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.

翻译：物理信息神经网络（PINNs）是一种通过基于控制方程、边界条件和初始条件定义神经网络损失函数来求解边值问题的新工具。近期研究表明，在为许多工程问题设计损失函数时，采用一阶导数并结合强形式与弱形式的方程，能够显著提升精度，尤其是在域内存在非均匀性和变量突变的场景下。这种新方法借鉴了混合有限元法的思想，被称为PINNs的混合格式。在该方法中，偏微分方程被重构为一组方程组，其中主未知量为解的流量或梯度，而次未知量则为解本身。本研究提出将混合格式应用于求解多物理场问题，具体针对稳态热力耦合方程组。此外，我们讨论了顺序训练与全耦合无监督训练，并比较了二者的精度与计算成本。为提高网络精度，我们引入硬边界约束以保障预测有效性，随后探究不同优化器与架构对精度和效率的影响。最后，我们提出一种类似于迁移学习的参数化学习简易方法，该方法结合数据与物理知识，旨在弥补PINNs在计算成本方面的局限性，并提升网络对未见工况下系统响应的预测能力。本研究成果将对其他采用深度学习进行多耦合方程组快速可靠计算的工程应用具有重要参考价值。