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.

本研究探讨了深度学习方法在多物理问题中的应用，特别是通过 PINN 解决热力学力学相互作用等问题，提出了顺序无监督训练和迁移学习的思想，以提高网络的准确性和计算效率。