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 \[ \|u_{\mathrm{loc}}-u_{\delta,p}\|_{L^2(\Omega)} \le C\!\left(\frac{\varepsilon_f}{\sqrt\delta} +\frac{\varepsilon_g}{\delta} +\frac{\varepsilon_u}{\sqrt\delta} +\eta_{\mathrm{opt}}\right) +C\sqrt\delta, \] where $\varepsilon_f$, $\varepsilon_g$ and $\varepsilon_u$ are the data and trial-class approximation errors and $\eta_{\mathrm{opt}}$ is the optimization residual. For the Neumann boundary condition, the $L^2$ estimate is improved to $O(\varepsilon_f+\varepsilon_g/\sqrt\delta+\varepsilon_u +\eta_{\mathrm{opt}}+\delta)$, 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.
From: Ziyue Cai [view email]
[v1]
Sun, 7 Jun 2026 15:37:55 UTC (2,359 KB)
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