Physics-informed graph convolutional neural network for modeling fluid flow and heat convection

Abstract

This paper introduces a novel surrogate model for two-dimensional adaptive steady-state thermal convection fields based on deep learning technology. The proposed model aims to overcome limitations in traditional frameworks caused by network types, such as the requirement for extensive training data, accuracy loss due to pixelated preprocessing of original data, and inability to predict information near the boundaries with precision. We propose a new framework that consists primarily of a physical-informed neural network (PINN) and a graph convolutional neural network (GCN). The GCN serves as the prediction module and predicts thermal convection in the two-dimensional computational domain by considering the mutual influence between unstructured nodes and their neighbors. On the other hand, the PINN acts as the physical constraint module of the framework by embedding the control equation of thermal convection into the loss function of the neural network, ensuring that the inference and prediction results of the GCN comply with the constraints of the control equation. The advantages of this framework lie in two aspects. First, the computation mechanism of the GCN is more in line with the actual evolution of temperature fields. Second, the PINN enhances the cognitive ability of the surrogate model toward the convection field information. It accurately describes the changes of temperature gradient information at the boundary position and reduces the model's demand for training data. To validate the advantages of the proposed model, we gradually analyzed the model's geometric adaptability and predictive accuracy from the single cylinder case to the double cylinder case. We also investigated the impact of the number of sampling points on model training and compared the model's prediction results with those of a purely data-driven model. The results show that the proposed model exhibits good geometric adaptability and stability. With only 20 training data, the mean error of the proposed model in predicting the velocity and temperature field is less than 1% and 0.6% for the single cylinder, and less than 2% and 1% for the double cylinder case, while the mean error of the purely data-driven GCN model in predicting the velocity and temperature field is 9.4% and 6.4% for the double cylinder case. These findings demonstrate the effectiveness of the proposed physics-informed graph convolutional neural network, allowing for more accurate prediction of fluid flow and heat convection using surrogate model.

A novel discretized physics-informed neural network model applied to the Navier–Stokes equations

“…A notable development is the introduction of automatic differentiation [1], which makes it possible to bypass numerical discretization and mesh generation, leading to a new machine learning-based technique known as PINN [2,3]. Since the introduction of the PINN, it has been widely used to approximate solution to PDE [4][5][6][7][8][9][10][11][12][13][14][15][16], particularly highly nonlinear ones with non-convex and oscillating behavior, such as Navier-Stokes equations, that pose challenges for traditional numerical discretization techniques. The problematic part of Navier-Stokes equations is because of the convective term in the equations introduces a nonlinearity due to the product of velocity components.…”