基于物理信息的神经网络求解曲面上对流扩散方程

Physics-informed Neural Networks for solving convection-diffusion equations on surfaces

引入基于物理信息的神经网络PINNs(Physics-informed Neural Networks)并将其用于求解曲面对流扩散方程。区别于传统的神经网络模型,PINNs在建立模型过程中引入了自动微分技术,并将物理信息即偏微分方程信息编译其中,通过定义损失函数得到关于该模型中神经网络参数即权重和偏置的优化目标,随后利用已有的优化算法进行求解。显而易见,PINNs通过添加额外的物理信息约束放宽了对于数据量的要求,对于一个确定性模型显示出更好的鲁棒性。本文基于曲面微分算子与欧氏空间下标准微分算子的解析关系,引入两种曲面微分算子处理技术,即非本征技术和嵌入技术,并结合PINNs针对定义在高维复杂曲面上的对流扩散方程进行求解,多个数值算例证明了该方法的有效性、鲁棒性以及其在求解此类问题的潜力。

This paper introduces Physics-informed Neural Networks and applies them to surface partial differential equations.Different from the traditional Neural Network model,PINNs introduce an automatic differentiation technology in the process of establishing the model and encoding the physical information.By defining the loss function,the optimization goal of the neural network parameters in the model including weightings and bias can be obtained explicitly and solved by existing optimization algorithms.Obviously,PINNs reduce the requirements for the amount of data by adding additional physical information constraints,and show better robustness for some deterministic models.Based on the analytical relationship between surface differential operators and standard differential operators in Euclidean space,this paper introduces two techniques including the extrinsic way and embedding way and combines them with PINNs to solve the convection-diffusion equations defined on high-dimensional complex surfaces.Numerous numerical examples prove the effectiveness,robustness and potential of this method in solving such problems.