Physics-informed graph convolutional neural network for modeling geometry-adaptive steady-state natural convection

This paper presents a novel deep learning-based surrogate model for steady-state natural convection problem with variable geometry. Traditional deep learning based surrogate models are more or less limited by the requirement for large amounts of training data, loss of accuracy due to pixelization of the original data, the low accurate prediction near the boundaries and low geometry adaptive capability. To overcome the above challenges, the proposed natural convection prediction framework is mainly composed of a physics-informed neural network (PINN) and a graph convolutional neural network (GCN), called natural convection prediction model based on physics-informed graph convolutional network (NCV-PIGN). The GCN serves as the prediction module, inferring and predicting natural convection phenomena by considering the interactions between unstructured nodes and their neighbor; the PINN incorporates the governing equations of natural convection into the loss function of the neural network, allowing the predictions from GCN to satisfy the constraints imposed by the physical laws. The advantages of this framework are twofold: the operation principles of the GCN better align with the development of the temperature field in real situations, and the embedding of physical information strengthens the model's understanding of the flow field, accurately describing the variations of temperature gradients at the boundary positions while reducing the model's reliance on training data. Finally, to validate the superiority of the NCV-PIGN, we analyze its geometric adaptability and accuracy of prediction using single and dual heat source cases. We compare the model's prediction results at different sampling point quantities and contrast them with those of purely data-driven models. The results demonstrate that the excellent geometric adaptability and prediction capability of the proposed model can be achieved with only 20 training data and once the fully trained the model can solve natural convection problems within 3 ms. The max and mean relative errors in predicting the temperature field are less than 2% and 0.4% for both single and dual heat source cases. Compared to the pure data-driven model, the proposed model has reduced the maximum error by 65.5% and the mean error by 72%. These results validate the effectiveness of the developed NCV-PIGN model, enabling better performance of the deep learning-based surrogate models for natural convection problems.