产率比增大。该模型和方法可以定量评价井间干扰程度和最优产率比，为水平井压裂和参数优化设计提供指导。

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引用次数: 0

Multi-point constraints enhanced discontinuous deformation analysis for low-velocity impact on low composite materials 多点约束增强的低速冲击复合材料不连续变形分析

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-03-01 Epub Date: 2026-01-09 DOI: 10.1016/j.enganabound.2025.106621

Ziheng Li, Hong Zheng, Shouyang Huang

The original Discontinuous Deformation Analysis (DDA) faces limitations due to its linear displacement assumption and contact solution approach, making it challenging to handle nonlinear contact problems involving localized small deformations, particularly in cases like low-velocity impact on composite materials. This study presents an multi-point constraints enhanced DDA (MPC-DDA) specifically designed for modeling low-velocity impact scenarios in composites. The proposed modification incorporates two key features: (1) removal of the penalty spring mechanism preserves residual block penetration states in OC iterative results, (2) Iterative residuals are reconstructed as surface indentations through multi-point constraints, enabling full historical contact trace visualization. A novel contact algorithm utilizing virtual entrance points is developed, enabling MPC-DDA to dynamically capture the master and slave points on the contact boundary. This approach effectively represents surface indentations on composite materials through virtual entrance point displacements, significantly reducing the computational complexity associated with resolving localized small deformations characteristic of original DDA implementations. The MPC-DDA method has been implemented in a Matlab program, and its accuracy has been validated by comparing the computational results with those published in existing literature. Comparative analyses demonstrate that MPC-DDA outperforms original DDA in addressing nonlinear contact problems, particularly in terms of solution accuracy and computational efficiency.

原始的非连续变形分析（DDA）由于其线性位移假设和接触求解方法的局限性，使得局部小变形的非线性接触问题难以处理，特别是在复合材料低速撞击等情况下。本研究提出了一种多点约束增强型DDA (MPC-DDA)，专门用于模拟复合材料中的低速碰撞场景。提出的改进包含两个关键特征：(1)去除惩罚弹簧机制，保留OC迭代结果中的残余块穿透状态；(2)通过多点约束将迭代残差重构为表面压痕，实现完整的历史接触轨迹可视化。提出了一种利用虚拟入口点的接触算法，使MPC-DDA能够动态捕获接触边界上的主从点。该方法通过虚拟入口点位移有效地表示复合材料的表面压痕，大大降低了与解决原始DDA实现的局部小变形特征相关的计算复杂性。在Matlab程序中实现了MPC-DDA方法，并将计算结果与现有文献的计算结果进行了比较，验证了其准确性。对比分析表明，MPC-DDA在解决非线性接触问题方面优于原始DDA，特别是在求解精度和计算效率方面。

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引用次数: 0

Crack analysis of the foundation gallery within an asphalt concrete core dam based on 3D SBFEM-PFM 基于三维SBFEM-PFM的沥青混凝土心坝基础廊裂缝分析

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-03-01 Epub Date: 2026-01-13 DOI: 10.1016/j.enganabound.2026.106641

Yue Zhuo , Kai Chen , Degao Zou , Shanlin Tian , Shiyong Wu , Shi Zhang

Fracture analysis of anti-seepage systems in earth-rock dams involves pronounced material nonlinearity, frictional contact behavior, and convergence difficulties, which remain long-standing challenges in geotechnical engineering. In this study, A nonlinear 3D SBFEM–PFM formulation was created by embedding the phase-field model (PFM) into the scaled boundary finite element method (SBFEM) and implementing an intra-element phase-field interpolation scheme. Base on independently developed GEODYNA, a unified and parallelized framework was established through various coupling schemes, including FEM-SBFEM, phase-field, and nonlinear contact algorithm. The method is validated against a classical benchmark and subsequently applied to the world’s highest asphalt concrete core rockfill dam (ACCRD) on the overburden to simulate full-process cracking of gallery. Cracks were identified along the inner surface of the gallery and on the outer surfaces of both banks, with reservoir impoundment exhibiting opposing effects on crack width. Additionally, the severity of structural damage was significantly influenced by the interface characteristics between the gallery and surrounding geomaterial. This study signifies the inaugural implementation of PFM to achieve a full-process simulation of stress accumulation, crack initiation, and progressive propagation within dam anti-seepage systems. High risk zones were precisely identified, and the practical optimization measure was suggested, providing an innovative and effective approach for structural analysis in related geotechnical engineering applications.

