[论文解读] Mixed formulation of physics-informed neural networks for thermo-mechanically coupled systems and heterogeneous domains

Ali Harandi, Ahmad Moeineddin+3 more|arXiv (Cornell University)|Feb 9, 2023

Model Reduction and Neural Networks参考文献 70被引用 8

一句话总结

该论文开发了一种用于异质固体静态热-力学问题的物理信息神经网络（PINN）混合型表述，比较了顺序训练与耦合训练，并引入硬约束与参量学习以提升精度与效率。

ABSTRACT

Physics-informed neural networks (PINNs) are a new tool for solving boundary value problems by defining loss functions of neural networks based on governing equations, boundary conditions, and initial conditions. Recent investigations have shown that when designing loss functions for many engineering problems, using first-order derivatives and combining equations from both strong and weak forms can lead to much better accuracy, especially when there are heterogeneity and variable jumps in the domain. This new approach is called the mixed formulation for PINNs, which takes ideas from the mixed finite element method. In this method, the PDE is reformulated as a system of equations where the primary unknowns are the fluxes or gradients of the solution, and the secondary unknowns are the solution itself. In this work, we propose applying the mixed formulation to solve multi-physical problems, specifically a stationary thermo-mechanically coupled system of equations. Additionally, we discuss both sequential and fully coupled unsupervised training and compare their accuracy and computational cost. To improve the accuracy of the network, we incorporate hard boundary constraints to ensure valid predictions. We then investigate how different optimizers and architectures affect accuracy and efficiency. Finally, we introduce a simple approach for parametric learning that is similar to transfer learning. This approach combines data and physics to address the limitations of PINNs regarding computational cost and improves the network's ability to predict the response of the system for unseen cases. The outcomes of this work will be useful for many other engineering applications where deep learning is employed on multiple coupled systems of equations for fast and reliable computations.

研究动机与目标

• 动机与解决具有异质材料属性的热弹性问题；使用PINN进行求解。
• 引入使用一阶导数和强/弱形式混合的混合PINN表述。
• 在精度与计算成本方面比较顺序和全耦合训练策略。
• 纳入硬边界约束以提升预测的有效性与鲁棒性。
• 提出一种轻量级的参量学习方法以提升对未见材料配置的泛化能力。

提出的方法

• 将具有空间变异的弹性常数和热特性问题建模为静态热弹性问题并推导相应的弱形式。
• 采用混合PINN结构，通过独立的神经网络预测位移、温度及其梯度/应力等主场变量及其导出量。
• 定义结合能量与守恒方程的物理损失项，用于力学和热场的边界与连接损失等。
• 实现两种训练策略：顺序（热-力损失交替最小化）和耦合（同时最小化所有损失）。
• 仅使用一阶导数来计算所需量，从而能够使用类转移的参量学习和硬边界约束以强制物理有效性。
• 引入硬约束与连接损失，将预测的应力和热通量与其本构对应项联系起来，避免鞍点问题。

实验结果

研究问题

• RQ1 一种使用一阶导数的混合PINN表述是否能在异质域中的热力耦合问题上比标准PINN获得更高的精度？
• RQ2 顺序与耦合两种训练策略在多物理PINN的精度与计算成本方面有何差异？
• RQ3 硬边界约束与网络结构对预测的物理有效性与鲁棒性有何影响？
• RQ4 一种参数学习方法（类似迁移学习）是否可以在降低计算成本的同时将预测推广到未见材料属性？

主要发现

• 提出一个混合PINN框架，通过预测位移、温度及其相关通量/应力，并配备专门的子网络来处理。
• 展示了两种训练方案——顺序与耦合，并讨论了它们在精度与计算上的权衡。
• 证明一阶表述与硬约束有助于在异质域中提高收敛性与预测有效性。
• 提出一种简单的参量学习方法，在降低计算成本的同时实现对未见材料配置的泛化。
• 指出材料异质性（空间变化的E、ν、k、α）通过将材料参数作为网络输入来处理，从而实现对不同配置的学习。