基于数值稳定型神经网络的Villain-Lai-Das Sarma方程的动力学标度行为研究  

Study on dynamic scaling behavior of Villain-Lai-Das Sarma equation based on numerically stable nueral networks

作　　者：宋天舒 夏辉[2] Song Tian-Shu;Xia Hui(School of Information and Control Engineering,China University of Mining and Technology,Xuzhou 221116,China;School of Materials Science and Physics,China University of Mining and Technology,Xuzhou 221116,China)

机构地区：[1]中国矿业大学信息与控制工程学院,徐州221116 [2]中国矿业大学材料与物理学院,徐州221116

出　　处：《物理学报》2024年第16期28-35,共8页Acta Physica Sinica

基　　金：中央高校基本科研业务费专项资金(批准号:2024QN11021)资助的课题.

摘　　要：Villain-Lai-Das Sarma(VLDS)方程因其能够有效描述分子束外延生长过程而在表面生长动力学等领域中备受关注.然而,长程关联噪声驱动下的VLDS方程的标度结果尚不明确,不同解析近似方法所得的标度结果仍不自洽.在数值模拟方面,由于非线性项的存在,VLDS方程一直存在数值发散的问题.当前主要引入指数衰减技术替换非线性项以缓解数值发散的问题,但是最近研究表明,这种方法会导致所获得的标度指数发生歧变.因此本文基于深度神经网络来表征VLDS方程中的各个确定项,并基于数值稳定型神经网络分别对含长程时间和空间关联噪声的VLDS系统进行有效的数值模拟.结果表明,我们所构建的深度神经网络具有良好的数值计算稳定性和泛化性,可以获得不同关联噪声驱动下的VLDS方程的可靠标度指数.同时,本文还发现长程时间关联噪声驱动的VLDS系统在时间关联指数较大时呈现谷堆状的表面形貌,而空间关联噪声驱动下的表面形貌则仍然呈现自仿射分形结构.The Villain-Lai-Das Sarma(VLDS)equation has received much attention in surface growth dynamics due to its effective description of molecular beam epitaxy(MBE)growth process.However,the scaling exponent of the VLDS equation driven by long-range correlated noise is still unclear,because different analytical approximation methods yield inconsistent results.The nonlinear term in the VLDS equation challenges the numerical simulation methods,which often leads to the problem of numerical divergence.In the existing numerical approaches,the exponential decay techniques are mainly used to replace nonlinear terms to alleviate the numerical divergence.However,recent studies have shown that these methods may change the scaling exponent and universality class of the growth system.Therefore,we propose a novel deep neural network-based method to address this problem in this work.First,we construct a fully convolutional neural network to characterize the deterministic terms in the VLDS equation.To train the neural network,we generate training data by using the traditional finite-difference method before numerical divergence occurs.Then,we train the neural network to represent the deterministic terms,and perform simulations of VLDS driven by long-range temporally and spatially correlated noises based on the neural networks.The simulation results demonstrate that the deep neural networks constructed here possess good numerical stability.It can obtain reliable scaling exponents of the VLDS equation driven by different uncorrelated noise and correlated noise.Furthermore,in this work,it is also found that the VLDS system driven by long-range correlated noise exhibits a mound-like morphology when the temporal correlation exponent is large enough,while the growing surface morphology driven by spatially correlated noise still presents a self-affine fractal structure,independent of the spatial correlation exponent.

关 键 词：神经网络 分子束外延生长 Villain-Lai-Das Sarma方程 动力学标度 

分 类 号：O175[理学—数学] TP183[理学—基础数学]