土石坝防渗系统的断裂分析涉及明显的材料非线性、摩擦接触行为和收敛困难，是岩土工程中长期存在的难题。本研究通过将相场模型（PFM）嵌入到比例边界有限元法（SBFEM）中，并实现单元内相场插值方案，建立了三维SBFEM - PFM非线性公式。在自主开发的GEODYNA基础上，通过FEM-SBFEM、相场法、非线性接触算法等多种耦合方案，建立了统一的并行化框架。通过经典基准验证了该方法的有效性，并将其应用于世界上最高的沥青混凝土堆芯坝（ACCRD）覆盖层上，模拟了廊道的全过程开裂。沿廊道的内表面和两岸的外表面发现了裂缝，水库蓄水对裂缝宽度的影响相反。此外，廊道与周围岩土材料之间的界面特征对结构损伤的严重程度有显著影响。该研究标志着PFM首次实现了大坝防渗系统内应力积累、裂缝萌生和渐进扩展的全过程模拟。准确识别出高风险区，并提出切实可行的优化措施，为相关岩土工程应用中的结构分析提供了一种创新有效的方法。

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引用次数: 0

Scattering of capillary-gravity waves by surface-piercing porous barriers in the presence of uniform current over a porous sea bed 在多孔海床上存在均匀水流时，穿透表面的多孔屏障对毛细管重力波的散射

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-03-01 Epub Date: 2026-01-15 DOI: 10.1016/j.enganabound.2026.106646

Gagan Sahoo, Harekrushna Behera, Tai-Wen Hsu

Capillary–gravity waves, influenced by both surface tension and gravity, interact strongly with marine structures, especially in the presence of uniform currents. Despite extensive studies on wave scattering by porous structures, the combined effects of surface tension, current, and porous barriers over a porous bottom remain insufficiently explored. This study examines the scattering of such waves by two thin surface-piercing porous barriers in the presence of a uniform current over a porous sea bed. A linear wave–structure interaction model is solved numerically through a hybrid Boundary Element-Finite Difference Method (BEM–FDM) and analytically through an eigenfunction expansion combined with a least-squares approach. The hybrid BEM–FDM efficiently handles higher-order boundary conditions that cannot be directly addressed by conventional BEM, while the analytical method eliminates the need for eigenfunction orthogonality and explicit mode coupling. The effects of surface tension, current velocity and direction, porous effect parameters of barriers as well as bottom, barrier length and spacing between them on reflection, transmission, and energy dissipation are analyzed. Results show that surface tension enhances reflection and dissipation while reducing transmission. Current direction strongly affects scattering: following currents enhance transmission, whereas opposing currents increase reflection and dissipation. Longer barriers and larger porous-effect parameters of both porous barriers and porous bottom enhance energy dissipation, while spacing between porous barriers induce interference driven oscillations.

毛细重力波受表面张力和重力的影响，与海洋结构有强烈的相互作用，特别是在有均匀洋流的情况下。尽管对多孔结构的波散射进行了广泛的研究，但表面张力、电流和多孔底部上的多孔屏障的综合影响仍然没有得到充分的探索。本研究考察了在多孔海床上均匀水流存在的情况下，两个薄的穿透表面的多孔屏障对这种波的散射。采用混合边界元-有限差分法（BEM-FDM）对线性波-结构相互作用模型进行数值求解，采用特征函数展开结合最小二乘方法对线性波-结构相互作用模型进行解析求解。该方法有效地处理了传统边界元法无法直接解决的高阶边界条件，而解析法消除了特征函数正交性和显式模态耦合的需要。分析了表面张力、电流速度和方向、阻挡层及底部的多孔效应参数、阻挡层长度和阻挡层间距对反射、透射和能量耗散的影响。结果表明，表面张力增强了反射和耗散，降低了透射。电流方向强烈影响散射：顺电流增强透射，相反电流增强反射和耗散。更长的势垒和更大的孔效应参数增强了能量耗散，而多孔势垒之间的间距引起干涉驱动振荡。

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引用次数: 0

Tanh-B6 KAN-based PINN for solving thin plate bending problems 用于解决薄板弯曲问题的基于Tanh-B6 kan的PINN

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-03-01 Epub Date: 2026-01-23 DOI: 10.1016/j.enganabound.2026.106653

