Machine learning (ML) has been used to solve multiphysics problems like thermoelasticity through multi-layer perceptron (MLP) networks. However, MLPs have high computational costs and need to be trained for each prediction instance. To overcome these limitations, we introduced an integrated finite element neural network (I-FENN) framework to solve transient thermoelasticity problems in Abueidda and Mobasher (2024). This approach used a physics-informed temporal convolutional network (PI-TCN) within a finite element scheme for solving transient thermoelasticity problems. In this paper, we introduce an I-FENN framework using a new variational TCN model trained to minimize the thermoelastic variational form rather than the strong form of the energy balance. We mathematically prove that the I-FENN setup based on minimizing the variational form of transient thermoelasticity still leads to the same solution as the strong form. Introducing the variational form to the ML model brings the advantages of lower requirement for the differentiability of the basis function and, thus, lower memory requirement and higher computational efficiency. Also, it automatically satisfies zero Neumann boundary conditions, thus reducing the complexity of the loss function. The formulation based on the variational form complies with thermodynamic requirements. The proposed loss function reduces the difference between predicted and target data while minimizing the variational form of thermoelasticity equations, combining the benefits of both data-driven and variational methods. In addition, this study uses finite element shape functions for spatial gradient calculations and compares their performance against automatic differentiation. Our results reveal that models leveraging shape functions exhibit higher accuracy in capturing the behavior of the thermoelasticity problem and faster convergence. Adding the variational term and using shape functions for gradient calculations ensure better adherence to the underlying physics. We demonstrate the capabilities of this I-FENN framework through multiple numerical examples. Additionally, we discuss the convergence of the proposed variational TCN model and the impact of hyperparameters on its performance. The proposed approach offers a well-founded and flexible platform for solving fully coupled thermoelasticity problems while retaining computational efficiency, where the efficiency scales proportional to the model size.

中文翻译：

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I-FENN 热弹性的变分时间卷积网络


机器学习 (ML) 已被用于通过多层感知器 (MLP) 网络解决热弹性等多物理问题。然而，MLP 的计算成本很高，并且需要针对每个预测实例进行训练。为了克服这些限制，我们在 Abueidda 和 Mobasher (2024) 中引入了集成有限元神经网络 (I-FENN) 框架来解决瞬态热弹性问题。该方法在有限元方案中使用物理信息时间卷积网络 (PI-TCN) 来解决瞬态热弹性问题。在本文中，我们引入了一个 I-FENN 框架，该框架使用新的变分 TCN 模型，经过训练可以最小化热弹性变分形式而不是能量平衡的强形式。我们在数学上证明，基于最小化瞬态热弹性变分形式的 I-FENN 设置仍然会产生与强形式相同的解。将变分形式引入机器学习模型具有对基函数可微性要求较低的优点，从而降低了内存需求，提高了计算效率。此外，它自动满足零诺依曼边界条件，从而降低了损失函数的复杂性。基于变分形式的公式符合热力学要求。所提出的损失函数减少了预测数据和目标数据之间的差异，同时最小化热弹性方程的变分形式，结合了数据驱动和变分方法的优点。此外，本研究使用有限元形状函数进行空间梯度计算，并将其性能与自动微分进行比较。 我们的结果表明，利用形状函数的模型在捕获热弹性问题的行为方面表现出更高的准确性，并且收敛速度更快。添加变分项并使用形状函数进行梯度计算可确保更好地遵循基础物理原理。我们通过多个数值示例展示了该 I-FENN 框架的功能。此外，我们还讨论了所提出的变分 TCN 模型的收敛性以及超参数对其性能的影响。所提出的方法提供了一个基础良好且灵活的平台，用于解决完全耦合的热弹性问题，同时保持计算效率，其中效率与模型大小成正比。