# PCG-Informed Neural Solvers for High-Resolution Homogenization of Periodic Microstructures

arXiv: https://arxiv.org/abs/2506.17087 · HTML: https://mlsanity.com/paper/2506.17087/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jun 2025.

## Built on by (most cited first, 1 of 1)

- [AutoMS: Multi-Agent Evolutionary Search for Cross-Physics Inverse Microstructure Design](https://mlsanity.com/paper/2603.27195/) (2026)

## Abstract (arXiv)

The mechanical properties of periodic microstructures are pivotal in various engineering applications. Homogenization theory is a powerful tool for predicting these properties by averaging the behavior of complex microstructures over a representative volume element. However, traditional numerical solvers for homogenization problems can be computationally expensive, especially for high-resolution and complicated topology and geometry. Existing learning-based methods, while promising, often struggle with accuracy and generalization in such scenarios. To address these challenges, we present CGINS, a preconditioned-conjugate-gradient-solver-informed neural network for solving homogenization problems. CGINS leverages sparse and periodic 3D convolution to enable high-resolution learning while ensuring structural periodicity. It features a multi-level network architecture that facilitates effective learning across different scales and employs minimum potential energy as label-free loss functions for self-supervised learning. The integrated preconditioned conjugate gradient iterations ensure that the network provides PCG-friendly initial solutions for fast convergence and high accuracy. Additionally, CGINS imposes a global displacement constraint to ensure physical consistency, addressing a key limitation in prior methods that rely on Dirichlet anchors. Evaluated on large-scale datasets with diverse topologies and material configurations, CGINS achieves state-of-the-art accuracy (relative error below 1%) and outperforms both learning-based baselines and GPU-accelerated numerical solvers. Notably, it delivers 2 times to 10 times speedups over traditional methods while maintaining physically reliable predictions at resolutions up to 5123.