Physics-Informed Functional Link Constrained Framework with Domain Mapping for Solving Bending Analysis of an Exponentially Loaded Perforated Beam

TL;DR

This paper introduces a novel physics-informed functional link framework with domain mapping for analyzing the bending behavior of perforated beams under exponential load, outperforming PINN in accuracy and efficiency.

Contribution

The work develops a new DFL-TFC method that combines domain mapping, functional expansion, and neural networks to solve complex differential equations more effectively.

Findings

DFL-TFC achieves faster convergence than PINN.

The proposed method reduces computational cost.

Results are validated against Galerkin and PINN solutions.

Abstract

This article presents a novel and comprehensive approach for analyzing bending behavior of the tapered perforated beam under an exponential load. The governing differential equation includes important factors like filling ratio ($\alpha$), number of rows of holes ($N$), tapering parameters ($\phi$ and $\psi$), and exponential loading parameter ($\gamma$), providing a realistic and flexible representation of perforated beam configuration. Main goal of this work is to see how well the Domain mapped physics-informed Functional link Theory of Functional Connection (DFL-TFC) method analyses bending response of perforated beam with square holes under exponential loading. For comparison purposes, a corresponding PINN-based formulation is developed. Outcomes clearly show that the proposed DFL-TFC framework gives better results, including faster convergence, reduced computational cost, and improved solution accuracy when compared to the PINN approach. These findings highlight effectiveness and potential of DFL-TFC method for solving complex engineering problems governed by differential equations. Within this framework, hidden layer is replaced by a functional expansion block that enriches input representation via orthogonal polynomial basis functions, and the domain of DE mapped to corresponding domain of orthogonal polynomials. A Constrained Expression (CE), constructed through the Theory of Functional Connections (TFC) using boundary conditions, ensures that constraints are exactly satisfied. In CE, free function is represented using a Functional Link Neural Network (FLNN), which learns to solve resulting unconstrained optimization problem. The obtained results are further validated through the Galerkin and PINN solutions.