ND-TNN: Tensor-Neural-Network Approximation for High-Dimensional Nonlocal Diffusion Models

We study a numerical method, built on the tensor neural network (TNN) architecture introduced in \cite{wang2022tensor}, for solving nonlocal diffusion models in high-dimensional spaces. The tensor-product structure of the TNN ansatz, combined with the separability of the Gaussian kernel, reduces the high-dimensional integrals in the nonlocal energy to products of low-dimensional integrals, which are evaluated by Gauss–Legendre quadrature; nonseparable source and boundary data are handled by a TNN-based preconditioning step. For the Dirichlet boundary condition, we establish the asymptotically compatible $L^2$ error estimate $$\mathrm{\parallel }{u}_{\mathrm{l}\mathrm{o}\mathrm{c}}-u_{\delta ,p}{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}\le C\text{\hspace{0.17em}⁣}(\frac{{\epsilon }_f}{\sqrt{\delta }}+\frac{{\epsilon }_g}{\delta }+\frac{{\epsilon }_u}{\sqrt{\delta }}+{\eta }_{\mathrm{o}\mathrm{p}\mathrm{t}})+C\sqrt{\delta },$$ where ${\epsilon }_f$, ${\epsilon }_g$ and ${\epsilon }_u$ are the data and trial-class approximation errors and ${\eta }_{\mathrm{o}\mathrm{p}\mathrm{t}}$ is the optimization residual. For the Neumann boundary condition, the $L^2$ estimate is improved to $O({\epsilon }_f+{\epsilon }_g\mathrm{/}\sqrt{\delta }+{\epsilon }_u+{\eta }_{\mathrm{o}\mathrm{p}\mathrm{t}}+\delta )$, and an $H^1$ gradient estimate is further obtained through a smoothing post-processing step. Numerical experiments on tensor-product domains up to $d=20$ support the theoretical results, and additional tests on two- and three-dimensional $L$-shaped domains demonstrate the practical robustness of the method beyond the smooth-domain setting covered by the analysis.