Accelerating Conjugate Gradient Solvers for Homogenization Problems with Unitary Neural Operators

TL;DR

UNO-CG is introduced, a hybrid solver that accelerates conjugate gradient (CG) solvers using specially designed machine-learned preconditioners, while ensuring convergence by construction and maintaining strong performance across a variety of boundary conditions.

Abstract

Rapid and reliable solvers for parametric partial differential equations (PDEs) are needed in many scientific and engineering disciplines. For example, there is a growing demand for composites and architected materials with heterogeneous microstructures. Designing such materials and predicting their behavior in practical applications requires solving homogenization problems for a wide range of material parameters and microstructures. While classical numerical solvers offer reliable and accurate solutions supported by a solid theoretical foundation, their high computational costs and slow convergence remain limiting factors. As a result, scientific machine learning is emerging as a promising alternative. However, such approaches often lack guaranteed accuracy and physical consistency. This raises the question of whether it is possible to develop hybrid approaches that combine the advantages of both data-driven methods and classical solvers. To address this, we introduce UNO-CG, a hybrid solver that accelerates conjugate gradient (CG) solvers using specially designed machine-learned preconditioners, while ensuring convergence by construction. As a preconditioner, we propose Unitary Neural Operators as a modification of Fourier Neural Operators. Our method can be interpreted as a data-driven discovery of Green's functions, which are then used to accelerate iterative solvers. We evaluate UNO-CG on various homogenization problems involving heterogeneous microstructures and millions of degrees of freedom. Our results demonstrate that UNO-CG enables a substantial reduction in the number of iterations and is competitive with handcrafted preconditioners for homogenization problems that involve expert knowledge. Moreover, UNO-CG maintains strong performance across a variety of boundary conditions, where many specialized solvers are not applicable, highlighting its versatility and robustness.

Figures (21)

• Figure 1: Different techniques for solving parametric PDEs, including a direct solver, iterative solver (\ref{['ssec:introduction-iterative-solvers']}), machine-learned surrogate (\ref{['ssec:introduction-inexact']}), and a hybrid solver with a machine-learned preconditioner (\ref{['ssec:introduction-hybrid-solvers']}).
• Figure 2: Architecture of a neural operator model with a lifting operator $\mathsf{V}_{\mathrm{lift}}$, several neural operator (NO) layers, and a projection operator $\mathsf{V}_{\mathrm{proj}}$. Figure inspired by Li_fno_2021.
• Figure 3: Architecture of a Fourier layer based on the Fourier transform ${\cal F}$ that features a learnable kernel $\ul{\ul{K}} {}^{\{l\}}$ in Fourier space and a learnable bypass $\ul{\ul{W}} {}^{\{l\}}$. Figure inspired by Li_fno_2021.
• Figure 4: Figures of a regular grid in $d=2$ dimensions with free nodes (green), periodic nodes (blue), and fixed nodes (red) for periodic BC (left), Dirichlet BC (middle), and mixed BC (right).
• Figure 5: Structure of the FANS preconditioner based on the Fourier transform ${\cal F}$ and a precomputed fundamental solution in Fourier space $\ul{\ul{\widehat{\Phi}}}$. Note the analogy to \ref{['fig:fno-architecture']}.

Theorems & Definitions (4)

• lemma 1
• Remark
• Remark
• proof : Proof of \ref{['spectrum-lemma']}