math.DSApr 8, 2026

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

Authors: Iswari Sahu, Ramanath Garai, S. Chakraverty

Organizations: Department of Mathematics, National Institute of Technology Rourkela, India

Read PDF
Ask AI
Cite

Copy citation

DOINot available for this paper

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 (αα), number of rows of holes (NN), tapering parameters (φφ and ψψ), and exponential loading parameter (γγ), 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.

Figures & tables

Figure 1: Geometric model of a tapered perforated beam
Figure 2: Schematic diagram of PINN
Figure 3: Schematic diagram of Domain mapped FL-TFC
Critical DeflectionKp~\widetilde{K_{p}}DFL-TFCGalerkinExisting [ 20 ]Existing [ 32 ]
W~p(0.5)\widetilde{W}_{p}(0.5)001.30211.30211.30211.3023
10100.64480.64480.64470.6448
25250.36610.36610.36610.3661
Table 1: Comparison of maximum bending deflection of a solid beam with uniform load q0~=1\widetilde{q_{0}}=1 corresponding to α=1,γ=0,ϕ=0,ψ=0,N=0\alpha=1,\gamma=0,\phi=0,\psi=0,N=0
W~p(X)\widetilde{W}_{p}(X)S-SC-S
DFL-TFCPINNGalerkinDFL-TFCPINNGalerkin
W~p(0.1)\widetilde{W}_{p}(0.1)4.42814.42804.42811.05521.10401.0552
W~p(0.5)\widetilde{W}_{p}(0.5)14.642514.642514.64259.59969.61239.5996
W~p(0.9)\widetilde{W}_{p}(0.9)5.02265.02275.02263.88353.84433.8835
Table 2: Comparison of bending deflection of a tapered perforated beam for both S-S and C-S boundary conditions corresponding to α=0.3,N=4,γ=1,ϕ=0,ψ=0,Kp~=10,and q0~=10\alpha=0.3,N=4,\gamma=1,\phi=0,\psi=0,\widetilde{K_{p}}=10,\text{and}\ \widetilde{q_{0}}=10
W~p(X)\widetilde{W}_{p}(X)S-SC-S
DFL-TFCPINNGalerkinDFL-TFCPINNGalerkin
W~p(0.1)\widetilde{W}_{p}(0.1)0.79800.80050.79800.17870.20900.1787
W~p(0.5)\widetilde{W}_{p}(0.5)2.10032.10012.10031.17141.18161.1714
W~p(0.9)\widetilde{W}_{p}(0.9)0.56100.55740.56100.35310.33130.3531
Table 3: Comparison of bending deflection of a tapered perforated beam for both S-S and C-S boundary conditions corresponding to α=0.5,N=2,γ=0,ϕ=0.5,ψ=0.5,Kp~=10,and q0~=5\alpha=0.5,N=2,\gamma=0,\phi=0.5,\psi=0.5,\widetilde{K_{p}}=10,\text{and}\ \widetilde{q_{0}}=5
Figure 4: DFL-TFC and PINN solution for α=0.8,γ=5,ϕ=0.5,ψ=0.5,N=3,Kp~=10,q0~=1\alpha=0.8,\gamma=5,\phi=0.5,\psi=0.5,N=3,\widetilde{K_{p}}=10,\widetilde{q_{0}}=1
α\alphaNNS-SC-S
W~p(0.1)\widetilde{W}_{p}(0.1)W~p(0.5)\widetilde{W}_{p}(0.5)W~p(0.9)\widetilde{W}_{p}(0.9)W~p(0.1)\widetilde{W}_{p}(0.1)W~p(0.5)\widetilde{W}_{p}(0.5)W~p(0.9)\widetilde{W}_{p}(0.9)
0.114.496614.86985.11111.08819.83933.9801
45.534818.24726.57451.819214.12315.7176
0.313.546211.71043.93490.71346.87602.7865
44.428114.64255.02261.05529.59963.8835
0.513.341611.03033.69300.65006.32602.5647
