标题： 一种通用的自动方法，用于构建各向异性函数表示的任意阶结构张量 显示英文标题

标题： A General, Automated Method for Building Structural Tensors of Arbitrary Order for Anisotropic Function Representations

摘要： 我们提出了一种通用的、构造性的方法，通过将问题转化为寻找线性算子的零空间来找到任意阶张量的基。 所提出的方法利用了标准的数值线性代数技术，这些技术高度优化且行为良好。 我们的主要应用是在力学中，模量张量和所谓的结构张量可以用来表征对其他输入（如应变）的功能依赖性的各向异性。 与模量张量一样，结构张量由其对称性群生成元变换的不变性定义，但具有更广泛的应用范围。 完全自动化的方法是经典的方法的替代方案，这些经典方法更依赖直觉，例如Pipkin-Rivlin多项式完整性基的构造。 我们通过以下方式展示了该过程的实用性：(a) 列举常见对称性的弹性模量张量，以及(b) 找到最低阶的结构张量，这些结构张量可以表示所有常见的点群/晶体类。 此外，我们使用这些结果在两个校准问题中，采用遵循经典函数表示理论的神经网络模型：(a) 给定应力-应变数据，学习超弹性材料的对称性类别和方向，以及(b) 在一系列单轴加载中，表示软基质-刚性纤维复合材料的应变相关各向异性应力响应。 这两个例子通过以下方式展示了该方法在模型选择和校准中的实用性：(a) 确定跨多个对称群的选定阶数的结构张量，以及(b) 确定一个给定群的基，该基允许表征所有子群。 在两种情况下使用共同的阶数使得稀疏回归可以在共同的函数表示上运行，以选择最适合数据的对称群。 显示英文摘要

摘要： We present a general, constructive procedure to find the basis for tensors of arbitrary order subject to linear constraints by transforming the problem to that of finding the nullspace of a linear operator. The proposed method utilizes standard numerical linear algebra techniques that are highly optimized and well-behaved. Our primary applications are in mechanics where modulus tensors and so-called structure tensors can be used to characterize anisotropy of functional dependencies on other inputs such as strain. Like modulus tensors, structure tensors are defined by their invariance to transformations by symmetry group generators but have more general applicability. The fully automated method is an alternative to classical, more intuition-reliant methods such as the Pipkin-Rivlin polynomial integrity basis construction. We demonstrate the utility of the procedure by: (a) enumerating elastic modulus tensors for common symmetries, and (b) finding the lowest-order structure tensors that can represent all common point groups/crystal classes. Furthermore, we employ these results in two calibration problems using neural network models following classical function representation theory: (a) learning the symmetry class and orientation of a hyperelastic material given stress-strain data, and (b) representing strain-dependent anisotropy of the stress response of a soft matrix-stiff fiber composite in a sequence of uniaxial loadings. These two examples demonstrate the utility of the method in model selection and calibration by: (a) determining structural tensors of a selected order across multiple symmetry groups, and (b) determining a basis for a given group that allows the characterization of all subgroups. Using a common order in both cases allows sparse regression to operate on a common function representation to select the best-fit symmetry group for the data.