Neural Approximate Inverse Preconditioners

TL;DR

This work tackles fast, robust preconditioning for large elliptic PDEs by learning an explicit Green's function with an adaptive multiscale neural network (\(\alpha\mathrm{MSNN}\)). The method integrates coarse-grid anchor data, a multi-$\varepsilon$ staged training protocol, and overlapping domain decomposition to capture near-, middle-, and far-field behavior, then converts the learned kernel into a sparse or \(\mathcal{H}\)-matrix preconditioner with near-linear setup and application costs. It demonstrates fast Krylov convergence across convection-dominated, indefinite, and anisotropic problems, and generalizes to parameterized PDE families, enabling a single model to serve as a robust preconditioner over a range of coefficients. The approach offers practical, geometry-driven preconditioners that preserve spectral properties and scale to large problems, with potential extensions to higher dimensions and time-dependent settings.

Abstract

In this paper, we propose a data-driven framework for constructing efficient approximate inverse preconditioners for elliptic partial differential equations (PDEs) by learning the Green's function of the underlying operator with neural networks (NNs). The training process integrates four key components: an adaptive multiscale neural architecture ($α$MSNN) that captures hierarchical features across near-, middle-, and far-field regimes; the use of coarse-grid anchor data to ensure physical identifiability; a multi-$\varepsilon$ staged training protocol that progressively refines the Green's function representation across spatial scales; and an overlapping domain decomposition that enables local adaptation while maintaining global consistency. Once trained, the NN-approximated Green's function is directly compressed into either a hierarchical ($\mathcal{H}$-) matrix or a sparse matrix-using only the mesh geometry and the network output. This geometric construction achieves nearly linear complexity in both setup and application while preserving the spectral properties essential for effective preconditioning. Numerical experiments on challenging elliptic PDEs demonstrate that the resulting preconditioners consistently yield fast convergence and small iteration counts.

Figures (14)

• Figure 1: Comparison between an approximation of the Green's function and the inverse of the stiffness matrix. Left: A NN approximation of the Green's function of the operator $\mathcal{L}=-\Delta - 50$ obtained, evaluated on a uniform grid over the domain $[0,1]$ with $64$ interior points. Right: Inverse of the finite difference discretization of the same operator on the same grid.
• Figure 1: Effect of approximation accuracy on the prediction accuracy of the learned Green's function for the convection–diffusion-reaction problem \ref{['eq:conv-diff']}. Approximations are compared against a numerical reference solution. Top: training loss history. Bottom left: numerical reference solution. Bottom center: training to a relative loss of $10^{-2}$ yields large errors. Bottom right: training to $10^{-4}$ substantially reduces the error and recovers the expected reaction-induced waves.
• Figure 1: Illustration of $\mathcal{H}$-matrix block structure for 1D (left) and 2D (right) problems.
• Figure 1: Problem size vs number of FGMRES(50) iterations for two problems: a convection-dominated problem with $a(x)=0.01$, $b(x)=1+x^2$, and $c(x)=0.0$; and a reaction-dominated problem with $a(x)=1.0$, $b(x)=0.0$, and $c(x)=-50(1+x^2)$. F indicates that the solver fails to converge within $500$ iterations.
• Figure 2: Gaussian source, residual after subtracting $\phi_1$, and residual after subtracting $\phi_1+\phi_2$ for operator $\mathcal{L}u=-\Delta u$ defined on $[0, 1]$. The progression illustrates how successive components cancel localized structure and refine the multiscale hierarchy at a fixed $y_0=0.5$.