Deep learning methods find a solution to a boundary value problem by defining loss functions of neural networks based on governing equations, boundary conditions, and initial conditions. Furthermore, the authors show that when it comes to many engineering problems, designing the loss functions based on first-order derivatives results in much better accuracy, especially when there is heterogeneity and variable jumps in the domain \cite{REZAEI2022PINN}. The so-called mixed formulation for PINN is applied to basic engineering problems such as the balance of linear momentum and diffusion problems. In this work, the proposed mixed formulation is further extended to solve multi-physical problems. In particular, we focus on a stationary thermo-mechanically coupled system of equations that can be utilized in designing the microstructure of advanced materials. First, sequential unsupervised training, and second, fully coupled unsupervised learning are discussed. The results of each approach are compared in terms of accuracy and corresponding computational cost. Finally, the idea of transfer learning is employed by combining data and physics to address the capability of the network to predict the response of the system for unseen cases. The outcome of this work will be useful for many other engineering applications where DL is employed on multiple coupled systems of equations.

翻译：深度学习方法通过基于控制方程、边界条件和初始条件定义神经网络的损失函数来求解边值问题。此外，作者指出，在众多工程问题中，基于一阶导数设计损失函数能显著提升精度，尤其当域中存在异质性和变量跳跃时\cite{REZAEI2022PINN}。这种所谓的PINN混合列式已应用于基本工程问题，如线性动量守恒和扩散问题。本文进一步将该混合列式拓展至多物理问题求解，重点研究可用于先进材料微结构设计的稳态热-力耦合方程组。讨论采用两种方法：其一为序贯无监督训练，其二为全耦合无监督学习。从精度和计算成本两方面对两种方法的结果进行对比。最后，结合数据与物理场融合迁移学习思想，评估网络对未见工况系统响应的预测能力。本研究成果对涉及多耦合方程组的深度学习工程应用具有重要参考价值。