微分方程(DE)被用作数值模型来描述整个工程和科学领域的物理现象,包括热量和流体流动,结构弯曲,和系统动力学。虽然有许多其他技术可以找到这些方程的近似解,本文旨在比较函数连接理论(TFC)与基于最小二乘支持向量机(LS-SVM)的应用。TFC方法使用约束表达式,始终满足DE约束的表达式,它将求解DE的过程转化为求解无约束优化问题,最终通过最小二乘(LS)求解。除了个人分析,这两种方法合并为一种新的方法论,称为约束SVM(CSVM),通过将LS-SVM方法纳入TFC框架来解决无约束问题。对四个样本问题进行了数值测试:一个一阶线性常微分方程(ODE),一个一阶非线性ODE,一个二阶线性ODE,和一个二维线性偏微分方程(PDE)。使用LS-SVM方法作为基准,通过安排训练时间对所有问题进行速度比较,并使用训练集和测试集上的最大误差和均方误差进行精度比较。总的来说,TFC显示比LS-SVM和CSVM方法稍微快(一个数量级或更小)并且更准确(多个数量级)。
Differential equations (DEs) are used as numerical models to describe physical phenomena throughout the field of engineering and science, including heat and fluid flow, structural bending, and systems dynamics. While there are many other techniques for finding approximate solutions to these equations, this paper looks to compare the application of the Theory of Functional Connections (TFC) with one based on least-squares support vector machines (LS-SVM). The TFC method uses a constrained expression, an expression that always satisfies the DE constraints, which transforms the process of solving a DE into solving an unconstrained optimization problem that is ultimately solved via least-squares (LS). In addition to individual analysis, the two methods are merged into a new methodology, called constrained SVMs (CSVM), by incorporating the LS-SVM method into the TFC framework to solve unconstrained problems. Numerical tests are conducted on four sample problems: One first order linear ordinary differential equation (ODE), one first order nonlinear ODE, one second order linear ODE, and one two-dimensional linear partial differential equation (PDE). Using the LS-SVM method as a benchmark, a speed comparison is made for all the problems by timing the training period, and an accuracy comparison is made using the maximum error and mean squared error on the training and test sets. In general, TFC is shown to be slightly faster (by an order of magnitude or less) and more accurate (by multiple orders of magnitude) than the LS-SVM and CSVM approaches.