Shuo Liu, Yonghong Cao

Solving high-order partial differential equations, such as the governing equations of Kirchhoff thin plate bending, remains a significant challenge for traditional Physics-Informed Neural Networks (PINNs). Conventional Multi-Layer Perceptron (MLP) architectures often suffer from spectral bias and gradient instability when computing high-order derivatives. To address these limitations, this study introduces a novel framework: the Tanh-B6 KAN-based PINN. This method integrates Kolmogorov-Arnold Networks (KAN) featuring learnable Tanh activation functions and sixth-order B-spline basis functions into the PINN architecture. Specifically, the sixth-order B-splines ensure C⁴ continuity, providing stable analytical computation for high-order derivatives, while the Tanh activation captures global trends. The effectiveness of this approach is validated through comprehensive numerical experiments on elliptical, triangular, rectangular, and L-shaped thin plates subject to varying boundary and load conditions. Comparative results demonstrate that the Tanh-B6 KAN-based PINN significantly outperforms the traditional MLP-PINN, reducing the relative L₂ norm error and Mean Absolute Error (MAE) of the displacement field and boundaries by one to three orders of magnitude, while reducing the number of parameters by two to three orders of magnitude. The proposed method offers a robust, interpretable, and highly efficient solution for high-order mechanics problems.

求解高阶偏微分方程，如基尔霍夫薄板弯曲的控制方程，仍然是传统物理信息神经网络（pinn）面临的一个重大挑战。传统的多层感知器（MLP）结构在计算高阶导数时往往存在谱偏差和梯度不稳定性。为了解决这些限制，本研究引入了一种新的框架：基于Tanh-B6 kan的PINN。该方法将具有可学习Tanh激活函数和六阶b样条基函数的Kolmogorov-Arnold网络（KAN）集成到PINN体系结构中。具体来说，六阶b样条确保C4连续性，为高阶导数提供稳定的解析计算，而Tanh激活捕获全局趋势。通过椭圆、三角形、矩形和l型薄板在不同边界和载荷条件下的综合数值实验，验证了该方法的有效性。对比结果表明，基于Tanh-B6 kan的PINN显著优于传统的MLP-PINN，将位移场和边界的相对L2范数误差和平均绝对误差（MAE）降低了1 ~ 3个数量级，同时将参数数量减少了2 ~ 3个数量级。该方法为高阶力学问题提供了鲁棒性、可解释性和高效率的求解方法。

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引用次数: 0

A spectral approach for fast evaluation of 3D thermomagnetoelectroelastic Green’s functions and their derivatives in the boundary element method 边界元法中快速求解三维热磁电弹性格林函数及其导数的谱法

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-03-01 Epub Date: 2026-01-13 DOI: 10.1016/j.enganabound.2026.106647

Viktoriya Pasternak , Heorhiy Sulym , Andrii Korniichuk , Iaroslav Pasternak

This paper presents a spectral method for the efficient evaluation of Green’s functions in three-dimensional anisotropic thermoelastic and thermomagnetoelectroelastic problems. The method expands the Green’s function kernel in spherical harmonics, reducing its integral representation to a finite sum containing only odd- or even-degree harmonic coefficients. This eliminates the need for mesh-based evaluation and interpolation, significantly improving computational speed and numerical robustness. The approach requires precomputation of only a small set of spectral coefficients, after which Green’s functions and their spatial derivatives can be evaluated rapidly at arbitrary target points. Moreover, the precomputed coefficients depend only on the material properties, can be calculated to the desired accuracy, and can be reused to analyze different solid geometries composed of the same material. The resulting formulation is well suited for boundary element methods (BEM) and other integral equation schemes. Numerical experiments demonstrate fast spectral convergence of the spherical harmonic series, while performance benchmarks show that the proposed approach reduces the overall BEM computation time for the considered problems by approximately a factor of two. The method provides an efficient and scalable tool for simulating complex multiphysics phenomena in anisotropic solids.

本文提出了三维各向异性热弹性和热磁电弹性问题中格林函数的有效求值的谱方法。该方法将球面谐波中的格林函数核展开，将其积分表示化为只包含奇次或偶次谐波系数的有限和。这消除了基于网格的评估和插值的需要，显著提高了计算速度和数值鲁棒性。该方法只需要预先计算一小部分谱系数，然后可以在任意目标点快速计算格林函数及其空间导数。此外，预先计算的系数仅取决于材料的性质，可以计算到所需的精度，并且可以重复使用来分析由相同材料组成的不同固体几何形状。所得公式非常适合于边界元法（BEM）和其他积分方程格式。数值实验表明，该方法具有快速的谱收敛性，而性能基准测试表明，该方法将所考虑问题的总边界元计算时间减少了大约两倍。该方法为模拟各向异性固体中复杂的多物理场现象提供了一种有效且可扩展的工具。

