Data sparsity is a common issue to train machine learning tools such as neural networks for engineering and scientific applications, where experiments and simulations are expensive. Recently physics-constrained neural networks (PCNNs) were developed to reduce the required amount of training data. However, the weights of different losses from data and physical constraints are adjusted empirically in PCNNs. In this paper, a new physics-constrained neural network with the minimax architecture (PCNNMM) is proposed so that the weights of different losses can be adjusted systematically. The training of the PCNN-MM is searching the high-order saddle points of the objective function. A novel saddle point search algorithm called Dual-Dimer method is developed. It is demonstrated that the Dual-Dimer method is computationally more efficient than the gradient descent ascent method for nonconvex-nonconcave functions and provides additional eigenvalue information to verify search results. A heat transfer example also shows that the convergence of PCNN-MMs is faster than that of traditional PCNNs.

本文提出了一种使用极小极大结构的物理约束神经网络（PCNN-MM），同时开发了一种新的鞍点搜索算法Dual-Dimer方法来调节不同约束带来的损失权重。结果表明，该方法的收敛速度快于传统的PCNNs，并能提供额外的特征值信息以验证搜索结果，通过热传递的案例展示了PCNN-MMs的应用。

一种双二聚法用于训练具有Minimax网络结构的受物理约束的神经网络