PCG-Informed Neural Solvers for High-Resolution Homogenization of Periodic Microstructures
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912624798597120 |
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| author | Xing, Yu Liu, Yang Chen, Lipeng Tang, Huiping Lu, Lin |
| author_facet | Xing, Yu Liu, Yang Chen, Lipeng Tang, Huiping Lu, Lin |
| contents | 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 $512^3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_17087 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | PCG-Informed Neural Solvers for High-Resolution Homogenization of Periodic Microstructures Xing, Yu Liu, Yang Chen, Lipeng Tang, Huiping Lu, Lin Computational Physics Graphics 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 $512^3$. |
| title | PCG-Informed Neural Solvers for High-Resolution Homogenization of Periodic Microstructures |
| topic | Computational Physics Graphics |
| url | https://arxiv.org/abs/2506.17087 |