用双形式神经网络提升复合材料均质化精度与可靠性
Homogenization with Guaranteed Bounds via Primal-Dual Physically Informed Neural Networks
- 提出物理信息神经网络的对偶形式，解决材料参数突变问题
- 可生成上下界误差估计，有效检测模型失效
- 适合微力学均质化研究者，尤其关注可靠性验证
物理信息神经网络（PINNs）在多尺度建模相关偏微分方程求解中表现良好，但在处理具有间断系数的材料（如分段常数性质介质）时常失效。本文提出一种针对强形式与变分（弱）形式的双形式框架，提升周期性热导复合材料均质化的可靠性。该方法能导出保证的上下界误差，增强对PINN失败的鲁棒检测能力。通过对比平滑近似下的标准PINNs与使用谱基和神经网络测试函数的变分PINNs（VPINNs），发现强形式PINNs在受控环境下可能更优，但对材料不连续敏感且缺乏明确诊断信号；而VPINNs可直接处理分段常数参数，但需谨慎选择测试函数以防不稳定。对偶形式成为收敛质量的可靠指标，其集成显著增强了PINN在微力学均质化中的适用性。
原文摘要 · Abstract (English)
Physics-informed neural networks (PINNs) have shown promise in solving partial differential equations (PDEs) relevant to multiscale modeling, but they often fail when applied to materials with discontinuous coefficients, such as media with piecewise constant properties. This paper introduces a dual formulation for the PINN framework to improve the reliability of the homogenization of periodic thermo-conductive composites, for both strong and variational (weak) formulations. The dual approach facilitates the derivation of guaranteed upper and lower error bounds, enabling more robust detection of PINN failure. We compare standard PINNs applied to smoothed material approximations with variational PINNs (VPINNs) using both spectral and neural network-based test functions. Our results indicate that while strong-form PINNs may outperform VPINNs in controlled settings, they are sensitive to material discontinuities and may fail without clear diagnostics. In contrast, VPINNs accommodate piecewise constant material parameters directly but require careful selection of test functions to avoid instability. Dual formulation serves as a reliable indicator of convergence quality, and its integration into PINN frameworks enhances their applicability to homogenization problems in micromechanics.
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