论文标题
功能连接的多元理论：部分微分方程中的理论，证明和应用
The Multivariate Theory of Functional Connections: Theory, Proofs, and Application in Partial Differential Equations
论文作者
论文摘要
本文介绍了功能连接理论的重新制定：功能插值的一般方法，可以嵌入一组用户指定的线性约束。本文介绍的重新制定利用了有关功能连接理论的基础功能结构，以简化这些插值功能的推导 - 被称为约束的表达式，并提供了严格的术语，提供了对这些约束表达属性的数学证据的直接衍生作用。此外，通过单变量公式的递归应用，该技术将技术扩展到N维度和证明是立即的。总的来说，将这种重新制定的结果与先前的工作进行了比较，以强调使用这种方法的新颖性和数学便利性。最后，本文介绍的方法应用于具有不同边界条件的两个部分微分方程，并且在可用数据时，将结果与最新方法进行比较。
This article presents a reformulation of the Theory of Functional Connections: a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The reformulation presented in this paper exploits the underlying functional structure presented in the seminal paper on the Theory of Functional Connections to ease the derivation of these interpolating functionals--called constrained expressions--and provides rigorous terminology that lends itself to straightforward derivations of mathematical proofs regarding the properties of these constrained expressions. Furthermore, the extension of the technique to and proofs in n-dimensions is immediate through a recursive application of the univariate formulation. In all, the results of this reformulation are compared to prior work to highlight the novelty and mathematical convenience of using this approach. Finally, the methodology presented in this paper is applied to two partial differential equations with different boundary conditions, and, when data is available, the results are compared to state-of-the-art methods.
