With modern computational advancements and statistical analysis methods, machine learning algorithms have become a vital part of engineering modeling. Neural Operator Networks (ONets) is an emerging machine learning algorithm as a "faster surrogate" for approximating solutions to partial differential equations (PDEs) due to their ability to approximate mathematical operators versus the direct approximation of Neural Networks (NN). ONets use the Universal Approximation Theorem to map finite-dimensional inputs to infinite-dimensional space using the branch-trunk architecture, which encodes domain and feature information separately before using a dot product to combine the information. ONets are expected to occupy a vital niche for surrogate modeling in physical systems and Digital Twin (DT) development. Three test cases are evaluated using ONets for operator approximation, including a 1-dimensional ordinary differential equations (ODE), general diffusion system, and convection-diffusion (Burger) system. Solutions for ODE and diffusion systems yield accurate and reliable results (R2>0.95), while solutions for Burger systems need further refinement in the ONet algorithm.
翻译:借助现代的计算进步和统计分析方法,机器学习算法已成为工程模型的一个重要部分。神经操作员网络(ONets)是一个新兴的机器学习算法,作为替代部分差异方程(PDEs)的近似解决方案的“较快代谢器 ”, 因为它们能够接近数学操作员,而神经网络(NN)直接近似。 ONETs使用通用通用应用理论,用分支盘形结构将有限维投入绘制到无限空间的地图,该结构在使用点形产品合并信息之前,将域名和特征信息分别编码。 ONets预计将在物理系统和数字双元开发中占据替代模型的重要位置。 三个测试案例使用ODets进行操作员近似化,包括1维普通差异方程、一般扩散系统以及调频集(Burger)系统。 Odece 和扩散系统的解决方案产生准确和可靠的结果(R2>0.95), 而Burger系统的解决办法则需要进一步改进ONet算法。