A Tensor Neural Network Method for High-Order Homogenization of Locally Periodic Elliptic Problems
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.