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arxiv: 2604.07025 · v1 · submitted 2026-04-08 · 🧮 math.DS · cs.LG· cs.NA· math.NA

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

Iswari Sahu , Ramanath Garai , S. Chakraverty This is my paper

Pith reviewed 2026-05-10 17:56 UTC · model grok-4.3

classification 🧮 math.DS cs.LGcs.NAmath.NA
keywords perforated beambending analysisexponential loadtheory of functional connectionsfunctional link neural networkphysics-informed neural networksdomain mappingtapered beam
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The pith

A domain-mapped functional link method solves perforated beam bending more accurately and with lower cost than PINN.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops the DFL-TFC framework to compute the deflection of a tapered beam with square holes under an exponentially varying distributed load. The governing differential equation directly includes the filling ratio, number of hole rows, two tapering parameters, and the exponential load decay rate. The method maps the physical interval to the domain of chosen orthogonal polynomials, forms a constrained expression that satisfies the boundary conditions exactly through the Theory of Functional Connections, and approximates the remaining free function with a functional link network whose expansion uses polynomial basis functions. Side-by-side tests against a physics-informed neural network formulation show faster convergence, lower computational cost, and smaller pointwise errors for the same physical parameters. Readers working on lightweight structural design would care because the approach supplies an exact-constraint alternative to penalty-based or mesh-based solvers for this class of variable-coefficient beam problems.

Core claim

The DFL-TFC framework replaces the hidden layer with a functional expansion block using orthogonal polynomial basis functions, maps the differential equation domain to the corresponding polynomial domain, constructs a constrained expression via the Theory of Functional Connections so that boundary conditions are satisfied exactly, and lets a Functional Link Neural Network learn the free function that solves the resulting unconstrained problem; the resulting solutions for the perforated beam deflection are more accurate, converge faster, and require less computation than those obtained from a comparable PINN formulation, with further agreement against independent Galerkin solutions.

What carries the argument

The DFL-TFC framework, which maps the physical domain to the orthogonal polynomial domain, replaces the neural hidden layer with a functional expansion block, and enforces boundary conditions exactly through a Theory of Functional Connections constrained expression whose free function is represented by a Functional Link Neural Network.

If this is right

  • Boundary conditions are satisfied exactly by construction, eliminating the need for penalty terms in the loss function.
  • The same framework applies without reformulation to any combination of the parameters α, N, φ, ψ, and γ.
  • Computational cost is lower than PINN while error norms are also smaller.
  • Results remain consistent with independent Galerkin solutions across the tested configurations.
  • The method extends naturally to other ordinary differential equations that arise in tapered or perforated structural members.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Domain mapping combined with exact-constraint functional expressions may reduce the depth or width needed for neural solvers on other variable-coefficient beam or plate problems.
  • If the improvement is driven by the polynomial expansion rather than the network itself, swapping different orthogonal bases could further tune accuracy for extreme taper values.
  • The approach supplies a concrete test case for whether functional-link methods systematically outperform standard PINNs on problems whose coefficients vary smoothly but whose geometry is piecewise.

Load-bearing premise

The specific choices of orthogonal polynomial basis, domain mapping, and network architecture produce a solution whose error is uniformly smaller than PINN across the full range of physical parameters without hidden fitting or selective reporting.

What would settle it

A side-by-side run of DFL-TFC and PINN on identical parameter sets for filling ratio, hole rows, taper rates, and load exponent, trained to the same tolerance, followed by direct comparison of maximum absolute deflection error against a high-resolution Galerkin reference solution.

Figures

Figures reproduced from arXiv: 2604.07025 by Iswari Sahu, Ramanath Garai, S. Chakraverty.

Figure 1
Figure 1. Figure 1: Geometric model of a tapered perforated beam [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic diagram of PINN 4.2 Galerkin Approximation Procedure The transverse displacement of the perforated tapered beam, denoted by Wp(X), is approx￾imated using a finite series of basis functions as [26, 21]: W p(X) = Xn i=1 ηiΘi(X), (14) where n is number of basis functions, ηi are unknown coefficients, and Θi(X) represent orthonormal polynomial functions, which comes from simple polynomial Xi−1 , i = … view at source ↗
Figure 3
Figure 3. Figure 3: Schematic diagram of Domain mapped FL-TFC [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: DFL-TFC and PINN solution for α = 0.8, γ = 5, ϕ = 0.5, ψ = 0.5, N = 3, Kfp = 10, qe0 = 1 5.2 New Outcomes The bending deflection of a tapered perforated beam is obtained using the DFL-TFC method, which demonstrates faster convergence, higher accuracy, and greater computational efficiency compared with the traditional physics-informed neural network (PINN) approach [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Variation of bending deflection with respect to filling ratio [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Variation of bending deflection with respect to exponential load parameter [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Effect of tapering parameter ϕ on the bending behaviour of a tapered perforated beam resting on a foundation 19 [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Effect of tapering parameter ψ on the bending behaviour of a tapered perforated beam resting on a foundation 5.2.4 Effect of foundation parameter It is important to mention that the foundation parameter Kfp strongly influences the bending deflection of the tapered perforated beam. The effect of this foundation parameter Kfp is presented in [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Effect of elastic foundation parameter Kfp on the bending behaviour of a tapered perforated beam with exponential load 20 [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Loss figures of PINN and DFL-TFC frameworks at different parameters for S-S [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Loss figures of PINN and DFL-TFC frameworks at different parameters for C-S [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces the Domain-mapped physics-informed Functional Link Constrained Framework (DFL-TFC) for analyzing the bending of a tapered perforated beam under exponential load. The governing DE incorporates parameters including filling ratio α, number of hole rows N, tapering parameters φ and ψ, and exponential load parameter γ. The method replaces the hidden layer with a functional expansion using orthogonal polynomial basis functions, maps the physical domain to the polynomial interval, employs the Theory of Functional Connections (TFC) to build a constrained expression that exactly satisfies boundary conditions, and uses a Functional Link Neural Network (FLNN) as the free function to solve the resulting unconstrained problem. Numerical outcomes are compared to a corresponding PINN formulation and Galerkin method, with the claim of faster convergence, lower computational cost, and higher accuracy for DFL-TFC.

