用于局部周期椭圆问题高阶均匀化的张量神经网络方法
A Tensor Neural Network Method for High-Order Homogenization of Locally Periodic Elliptic Problems
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中文总结 AI 辅助
针对系数同时依赖快慢变量的局部周期椭圆多尺度问题，提出高阶张量神经网络方法，推导双尺度展开并证明收敛性，构建TNN框架避免积分误差，实验验证其高阶精度。
中文摘要 AI 辅助
我们针对形如 $-\nabla\cdot(A(x,x/\varepsilon)\nabla u_\varepsilon)=f$ 的局部周期椭圆多尺度问题，提出了一种高阶张量神经网络（TNN）方法。由于系数同时依赖于慢变量 $x$ 和快周期变量 $y=x/\varepsilon$，高阶单元问题与宏观校正方程比经典情形 $A=A(y)$ 更为复杂，且校正子参数依赖于 $x$。我们推导了可计算的高阶双尺度展开，并在无边界层设置（包括周期域和理想边界匹配构型）下，证明了该部分展开的 $H^1$ 收敛估计，证明过程利用了校正子层级的递归相容结构和 $H^{-1}$ 中的零均值振荡估计。随后，我们为高维校正子问题构建了 TNN 框架，其张量积结构允许对单元问题、均匀化系数、宏观源项和损失函数进行确定性一维求积，避免了蒙特卡洛积分误差。数值实现假设系数项与组装数据具有有限或可控的张量积表示，该计算假设与分析所用的一般矩阵值系数类相互独立。对标量局部周期系数的实验表明，高阶校正子精度较高，$H^1$ 半范数误差与所证明的估计一致，而点归一化的 $L^2$ 误差呈现出形式展开所预测的名义高阶行为。
英文摘要
We develop a high-order tensor neural network (TNN) method for locally periodic elliptic multiscale problems of the form $-\nabla\cdot(A(x,x/\varepsilon)\nabla u_\varepsilon)=f$. Because the coefficient depends on both the slow variable $x$ and the fast periodic variable $y=x/\varepsilon$, the high-order cell problems and macroscopic corrector equations are more involved than in the classical case $A=A(y)$, and the correctors depend parametrically on $x$. We derive a computable high-order two-scale expansion and prove an $H^1$ convergence estimate for the partial expansion in boundary-layer-free settings, including periodic domains and ideal boundary-matching configurations. The proof uses the recursive compatibility structure of the corrector hierarchy and a zero-mean oscillation estimate in $H^{-1}$. We then construct a TNN framework for the high-dimensional corrector problems. Its tensor-product structure permits deterministic one-dimensional quadrature for the cell problems, homogenized coefficients, macroscopic source terms, and loss functions, avoiding Monte Carlo integration error. The numerical realization assumes that the coefficient entries and assembled data admit finite or controlled tensor-product representations; this computational assumption is separate from the general matrix-valued coefficient class used in the analysis. Experiments with scalar locally periodic coefficients show accurate high-order correctors. The $H^1$ semi-norm errors are consistent with the proved estimate, while point-normalized $L^2$ errors display the nominal high-order behavior predicted by the formal expansion.
发表机构
- Beijing Institute of Technology（北京理工大学）
- Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学研究所）
- University of Chinese Academy of Sciences（中国科学院大学）
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