Recent advances in deep learning for solving partial differential equations (PDEs) have introduced physics-informed neural networks (PINNs), which integrate machine learning with physical laws. Physics-informed convolutional neural networks (PICNNs) extend PINNs by leveraging CNNs for enhanced generalization and efficiency. However, current PICNNs depend on manual design, and inappropriate designs may not effectively solve PDEs. Furthermore, due to the diversity of physical problems, the ideal network architectures and loss functions vary across different PDEs. It is impractical to find the optimal PICNN architecture and loss function for each specific physical problem through extensive manual experimentation. To surmount these challenges, this paper uses automated machine learning (AutoML) to automatically and efficiently search for the loss functions and network architectures of PICNNs. We introduce novel search spaces for loss functions and network architectures and propose a two-stage search strategy. The first stage focuses on searching for factors and residual adjustment operations that influence the loss function, while the second stage aims to find the best CNN architecture. Experimental results show that our automatic searching method significantly outperforms the manually-designed model on multiple datasets.

翻译：近年来，利用深度学习求解偏微分方程（PDEs）的研究取得了进展，其中物理信息神经网络（PINNs）将机器学习与物理定律相结合。物理信息卷积神经网络（PICNNs）通过利用CNN增强了泛化能力和效率，从而扩展了PINNs。然而，现有的PICNNs依赖于人工设计，不当的设计可能无法有效求解PDEs。此外，由于物理问题的多样性，理想的网络架构和损失函数在不同PDEs间存在差异。针对每个具体的物理问题，通过大量人工实验寻找最优的PICNN架构和损失函数是不现实的。为克服这些挑战，本文采用自动化机器学习（AutoML）来自动且高效地搜索PICNNs的损失函数和网络架构。我们引入了新颖的损失函数和网络架构搜索空间，并提出了一种两阶段搜索策略。第一阶段专注于搜索影响损失函数的因子和残差调整操作，而第二阶段旨在寻找最佳的CNN架构。实验结果表明，我们的自动搜索方法在多个数据集上显著优于人工设计的模型。