基于偏微分方程的声波神经网络  

Sound wave neural network based on partial differential equation

作　　者：孙卫涛 杨志芳 晏信飞 Sun Weitao;Yang Zhifang;Yan Xinfei(School of Aerospace Engineering,Tsinghua University,Beijing 100084,China;Zhou Pei-Yuan Center for Applied Mathematics,Tsinghua University,Beijing 100084,China;Geophysical Department,RIPED,PetroChina,Beijing 100083,China)

机构地区：[1]清华大学航天航空学院,北京100084 [2]清华大学周培源应用数学研究中心,北京100084 [3]中国石油集团科学技术研究院有限公司,北京100083

出　　处：《地球与行星物理论评（中英文）》2024年第2期257-266,共10页Reviews of Geophysics and Planetary Physics

基　　金：国家自然科学基金资助项目(42074144,41874137);中国石油天然气集团有限公司“十四五”前瞻性基础性项目(2021DJ3502)。

摘　　要：神经网络是一种重要的机器学习算法,在地球物理学等领域的应用得到了迅速发展,这主要得益于其在数据建模、信号处理和图像识别等方面的强大能力.然而,神经网络的数学基础和物理解释仍然十分不足,模型内部复杂性使得难以解释其决策过程,限制了神经网络的进一步发展.利用数学和物理方法解释神经网络的行为仍然是一个具有挑战性的任务.本文目标是从声波偏微分方程和有限差分方法出发设计一个声波神经网络结构,该方法将一阶声波方程转化为基于有限差分的离散化声波方程,声波方程有限差分格式与神经网络传播函数具有近似的数学表达形式,可以构建一种基于声波传播物理模型的神经网络.声波神经网络的主要特点是:(1)具有压力-速度耦合结构和层间跳跃连接的神经网络;(2)主变量-伴随变量双流网络结构改善了训练中的梯度消失问题.从声波偏微分方程和有限差分算法出发建立的声波神经网络具有良好数学基础和清晰的物理解释,为在数学和物理方法框架内提高网络性能提供了可行性.数值计算结果表明,声波神经网络在CIFAR-10和CIFAR-100数据集的图像分类任务中性能有明显提升,优于传统残差神经网络.偏微分方程神经网络建模方法可以应用于许多其他类型的数学物理方程,并为深度神经网络算法提供数学和物理解释.Applications of neural network algorithms in rock physics have developed rapidly developed,mainly due to the neural network's powerful abilities in data modeling,signal processing,and image recognition.However,mathematical and physical explanations of neural networks remain limited,which makes it difficult to understand the behavior and mechanism of neural networks and limits their further development.Using mathematical and physical methods to explain the behavior of neural networks remains a challenging task.The goal of this study was to design a sound wave neural network(SWNN)structure based on sound wave partial differential equations and finite difference methods.The method transforms the first-order sound equations into the frequency domain and discretizes them using a central difference scheme.The differential formula takes the same form as the propagation function of a neural network,enabling the construction of a sound wave neural network.The main features of the SWNN are(1)a neural network with explicitly coupled pressure-velocity streams and inter-layer connections and(2)an adjoint variable method to improve the vanishing gradient problem in network training.The sound wave neural network established from the sound wave partial differential equation and finite difference algorithm has a solid mathematical modeling process and a clear physical explanation.This makes improving network performance within the framework of the mathematical and physical methods feasible.The numerical results showed that SWNN outperforms residual neural networks in image classification on CIFAR-10 and CIFAR-100 datasets.The partial differential equation neural network modeling method proposed in this paper can be applied to many other types of mathematical physics equations,providing a deep mathematical explanation for neural networks.

关 键 词：可解释神经网络 声波方程 伴随变量法 双流网络 

分 类 号：P315[天文地球—地震学]