IV-Net: A neural network for elliptic PDEs with random and highly varying coefficients

2026-05-24

AI Summary

This work proposes IV-Net, a neural operator architecture inspired by V-cycle multigrid methods, to solve linear elliptic partial differential equations with high-contrast, spatially varying coefficients. IV-Net is the first to integrate the V-cycle multigrid structure into neural operator design, employing physical-domain convolutional layers to establish an end-to-end mapping from input coefficients and source terms to the solution field. The method significantly outperforms proper orthogonal decomposition (POD) and existing neural operators in handling highly heterogeneous coefficients, excels in high-contrast scenarios, and matches the performance of Fourier neural operators on low-frequency Helmholtz problems. Comprehensive numerical experiments demonstrate IV-Net’s superior data efficiency and mesh adaptability across diverse settings.

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.

Problem

• elliptic PDEs
• random coefficients
• highly varying coefficients
• neural operator
• heterogeneous media

Innovation

• neural operator
• multigrid-inspired architecture
• elliptic PDEs
• high-contrast coefficients
• convolutional layers in physical domain