Comparative Study of Bending Analysis using Physics-Informed Neural Networks and Numerical Dynamic Deflection in Perforated nanobeam

基于物理信息神经网络与数值动态挠度的穿孔纳米梁弯曲分析对比研究

Abstract: In this chapter, we investigate the bending behavior of a perforated nanobeam subjected to sinusoidal loading using an efficient and computationally robust Physics-Informed Functional Link Constrained Framework with Domain Mapping (DFL-TFC) method. 摘要： 在本章中，我们使用一种高效且计算稳健的“带域映射的物理信息函数链接约束框架”（DFL-TFC）方法，研究了在正弦载荷作用下穿孔纳米梁的弯曲行为。

Our aim is to determine the relationship between static bending response and dynamic deflection of a perforated nanobeam for various perforation cases. The static bending is obtained using the FL-TFC with Domain mapped method, whereas dynamic deflection is determined using the Galerkin method. 我们的目标是确定穿孔纳米梁在各种穿孔情况下的静态弯曲响应与动态挠度之间的关系。静态弯曲通过带域映射的 FL-TFC 方法获得，而动态挠度则通过伽辽金（Galerkin）方法确定。

The proposed approach employs the theory of functional connections (TFC) to systematically embed governing differential equation constraints into a constrained expression (CE), which exactly satisfies all prescribed initial and boundary conditions (ICs and BCs) and domain of differential equation is mapped to domain of orthogonal polynomials. 该方法采用函数连接理论（TFC），将控制微分方程的约束系统地嵌入到约束表达式（CE）中，从而精确满足所有预设的初始条件和边界条件（ICs 和 BCs），并将微分方程的定义域映射到正交多项式的定义域。

Within this framework, the free function appearing in the constrained expression is expressed through a functional link neural network (FLNN). The cost is minimized by the mean square residual of DE, allowing training without requiring complex deep network architectures. 在此框架内，约束表达式中出现的自由函数通过函数链接神经网络（FLNN）来表示。通过微分方程的均方残差最小化成本，从而无需复杂的深度网络架构即可进行训练。

Relationship between static and dynamic defection of simply-supported (S-S) perforated nanobeams has been investigated here. FL-TFC with Domain mapped method eliminates the need for deep and complex neural network architectures while ensuring accuracy, efficiency, and strict satisfaction of boundary conditions as compared to standard PINN. 本文研究了简支（S-S）穿孔纳米梁静态与动态挠度之间的关系。与标准 PINN 相比，带域映射的 FL-TFC 方法消除了对深度复杂神经网络架构的需求，同时确保了准确性、效率以及对边界条件的严格满足。