In this paper, we employed a transfer learning technique to predict the Nusselt number for natural convection flows in enclosures. Specifically, we considered the benchmark problem of a two-dimensional square enclosure with insulated horizontal walls and constant-temperature vertical walls. The Rayleigh and Prandtl numbers are sufficient parameters to simulate this problem numerically. We adopted two approaches to this problem: Firstly, we made use of a multi-grid dataset in order to train our artificial neural network in a cost-effective manner. By monitoring the training losses for this dataset, we detected any significant anomalies that stemmed from an insufficient grid size, which we further corrected by altering the grid size or adding more data. Secondly, we aimed to endow our metamodel with the ability to account for additional input features by performing transfer learning using deep neural networks. We trained a neural network with a single input feature (Rayleigh number) and extended it to incorporate the effects of a second feature (Prandtl number). We also considered the case of hollow enclosures, demonstrating that our learning framework can be applied to systems with higher physical complexity, while bringing the computational and training costs down.

在本文中，我们采用迁移学习技术来预测外壳中自然对流的努塞尔数。具体来说，我们考虑了具有绝缘水平墙和恒温垂直墙的二维方形外壳的基准问题。Rayleigh 和 Prandtl 数足以在数值上模拟这个问题。我们采用了两种方法来解决这个问题：首先，我们利用多网格数据集以经济高效的方式训练我们的人工神经网络。通过监控该数据集的训练损失，我们检测到任何由网格大小不足引起的重大异常，我们通过更改网格大小或添加更多数据进一步纠正了这些异常。第二，我们的目标是通过使用深度神经网络执行迁移学习，使我们的元模型能够解释额外的输入特征。我们训练了一个具有单个输入特征（瑞利数）的神经网络，并将其扩展为包含第二个特征（普朗特数）的影响。我们还考虑了空心外壳的情况，证明我们的学习框架可以应用于具有更高物理复杂性的系统，同时降低计算和培训成本。