发表机构
Technische Universität Berlin（柏林工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对非均质复合材料模拟问题，提出基于网格的神经能量方法（M-NEM），通过形函数离散化、代数导数及高阶求积优化，在基准测试中误差更低、速度更快，且仅该方法能在极端刚度对比场景下收敛，RBFNNs在其中表现最优。
AI 中文摘要
对非均质材料进行建模仍是物理信息神经网络（如深度能量方法（DEM））面临的挑战。DEM及其变体在此统称为神经能量方法（NEM），提供了可微变分框架。然而，其传统的配点实现（C-NEM）常出现物理上不可接受的位移振荡、积分误差，以及自动微分带来的高计算成本。本研究引入基于网格的神经能量方法（M-NEM），通过对位移场进行基于网格的离散化扩展NEM。该方法利用形函数插值节点位移，施加抑制振荡的运动学约束；此外，用代数形函数导数替代自动微分计算应变，并采用高阶高斯求积实现精确的能量积分。在直接可比的基准问题上，M-NEM相较于C-NEM将应力误差降低了三个数量级，同时速度提升一到两个数量级；在涉及极端刚度对比的另外两个基准问题中，仅M-NEM能够收敛。对神经架构的对比研究显示，径向基函数神经网络（RBFNNs）在M-NEM中表现最优，其对材料界面处尖锐梯度的解析精度高于带有随机傅里叶特征（RFF）映射的多层感知器（MLPs），收敛速度快于柯尔莫哥洛夫-阿诺尔德网络（KANs）。
英文摘要
Modeling heterogeneous materials remains a challenge for physics-informed neural networks such as the deep energy method (DEM). The DEM and its variants, here collectively referred to as the neural energy method (NEM), offer a differentiable variational framework. However, their conventional collocation-based implementation (C-NEM) often suffers from physically inadmissible displacement oscillations, integration errors, and high computational costs from automatic differentiation. This work introduces the mesh-based neural energy method (M-NEM), extending the NEM through a mesh-based discretization of the displacement field. By interpolating nodal displacements via shape functions, the M-NEM imposes a kinematic constraint that suppresses oscillations. Furthermore, the method replaces automatic differentiation with algebraic shape function derivatives for strain computation and employs high-order Gaussian quadrature for accurate energy integration. On a directly comparable benchmark problem, the M-NEM reduces stress errors by up to three orders of magnitude relative to the C-NEM while being one to two orders of magnitude faster. On two further benchmarks involving extreme stiffness contrasts, only the M-NEM converges. A comparative study of neural architectures reveals that radial basis function neural networks (RBFNNs) yield optimal performance within the M-NEM, resolving sharp gradients at material interfaces with higher accuracy than multi-layer perceptrons (MLPs) with random Fourier feature (RFF) mapping and faster convergence than Kolmogorov-Arnold networks (KANs).
