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:此外,作者们还表明,在涉及许多工程问题时,设计以一阶衍生物为基础的损失函数时,设计以一阶衍生物为基础的损失函数的精确度要高得多,特别是当域上存在异质性和可变跳跃时。PINN所谓的混合配方适用于基本的工程问题,例如线性动力与传播问题之间的平衡。在这项工作中,拟议的混合配方进一步扩展,以解决多物理问题。特别是,我们侧重于一个固定的热机械结合的方程式系统,可用于设计先进材料的微结构。首先,讨论了连续的未经监督的培训,以及第二,完全互不监督的学习。每种方法的结果在准确性和相应的计算成本方面进行了比较。最后,通过将数据和物理结合起来,将转移学习的概念用于解决网络预测系统对无形案例的反应的能力问题。这项工作的结果将是用于多种复杂案例的其他系统。