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引用次数: 0

A novel quasi-interpolation radial integration BEM for non-homogeneous problems 非齐次问题的一种新的准插值径向积分边界元

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-03-01 Epub Date: 2026-01-16 DOI: 10.1016/j.enganabound.2026.106645

Bin Hu , Cong Li

A novel quasi-interpolation radial integration boundary element method (QIRIBEM) is proposed for solving non-homogeneous problems in this study. For domain integrals involving unknowns, the conventional methods use compactly supported radial basis functions (CSRBFs) to interpolate these unknowns directly. The direct interpolation scheme requires the introduction of an interpolation matrix, followed by matrix inversion and matrix multiplication operations. To overcome this limitation, a quasi-interpolator based on CSRBFs is first introduced to approximate the unknowns, where the unknown serves as the interpolation coefficient, eliminating the requirement for the interpolation matrix. Subsequently, a new interaction list is proposed to improve the computational efficiency of constructing the quasi-interpolator. Finally, the quasi-interpolator is incorporated into the radial integration method to transform domain integrals into boundary integrals. In contrast to the direct interpolation radial integration boundary element method (DIRIBEM), the proposed method achieves good accuracy by use a smaller supported domain and avoid a series of calculation and storage operations related to the interpolation matrix, which can save considerable computational time and memory spaces for large-scale models.

提出了一种求解非齐次问题的准插值径向积分边界元法。对于涉及未知数的域积分，传统方法使用紧支持径向基函数（csrbf）直接插值这些未知数。直接插补方案需要引入插补矩阵，然后进行矩阵反演和矩阵乘法运算。为了克服这一限制，首先引入基于csrbf的准插值器来逼近未知量，其中未知量作为插值系数，消除了对插值矩阵的要求。为了提高拟插值器的计算效率，提出了一种新的相互作用表。最后，将拟插值器引入到径向积分法中，将域积分转化为边界积分。与直接插值径向积分边界元法（DIRIBEM）相比，该方法利用更小的支持域，避免了与插值矩阵相关的一系列计算和存储操作，获得了较好的精度，可为大规模模型节省大量的计算时间和存储空间。

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引用次数: 0

Graph-based compactly supported radial basis function neural network 基于图的紧支持径向基函数神经网络

IF 4.1 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2026-03-01 Epub Date: 2026-01-15 DOI: 10.1016/j.enganabound.2026.106644

Hongjin Ren , Dengao Li , Hongen Jia , Ruiping Niu , Hongbin Wang

In this paper, a novel graph-based compactly supported radial basis function physics-informed neural network (G-CS-RBN) is proposed for partial differential equations. Compactly supported radial basis functions are employed to replace the linear interpolation to construct an efficient one-hidden-layer neural network. An adaptive support radius is proposed that allows each point to automatically learn its local support according to the loss function. This overcomes the drawback of the traditional numerical compactly supported radial basis functions using fixed empirical support radius, which restricts the accuracy of models. A graph structure is used to store the collocation points and their respective center points to improve the interpretability of the network, storage efficiency, and network learning. Besides, the adaptive center point is also suggested to aid the adaptive support radius, which can further boost the performance of G-CS-RBN. Finally, extensive numerical experiments on 2D and 3D PDEs demonstrate that G-CS-RBN achieves consistently better accuracy and efficiency compared with classical numerical CS-RBF methods and standard PINNs, while showing improved robustness across different PDEs.

本文提出了一种新的基于图的紧支持径向基函数物理信息神经网络（G-CS-RBN）。采用紧支持径向基函数代替线性插值，构造了高效的单隐层神经网络。提出了一种自适应支持半径，使每个点能够根据损失函数自动学习其局部支持。这克服了传统数值紧支撑径向基函数采用固定经验支持半径的缺点，限制了模型的精度。采用图结构存储并置点及其各自的中心点，提高了网络的可解释性、存储效率和网络学习能力。此外，还提出了自适应中心点来辅助自适应支撑半径，进一步提高了G-CS-RBN的性能。最后，在二维和三维偏微分方程上进行的大量数值实验表明，与经典的数值CS-RBF方法和标准pin相比，G-CS-RBN方法的精度和效率始终更高，同时在不同偏微分方程上表现出更好的鲁棒性。

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引用次数: 0