IV-Net: A neural network for elliptic PDEs with random and highly varying coefficients
Provides an efficient and accurate neural operator for solving PDEs with high-contrast coefficients, benefiting computational science and engineering applications like uncertainty quantification and inverse problems.
IV-Net, a neural operator inspired by V-cycle multigrid solvers, approximates solutions to elliptic PDEs with highly varying coefficients, outperforming POD and several neural operators for heterogeneous problems while matching Fourier neural operators for smooth Helmholtz problems.
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.