Abstract:We introduce a novel neural operator architecture designed to approximate solutions of linear elliptic partial differential equations with high-contrast, spatially varying coefficients. The network, termed the Iterated V-shaped Net (IV-Net), realizes a mapping from the input coefficients and righthand side to the corresponding solution field. The architecture of IV-Net is informed by, and closely resembles, a V-cycle multigrid solver. The IV-Net model is parameterized via convolutional layers defined in the physical domain. For coercive problems with highly heterogeneous coefficients, the proposed network exhibits superior performance relative to a proper orthogonal decomposition (POD) approach and several existing neural operator architectures. For low-frequency oscillatory Helmholtz problems with smooth coefficients, its performance is similar to that of a Fourier neural operator. We analyze the approximation error and convergence behavior of IV-Net, its data efficiency, and its dependence on the underlying discretization mesh. Furthermore, we demonstrate the practical effectiveness of the architecture through a series of numerical experiments, including applications to uncertainty quantification, inverse problems, and prediction of quantities of interest.
| Comments: | 36 pages |
| Subjects: | Numerical Analysis (math.NA); Machine Learning (cs.LG) |
| Cite as: | arXiv:2605.24876 [math.NA] |
| (or arXiv:2605.24876v1 [math.NA] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24876 arXiv-issued DOI via DataCite (pending registration) |
From: Shan Zhong [view email]
[v1]
Sun, 24 May 2026 05:37:11 UTC (2,197 KB)
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