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

Abstract

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 \[ \|u_{\mathrm{loc}}-u_{δ,p}\|_{L^2(Ω)} \le C\!\left(\frac{\varepsilon_f}{\sqrtδ} +\frac{\varepsilon_g}δ +\frac{\varepsilon_u}{\sqrtδ} +η_{\mathrm{opt}}\right) +C\sqrtδ, \] where $\varepsilon_f$, $\varepsilon_g$ and $\varepsilon_u$ are the data and trial-class approximation errors and $η_{\mathrm{opt}}$ is the optimization residual. For the Neumann boundary condition, the $L^2$ estimate is improved to $O(\varepsilon_f+\varepsilon_g/\sqrtδ+\varepsilon_u +η_{\mathrm{opt}}+δ)$, 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.