Table 4: Bending deflection of a tapered perforated beam for C-S boundary conditions with varying α\alpha and NN corresponding to γ=1,ϕ=0,ψ=0,Kp~=10,and q0~=10\gamma=1,\phi=0,\psi=0,\widetilde{K_{p}}=10,\text{and}\ \widetilde{q_{0}}=10
ϕ\phiψ\psiS-SC-S
  W~p(0.1)\widetilde{W}_{p}(0.1)W~p(0.5)\widetilde{W}_{p}(0.5)W~p(0.9)\widetilde{W}_{p}(0.9)W~p(0.1)\widetilde{W}_{p}(0.1)W~p(0.5)\widetilde{W}_{p}(0.5)W~p(0.9)\widetilde{W}_{p}(0.9)
0.10.11.04503.15880.95870.21351.78050.6363
0.50.96522.73680.76620.20151.52750.4918
0.90.89592.39480.63260.19091.32510.3954
0.50.10.86122.40300.68110.18911.34940.4392
0.50.79802.10030.56100.17871.17140.3531
Table 5: Bending deflection of a tapered perforated beam for S-S and C-S boundary conditions with varying ϕ\phi and ψ\psi corresponding to γ=0,α=0.5,N=2,Kp~=10,and q0~=5\gamma=0,\alpha=0.5,N=2,\widetilde{K_{p}}=10,\text{and}\ \widetilde{q_{0}}=5
γ\gammaKp~\widetilde{K_{p}}S-SC-S
  W~p(0.1)\widetilde{W}_{p}(0.1)W~p(0.5)\widetilde{W}_{p}(0.5)W~p(0.9)\widetilde{W}_{p}(0.9)W~p(0.1)\widetilde{W}_{p}(0.1)W~p(0.5)\widetilde{W}_{p}(0.5)W~p(0.9)\widetilde{W}_{p}(0.9)
110.32350.86990.23420.05920.42550.1322
50.27410.74420.20230.05520.39270.1225
100.22980.63070.17340.05090.35820.1122
310.95752.72920.77300.17890.17890.4723
50.80692.33910.67290.16581.30980.4390
Table 6: Bending deflection of a tapered perforated beam for S-S and C-S boundary conditions with varying γ\gamma and Kp~\widetilde{K_{p}} corresponding to α=0.8,N=3,ϕ=ψ=0.5,and q0~=1\alpha=0.8,N=3,\phi=\psi=0.5,\text{and}\ \widetilde{q_{0}}=1
Figure 5: Variation of bending deflection with respect to filling ratio α\alpha for a tapered perforated beam subjected to an exponential load
Figure 6: Variation of bending deflection with respect to exponential load parameter γ\gamma for a tapered perforated beam resting on foundation
Figure 7: Effect of tapering parameter ϕ\phi on the bending behaviour of a tapered perforated beam resting on a foundation
Figure 8: Effect of tapering parameter ψ\psi on the bending behaviour of a tapered perforated beam resting on a foundation
Figure 9: Effect of elastic foundation parameter Kp~\widetilde{K_{p}} on the bending behaviour of a tapered perforated beam with exponential load
Figure 10: Loss figures of PINN and DFL-TFC frameworks at different parameters for S-S boundary conditions
PINN  DFL-TFC  
No. of Hidden LayersLossTimeOrder of Chebyshev PolynomialLossTime
14.0750×10−64.0750\times 10^{-6}23.86 s133.1803×10−83.1803\times 10^{-8}4.94 s
25.3076×10−65.3076\times 10^{-6}39.43 s141.8148×10−101.8148\times 10^{-10}5.80 s
34.7255×10−64.7255\times 10^{-6}47.43 s154.8919×10−104.8919\times 10^{-10}7.08 s
Table 7: Comparison of PINN and DFL-TFC for parameters α=0.7\alpha=0.7 , γ=3\gamma=3 , ϕ=ψ=0.5\phi=\psi=0.5 , N=1N=1 , Kp~=10\widetilde{K_{p}}=10 , and q0~=2\widetilde{q_{0}}=2 (S-S boundary conditions)
PINN  DFL-TFC  
