Research On Convection Process Of Silicon Melt Based On Physics-informed Neural Neural Network

Silicon,as one of the most important semiconductor materials,is the primary material used in the manufacturing of integrated circuit chips.Achieving high-quality and large-size silicon monocrystals is a necessary approach to ensuring the sustainable and steady development of China’s semiconductor and integrated circuit industry.The Czochralski method is a common way of preparing silicon monocrystals.During the growth process of silicon monocrystals,various types of convection occur within the silicon melt,such as forced convection,natural convection,and capillary convection.The interactions between these complex convections affect the temperature distribution,concentration distribution,and impurity distribution of the silicon melt during growth.The temperature distribution is closely related to the amount of heat that drives crystal growth and directly affects the quality of crystal growth.Therefore,predicting the flow velocity and temperature at different points within the silicon melt is of great significance in optimizing crystal growth processes and improving crystal quality.This paper first establishes a convective model of silicon melt in the process of single crystal growth.Secondly,based on the study of the coupling effect between temperature and flow,a thermal flow coupling model of silicon melt is established.Finally,the correctness of the above models is verified by numerical simulation.Considering that traditional numerical simulation methods require the discretization and mesh division of equations when solving the above models,the obtained solution is in a discrete form.The solution at non-grid points depends on interpolation,introducing discretization error and interpolation error,and traditional numerical simulation methods are limited in their use in optimization,control,real-time monitoring and other scenarios.However,Physics-Informed Neural Network(PINN)satisfies the strong form of partial differential equations in the convective model of silicon melt,and does not require discretization of the model.Once training is completed,it can quickly predict the flow velocity of silicon melt at the corresponding position.Therefore,this paper adopts PINN to solve the above models.As traditional PINN is difficult to solve highly nonlinear partial differential equations and has drawbacks such as poor convergence and low accuracy,this article introduces spatial coordinate information into the hidden layer of the PINN network and proposes the Physics-Informed Neural Network with Spatial Information(SIPINN).SIPINN enhances the correlation between the network and the original input,which is helpful in recovering the derivative information of the initial spatial input during backpropagation.This article uses SIPINN to solve the convective model of silicon melt and establishes a silicon melt flow rate prediction model based on SIPINN.Comparative experiments under three different rotation conditions were conducted to verify that the proposed SIPINN has strong convergence ability and high prediction accuracy.In the study of the coupling effect between flow and temperature,the significant differences in loss magnitude between the equations of the silicon melt heat flow coupling model lead to imbalanced weights of each equation during the training process.However,SIPINN lacks a weight adjustment mechanism,and in the process of optimizing the network,it ignores equations with smaller weights,which may ultimately lead to solving errors.To address this issue,this paper proposes a Gradient Normalization Physics-Informed Neural Network(GNPINN),which adjusts the weight automatically by the size of the gradient update of the network’s shared layer parameters for each equation,ensuring equal contribution of all equations in the gradient descent process.Meanwhile,a silicon melt heat flow prediction model based on GNPINN is established to solve the temperature and flow velocity distribution of silicon melt under different Gr numbers and rotation conditions.In the experiments,GNPINN accurately captured the temperature changes caused by natural convection due to increasing Gr numbers,verifying the effectiveness of GNPINN in solving coupled equations.