标题： 基于PCG的神经求解器用于周期性微观结构的高分辨率均质化 显示英文标题

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

摘要： 周期微结构的力学性能在各种工程应用中至关重要。 均质化理论是通过在代表性体积单元上平均复杂微结构的行为来预测这些性能的强大工具。 然而，传统的均质化问题数值求解器计算成本较高，尤其是在高分辨率和复杂拓扑及几何情况下。 现有的基于学习的方法虽然有前景，但在这种情况下往往在准确性和泛化能力上存在困难。 为了解决这些挑战，我们提出了CGINS，这是一种预条件共轭梯度求解器启发的神经网络，用于求解均质化问题。 CGINS利用稀疏和周期性的三维卷积，以确保结构周期性的同时实现高分辨率学习。 它具有多级网络架构，有助于在不同尺度上进行有效学习，并采用最小势能作为无标签的损失函数进行自监督学习。 集成的预条件共轭梯度迭代确保网络提供适合PCG的初始解，从而实现快速收敛和高精度。 此外，CGINS施加全局位移约束以确保物理一致性，解决了之前方法依赖Dirichlet锚点的关键局限性。 在包含多种拓扑和材料配置的大规模数据集上评估，CGINS实现了最先进的准确性（相对误差低于1%），并且优于基于学习的基线方法和GPU加速的数值求解器。 值得注意的是，它在保持物理可靠预测的同时，在分辨率达到$512^3$的情况下，比传统方法快2到10倍。 显示英文摘要

摘要： 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 $512^3$.