No. of Hidden LayersLossTimeOrder of Chebyshev PolynomialLossTime
15.3951×10−55.3951\times 10^{-5}15.60 s133.4304×10−103.4304\times 10^{-10}5.84 s
24.1046×10−74.1046\times 10^{-7}34.86 s148.7719×10−118.7719\times 10^{-11}6.16 s
32.7828×10−62.7828\times 10^{-6}43.26 s159.9721×10−119.9721\times 10^{-11}8.21 s
Table 8: Comparison of PINN and DFL-TFC for parameters α=0.8\alpha=0.8 , γ=4\gamma=4 , ϕ=ψ=0.3\phi=\psi=0.3 , N=2N=2 , Kp~=5\widetilde{K_{p}}=5 , and q0~=1\widetilde{q_{0}}=1 (S-S boundary conditions)
PINN  DFL-TFC  
No. of Hidden LayersLossTimeOrder of Chebyshev PolynomialLossTime
17.9079×10−57.9079\times 10^{-5}36.55 s131.5806×10−81.5806\times 10^{-8}5.14 s
22.2729×10−32.2729\times 10^{-3}57.14 s143.9759×10−113.9759\times 10^{-11}6.72 s
31.4391×10−41.4391\times 10^{-4}58.31 s156.8713×10−136.8713\times 10^{-13}8.90 s
Table 9: Comparison of PINN and DFL-TFC for parameters α=0.2\alpha=0.2 , γ=5\gamma=5 , ϕ=ψ=0.2\phi=\psi=0.2 , N=1N=1 , Kp~=8\widetilde{K_{p}}=8 , and q0~=4\widetilde{q_{0}}=4 (S-S boundary conditions)
Figure 11: Loss figures of PINN and DFL-TFC frameworks at different parameters for C-S boundary conditions
PINN  DFL-TFC  
No. of Hidden LayersLossTimeOrder of Chebyshev PolynomialLossTime
12.6243×10−62.6243\times 10^{-6}35.78 s131.3258×10−71.3258\times 10^{-7}4.80 s
25.8697×10−55.8697\times 10^{-5}56.07 s145.5581×10−105.5581\times 10^{-10}7.11 s
31.2247×10−61.2247\times 10^{-6}49.03 s154.3449×10−114.3449\times 10^{-11}9.44 s
Table 10: Comparison of PINN and Domain mapped FL-TFC for parameters α=0.7\alpha=0.7 , γ=3\gamma=3 , ϕ=ψ=0.5\phi=\psi=0.5 , N=1N=1 , Kp~=10\widetilde{K_{p}}=10 , and q0~=2\widetilde{q_{0}}=2 (C-S boundary conditions)
PINN  DFL-TFC  
No. of Hidden LayersLossTimeOrder of Chebyshev PolynomialLossTime
11.6465×10−61.6465\times 10^{-6}27.36 s136.6098×10−106.6098\times 10^{-10}6.39 s
22.7963×10−52.7963\times 10^{-5}45.26 s141.9484×10−101.9484\times 10^{-10}8.09 s
31.1195×10−51.1195\times 10^{-5}41.58 s151.7089×10−131.7089\times 10^{-13}9.66 s
Table 11: Comparison of PINN and Domain mapped FL-TFC for parameters α=0.8\alpha=0.8 , γ=4\gamma=4 , ϕ=ψ=0.3\phi=\psi=0.3 , N=2N=2 , Kp~=5\widetilde{K_{p}}=5 , and q0~=1\widetilde{q_{0}}=1 (C-S boundary conditions)
PINN  DFL-TFC  
No. of Hidden LayersLossTimeOrder of Chebyshev PolynomialLossTime
18.8678×10−48.8678\times 10^{-4}35.76 s134.9020×10−84.9020\times 10^{-8}6.71 s
22.7195×10−42.7195\times 10^{-4}58.58 s142.7857×10−112.7857\times 10^{-11}8.23 s
33.7222×10−13.7222\times 10^{-1}71.00 s158.0515×10−128.0515\times 10^{-12}10.34 s
Table 12: Comparison of PINN and Domain mapped FL-TFC for parameters α=0.2\alpha=0.2 , γ=5\gamma=5 , ϕ=ψ=0.2\phi=\psi=0.2 , N=1N=1 , Kp~=8\widetilde{K_{p}}=8 , and q0~=4\widetilde{q_{0}}=4 (C-S boundary conditions)
More in this topic

