Simulation of flow field in silicon single-crystal growth using physics-informed neural network with spatial information

Melt convection plays a crucial role in the growth of silicon single crystals. In particular, melt flow transfers mass and heat, and it may strongly affect the crystal growth conditions. Understanding and controlling convection remains a significant challenge in industrial crystal production. Currently, numerical methods such as the finite element method and the finite volume method are mainly used to simulate melt convection in the crystal growth process. However, these methods are not suitable for most applications with real-time requirements. Physics-informed neural networks (PINNs) have the advantages of fast calculation and wide application. They provide a new concept for the numerical solutions of nonlinear partial differential equations (PDEs). This paper proposes a PINN with spatial information to solve the silicon melt flow model, which does not depend on any simulation data. As the network depth (number of layers) increases, the derivative information in the PDE loss becomes weak, which reduces the expression of the original features in the loss function. Therefore, this study introduces spatial information into the hidden layer of the network, thereby enhancing the correlation between the network and the original input and improving the expression ability of the network. Specifically, silicon melt flow models under three rotating conditions are considered. Compared with other methods, the proposed algorithm can accurately capture regions with complex local morphology. The experimental results reveal the flow characteristics of the silicon melt and confirm the effectiveness of the proposed algorithm. All codes and data attached to this manuscript are publicly available on the following websites: https://github.com/callmedrcom/SIPINN.