[论文解读] Physics-Consistent Neural Networks for Learning Deformation and Director Fields in Microstructured Media with Loss-Based Validation Criteria

Milad Shirani, Pete H. Gueldner+3 more|arXiv (Cornell University)|Mar 6, 2026

Model Reduction and Neural Networks被引用 0

一句话总结

本文提出一个带单一定向方向向量的Cosserat弹性物理信息神经网络框架，将基于能量的损失与基于物理的验证结合起来——通过准凸性、秩一凸性和 Legendre-Hadamard 不等式来确保能量稳定的平衡解。

ABSTRACT

In this work, we study the mechanical behavior of solids with microstructure using the framework of Cosserat elasticity with a single unit director. This formulation captures the coupling between deformation and orientational fields that arises in many structured materials. To compute equilibrium configurations of such media, we develop two complementary computational approaches: a finite element formulation based on variational principles and a neural network-based solver that directly minimizes the total potential energy. The neural architecture is constructed to respect the fundamental kinematic structure of the theory. In particular, it enforces frame invariance of the energy, satisfies the unit-length constraint on the director field, and represents deformation and director fields through separate networks to preserve their kinematic independence in the variational setting. Beyond satisfying balance laws, however, physically admissible solutions must also correspond to stable energy minimizers. To assess this requirement, we derive the quasiconvexity condition, rank-one convexity condition, and the Legendre-Hadamard inequalities for the Cosserat model and formulate them in a manner suitable for evaluating neural network predictions. These necessary stability conditions provide a physics-based validation framework: network outputs that violate these necessary conditions cannot correspond to stable energy minimizers and can therefore be rejected. In this way, we integrate classical variational stability theory with modern machine-learning solvers, establishing a computational workflow in which equilibrium solutions are not only learned but also assessed for energetic consistency.

研究动机与目标

• 使用单位定向向量的Cosserat弹性来表征微观结构材料以捕捉形变–方向耦合。
• 开发两种计算求解器：基于变分原理的有限元法和最小化总势能的神经网络求解器。
• 在神经网络中施加框架不变性和定向向量单位长度约束以保持运动学。
• 导出并应用必要的变分稳定性条件（准凸性、秩一凸性、Legendre-Hadamard）作为基于物理的验证框架。

提出的方法

• 通过Cosserat介质中的虚功原理来建立平衡定律和边界条件。
• 构建表示变形和定向向量场的神经网络结构，并强制实现能量框架不变性和单位长度约束。
• 使用总势能作为网络训练的损失函数，并通过推导得到的准凸性、秩一凸性和 Legendre-Hadamard 条件来评估稳定性。
• 从一阶变分和边界项导出基于能量的变分条件，并将其与神经网络输出联系起来。
• 提供一个基于变分原理的互补有限元公式用于基准测试和验证。
• 将稳定性条件以便于神经网络验证的形式表达，使得能够拒绝能量上非极小的预测。

实验结果

研究问题

• RQ1 如何在遵循理论运动学结构的同时，从数据中学习带单一定向向量的Cosserat弹性？
• RQ2 神经网络预测能否在超越平衡定律的能量极小化条件下进行验证？
• RQ3 准凸性、秩一凸性及 Legendre-Hadamard 不等式如何转化为对基于NN的求解器的实际检查？
• RQ4 在强制框架不变性和单位定向向量约束下，对学习解的稳定性和准确性有何影响？

主要发现

• 作者建立了Cosserat介质（单位定向向量）所需的稳定性条件——准凸性、秩一凸性和 Legendre-Hadamard 不等式。
• 展示了如何在满足平衡定律的同时，将这些条件作为物理基础的验证用于神经网络输出的评估。
• 提出一个与物理一致的NN求解器，在尊重运动学约束的同时最小化总势能。
• 该框架将变分力学与机器学习结合起来，确保平衡解在能量上一致。
• 基于变分原理的有限元公式用于对神经求解器进行互补的交叉验证。