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 problem is the 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）是将模型方程（如偏微分方程）直接编码到网络中的神经网络。尽管现有文献中大多数PINN算法通过最小化控制方程的局部残差进行求解，但存在基于能量的方法另辟蹊径，通过最小化模型的变分能量来求解。我们证明，在稳态热方程与磁场方程弱耦合的情况下，基于能量的方法相比标准残差型PINN具有多重优势：计算效率更高，所需导数阶次更低，且超参数更少。分析的基准问题是用于石墨板受控加热的感应器优化设计。优化装置涉及多物理场问题：时谐磁场问题与稳态热学问题。对于前者，我们利用有限元分析数据对求解正问题的深度神经网络进行监督训练。对于后者，其求解依赖一个超网络——该网络以感应器几何参数为输入，输出基于能量的PINN（简称ePINN）的模型权重。最终，ePINN预测石墨板内部的温度场分布。