发表机构
School of Civil and Hydraulic Engineering, Huazhong University of Science and Technology; National Center of Technology Innovation for Digital Construction, Huazhong University of Science and Technology; School of Artificial Intelligence and Automation, Huazhong University of Science and Technology（华中科技大学土木与水利工程学院; 华中科技大学数字建造技术创新中心; 华中科技大学人工智能与自动化学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对神经算子元扩展的结构缺陷，提出凸神经能量元方法，实现几何参数化神经算子的稳定整体有限元组装，在热传导等任务中获高精度与高速度，且保证与类型、维度无关。
AI 中文摘要
将神经算子元方法从单独训练的固定几何神经元扩展到可重复使用的几何参数化元类型库时会出现结构问题：通过值回归训练的场预测算子会诱导出一种能量，其组装后的海森矩阵是不定的，即使场预测精度达1%，牛顿法也会收敛到虚假极小值（误差达247%）。我们提出凸神经能量元：每个元输出标量能量E(g,U)，其在边界自由度U上具有架构凸性，并由几何g平滑参数化，实现为超网络生成的半正定二次型（针对非二次物理问题保留输入凸修正）。正则化零空间原则——正则化器的零空间必须包含物理零空间——可消除原本不可约的偏差，且组装后的元继承经典保证：奇异元刚度会产生正定全局系统。我们证明了条件误差界（能量到解的精度、元数量缩放、几何泛化）并通过实验验证。在带椭圆孔的热传导问题中，一个训练后的元可组装成2×2至8×8网格以及未见过几何的L形布局，相对L2误差为0.6-1.0%，在边界量工作负载下每个几何的设置速度快175倍。第二种训练后的元类型可与第一种在一次整体组装中自由混合，三维实例在8个元的组装上达到0.23%的误差——这些保证与类型和维度无关。平面应变弹性元的物理零空间为三维，达到分析预测的正则化下限。将能量作为学习对象，使神经算子从一次性代理转变为可重复使用的元，继承其扩展方法的组装保证。
英文摘要
Extending the neural-operator element method from individually trained, fixed-geometry neural elements to a library of reusable, geometry-parameterized element types fails structurally: a field-predicting operator trained by value regression induces an energy whose assembled Hessian is indefinite, and Newton converges to spurious minima (247% error) even with 1%-accurate field predictions. We introduce convex neural energy elements: each element exports a scalar energy E(g,U), architecturally convex in its boundary degrees of freedom U and smoothly parameterized by its geometry g, realized as a hypernetwork-generated positive-semidefinite quadratic form (an input-convex correction is reserved for non-quadratic physics). A regularization-nullspace principle--the regularizer's nullspace must contain the physics nullspace--removes an otherwise irreducible bias, and assembled elements inherit the classical guarantee that singular element stiffnesses yield a positive-definite global system. We prove conditional error bounds (energy-to-solution accuracy, element-count scaling, geometry generalization) and verify each experimentally. On heat conduction with elliptic holes, one trained element assembles into 2x2 to 8x8 grids and an L-shaped layout of unseen geometries at 0.6-1.0% relative L2 error, with 175x faster per-geometry setup for boundary-quantity workloads. A second trained element type mixes freely with the first in one monolithic assembly, and a three-dimensional instantiation reaches 0.23% on eight-element assemblies--the guarantees are type- and dimension-agnostic. A plane-strain elasticity element, whose physics nullspace is three-dimensional, lands on the analytically predicted regularization floors. Making the energy the learned object turns neural operators from single-use surrogates into reusable elements that inherit the assembly guarantees of the method they extend.
Comments19 pages, 10 figures, 1 table