Significance. If the performance advantages are substantiated with quantitative evidence across the parameter space, the combination of domain mapping, orthogonal polynomial enrichment, and exact TFC constraint satisfaction could provide an efficient alternative to standard PINNs for constrained differential equations in structural mechanics. The low number of free parameters in the FLNN component is a methodological strength that merits careful validation.

major comments (2)
  1. Abstract: The assertion that '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' is unsupported by any quantitative metrics such as L2 or maximum error norms, iteration counts, wall-clock times, or convergence plots for varying values of α, N, φ, ψ, and γ.
  2. Numerical results section: No tables or figures report error norms, computational costs, or performance metrics across the full range of physical parameters (α, N, φ, ψ, γ), making it impossible to verify the central claim of uniform superiority or to rule out selective reporting or problem-specific hyperparameter tuning.
minor comments (2)
  1. The specific choice of orthogonal polynomial family (Legendre, Chebyshev, etc.) and the explicit form of the domain mapping function should be stated with equations in the method section.
  2. A schematic figure of the tapered perforated beam geometry with square holes and the applied exponential load would clarify the physical setup and parameter definitions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. The feedback correctly identifies the need for stronger quantitative evidence to support the performance claims of the DFL-TFC framework relative to PINN. We will revise the manuscript to incorporate the requested metrics and comparisons across the parameter space.

read point-by-point responses

  1. Referee: Abstract: The assertion that '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' is unsupported by any quantitative metrics such as L2 or maximum error norms, iteration counts, wall-clock times, or convergence plots for varying values of α, N, φ, ψ, and γ.

    Authors: We agree that the abstract claim requires explicit quantitative backing. In the revised manuscript we will modify the abstract to reference specific supporting metrics and ensure the numerical section provides L2 and maximum error norms, iteration counts, wall-clock times, and convergence plots for representative values of α, N, φ, ψ, and γ. These additions will directly substantiate the stated advantages. revision: yes

  2. Referee: Numerical results section: No tables or figures report error norms, computational costs, or performance metrics across the full range of physical parameters (α, N, φ, ψ, γ), making it impossible to verify the central claim of uniform superiority or to rule out selective reporting or problem-specific hyperparameter tuning.

    Authors: We acknowledge that the current numerical results section lacks comprehensive error norms and performance metrics over the full parameter range. The revised version will include new tables and figures reporting L2 and maximum errors, iteration counts, and wall-clock times for multiple combinations of α, N, φ, ψ, and γ. This will enable verification of performance across the space and address concerns regarding selective reporting. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation or performance claims

full rationale

The paper presents a numerical solver (DFL-TFC) that maps the domain, uses orthogonal polynomial expansion inside a TFC-constrained expression whose free function is an FLNN, and minimizes the DE residual. This construction directly implements the standard physics-informed collocation approach and is compared to independent external methods (Galerkin and a separately formulated PINN). No equation or claim reduces the obtained solution, error norms, convergence rate, or computational cost to the input parameters or basis choices by definition. Validation against Galerkin and PINN supplies external benchmarks rather than self-referential fitting. The performance superiority statement is therefore an empirical outcome of the experiments, not a tautology.

Axiom & Free-Parameter Ledger

4 free parameters · 2 axioms · 0 invented entities

The framework rests on standard properties of orthogonal polynomials, the existence of a TFC constrained expression for the chosen boundary conditions, and the assumption that the functional-link network can represent the residual-minimizing function. No new physical entities are postulated.

free parameters (4)
  • filling ratio alpha
    Physical parameter controlling hole size fraction; appears in the governing DE and is varied in experiments.
  • number of hole rows N
    Discrete parameter defining perforation pattern; enters the DE coefficients.
  • tapering parameters phi and psi
    Coefficients controlling linear taper in beam height and width.
  • exponential load parameter gamma
    Controls the spatial decay rate of the applied load.
axioms (2)
  • standard math Orthogonal polynomials form a complete basis on the mapped domain.
    Invoked when replacing the hidden layer with functional expansion.
  • domain assumption The TFC constrained expression exactly satisfies the boundary conditions for any choice of the free function.
    Central to the method; stated in the description of the constrained expression.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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

    cs.LG 2026-04 unverdicted novelty 5.0

    A domain-mapped functional link neural network solves static bending of perforated nanobeams and is compared to Galerkin results for dynamic deflection.

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