Lagrange有限元函数的稀疏连接神经网络表示
Sparsely connected neural network representation of Lagrange finite element function
- Xiangtan University（湘潭大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种由网格诱导的稀疏连接神经网络，精确表示任意阶Lagrange有限元空间，实现函数值与梯度同步提取，并支持非匹配网格插值及自适应有限元求解抛物型偏微分方程。
AI中文摘要：
我们构建了一个由网格诱导的稀疏连接神经网络框架，该框架能够在单纯形网格上精确重现任意阶Lagrange有限元空间。与传统的黑箱神经代理不同，所提出的网络架构完全由有限元离散规则决定：局部计算源于单纯形几何和重心坐标变换，而全局一致性通过共享自由度来强制实现。对于线性Lagrange单元，局部基函数通过仿射重心层直接实现，高阶多项式基被显式分解为重心坐标乘积组合，并由专门设计的$\mathrm{ReLU}^p$模块实现。配备单元指示分支和乘法单元后，这些模块化的局部组件被全局组装成一个稀疏连接的神经网络，其函数空间与目标有限元空间完全一致，从而继承了完整的经典有限元逼近理论。通过为分段激活函数规定定制的反向微分规则，函数值及其空间梯度可以在统一的计算图中通过自动微分同时提取，消除了标准有限元实现中所需的单独梯度计算子程序。数值实验验证了Lagrange有限元神经网络表示的准确性。此外，由于该神经网络表示固有的无网格特性，有限元函数可以在不匹配的网格之间进行插值，并且所提出的方案可应用于求解抛物型偏微分方程的自适应有限元方法。所提出架构的开源代码实现已公开提供。
英文摘要：
We construct a mesh-induced sparsely connected neural network framework that exactly reproduces arbitrary-order Lagrange finite element spaces over simplicial meshes. Unlike conventional black-box neural surrogates, the proposed network architecture is fully dictated by finite element discretization rules: local computations stem from simplex geometry and barycentric coordinate transformations, while global consistency is enforced through shared degrees of freedom. For linear Lagrange elements, local basis functions are directly implemented via affine barycentric layers, and high-order polynomial bases are explicitly decomposed into barycentric product compositions realized by specially designed $\mathrm{ReLU}^p$ modules. Equipped with element indicator branches and multiplication units, these modular local components are globally assembled into a sparsely connected neural network whose function space coincides exactly with the target finite element space, thereby inheriting the complete classical finite element approximation theory. By prescribing customized backward differentiation rules for piecewise activations, function values and their spatial gradients can be simultaneously extracted via automatic differentiation within a unified computational graph, eliminating the separate gradient calculation subroutines required in standard finite element implementations. Numerical experiments verify the accuracy of the neural network representation of Lagrange finite elements. Furthermore, by virtue of the intrinsic mesh-free nature of this neural network representation, finite element functions can be interpolated between non-matching meshes, and the proposed scheme can be applied to adaptive finite element methods for solving parabolic partial differential equations. An open-source code implementation of the proposed architecture is made publicly available.
