Physics-informed neural networks (PINNs) are neural networks (NNs) that directly encode model equations, like Partial Differential Equations (PDEs), in the network itself. While most of the PINN algorithms in the literature minimize the local residual of the governing equations, there are energy-based approaches that take a different path by minimizing the variational energy of the model. We show that in the case of the steady thermal equation weakly coupled to magnetic equation, the energy-based approach displays multiple advantages compared to the standard residual-based PINN: it is more computationally efficient, it requires a lower order of derivatives to compute, and it involves less hyperparameters. The analyzed benchmark problems are the single- and multi-objective optimal design of an inductor for the controlled heating of a graphite plate. The optimized device is designed involving a multi-physics problem: a time-harmonic magnetic problem and a steady thermal problem. For the former, a deep neural network solving the direct problem is supervisedly trained on Finite Element Analysis (FEA) data. In turn, the solution of the latter relies on a hypernetwork that takes as input the inductor geometry parameters and outputs the model weights of an energy-based PINN (or ePINN). Eventually, the ePINN predicts the temperature field within the graphite plate.

翻译：物理信息神经网络（PINNs）是一种将模型方程（如偏微分方程）直接编码于网络内部的神经网络。现有文献中的大多数PINN算法主要通过对控制方程的局部残差进行最小化来实现，而基于能量的方法则另辟蹊径，通过最小化模型的变分能量来求解。本文研究表明，在处理与磁方程弱耦合的稳态热方程时，基于能量的方法相较于标准基于残差的PINN展现出多重优势：计算效率更高、所需导数阶数更低、且涉及的超参数更少。所分析的基准问题聚焦于石墨板可控加热感应器的单目标与多目标优化设计。该优化设备的设计涉及多物理场问题：时谐磁场问题与稳态热问题。针对前者，通过监督训练一个深度神经网络基于有限元分析数据求解正问题；针对后者，其求解依赖于一个超网络，该网络以感应器几何参数作为输入，并输出基于能量的PINN（或称ePINN）的模型权重。最终，该ePINN可预测石墨板内部的温度场分布。