Physics-Informed ML

Physics-informed machine learning combines data-driven models with physical laws to solve, predict, and infer scientific systems.

See all 833 papers

Explore similar work

CardsList
Apr 15, 2026

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

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. 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. 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. 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. 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.

Ramanath Garai, Iswari Sahu, S. Chakraverty

Department of Mathematics, National Institute of Technology Rourkela, Rourkela-769008, India.

Structural MechanicsPhysics-Informed MLPDE Solving
AskPDF
Aug 8, 2026

Finite basis physics-informed neural networks with hard constraints for viscous fluid flow in highly perforated domains

In this work, viscous fluid flow governed by the Stokes equations in highly perforated domains is studied using physics-informed neural networks (PINNs). Perforated microstructures induce complex boundary conditions and fine-scale flow features that are difficult for standard neural networks to resolve. Conventional PINNs, even when combined with advanced training techniques, can suffer from a loss of accuracy and efficiency as the number of perforations increases. One important source of this difficulty is the soft enforcement of boundary conditions through penalty terms, which can lead to stiffness, gradient conflicts, and poor resolution of near-boundary flow structures. Hard constraints provide an alternative by encoding boundary conditions exactly into the network ansatz, but may introduce undesirable non-local effects due to the global nature of the approximation. To address these challenges, finite basis PINNs (FBPINNs), which are based on domain decomposition and localisation principles, are used together with hard boundary constraints that efficiently encode perforation-related boundary conditions. This approach helps mitigate spectral bias, improves overall accuracy, and exhibits convergence that is only weakly affected by the number of perforations, thereby providing an efficient and highly parallelisable neural network framework. The proposed approach is further supported with theoretical arguments, specifically focusing on the localisation and approximation properties of FBPINNs.

Jeeeun Lee, Denis Korolev, Miro Duhovic +1

Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon, 34141, Republic of Korea · Weierstrass-Institute for Applied Analysis and Stochastics, Anton-Wilhelm-Amo-Str. 39, Berlin, 10117, Germany · Leibniz-Institut für Verbundwerkstoffe GmbH, Erwin-Schrödinger-Str. 58, Kaiserslautern, 67663, Germany

Neural PDE SolversPhysics-Informed MLComputational Fluid Dynamics
AskPDF
Jun 2, 2026

Critical evaluation of PINN for FWD inverse analysis and differentiable FEM as an alternative

Automatic-differentiation-based inverse analysis methods, including physics-informed neural networks (PINNs) and differentiable programming, have recently shown great promise due to their ability to compute accurate gradients and convergence efficiency. However, their applicability to falling weight deflectometer (FWD) backcalculation remains unexplored. This study critically evaluates PINN-based inverse analysis for a multilayer pavement system and investigates differentiable finite element method (DiffFEM) as an alternative based on a synthetic benchmark. The standard PINN does not recover layer moduli because of the sharp domain discontinuities inherent to layered pavement systems. Although we use an extended PINN with domain decomposition (XPINN), which shows better performance on discontinuous domains, its performance remains highly sensitive to loss weighting and network architecture, and degrades under measurement noise. By contrast, DiffFEM consistently achieves more accurate, stable, and computationally efficient inversion results. These results indicate that DiffFEM, which enforces the governing physics as a hard constraint, yields better accuracy, robustness, and computational efficiency than PINN-based approaches, in which the governing physics is imposed as a soft constraint through the loss function. More broadly, the findings suggest that the choice between PINN- and DiffFEM-based inverse analysis needs careful consideration, with DiffFEM offering practical advantages when an efficient and robust differentiable forward solver is available.

Yongjin Choi, Hyeonbin Moon, Seunghwa Ryu

KAIST InnoCORE PRISM-AI Center, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, 34141, Republic of Korea · Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, 34141, Republic of Korea · School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, 30332, GA, USA

Differentiable PhysicsPhysics-Informed MLFinite Element MethodPDE Inverse Problems
AskPDF