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

Iswari Sahu; Ramanath Garai; S. Chakraverty

arxiv: 2604.24768 · v1 · submitted 2026-04-15 · 💻 cs.LG · cs.AI· physics.comp-ph

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

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

Pith reviewed 2026-05-10 14:24 UTC · model grok-4.3

classification 💻 cs.LG cs.AIphysics.comp-ph

keywords physics-informed neural networksfunctional link neural networktheory of functional connectionsperforated nanobeambending analysisdomain mappingGalerkin methodnanostructures

0 comments

The pith

The FL-TFC domain-mapped method solves perforated nanobeam bending accurately by embedding boundary conditions exactly into a functional link network.

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

This paper presents a Physics-Informed Functional Link Constrained Framework with Domain Mapping to compute the static bending response of simply-supported perforated nanobeams under sinusoidal loading. Static solutions come from the new method while dynamic deflections are found separately with the Galerkin technique, allowing direct comparison across different perforation patterns. The framework builds a constrained expression via the theory of functional connections that satisfies all boundary conditions by construction, maps the physical domain onto orthogonal polynomial space, and lets a functional link neural network represent the remaining free function. Training minimizes only the mean-square residual of the governing differential equation, avoiding the need for deep or complex architectures required in standard physics-informed neural networks.

Core claim

The DFL-TFC approach, which embeds the governing differential equation constraints into a constrained expression using the theory of functional connections, maps the domain to orthogonal polynomials, and approximates the free function with a functional link neural network, produces accurate bending solutions for perforated nanobeams while exactly satisfying all initial and boundary conditions and requiring only modest network complexity.

What carries the argument

The constrained expression from the theory of functional connections, which exactly satisfies all boundary conditions, combined with domain mapping to orthogonal polynomial space and a functional link neural network for the free function, whose parameters are optimized by minimizing the mean-square residual of the differential equation.

If this is right

Static bending responses for simply-supported perforated nanobeams can be obtained efficiently for multiple perforation configurations without deep networks.
A direct numerical relationship can be established between the static bending and dynamic deflection of the same nanobeam structures.
Boundary conditions are satisfied exactly by construction rather than through penalty terms or additional loss components.
The method remains applicable when the perforation pattern changes, as long as the governing equation and boundary conditions are preserved.

Where Pith is reading between the lines

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

The same constrained functional-link construction could be applied to other beam or plate problems in nanomechanics where boundary conditions are known in advance.
Domain mapping to orthogonal polynomials may reduce the number of training points needed for convergence in similar residual-minimization schemes.
If the residual minimum reliably yields the correct solution, functional-link networks may suffice for a broader class of linear and weakly nonlinear boundary-value problems that currently rely on deeper architectures.

Load-bearing premise

Minimizing the mean square residual of the governing differential equation with a functional link neural network will produce accurate solutions to the perforated nanobeam bending problem for various perforation cases.

What would settle it

Computed static deflections from the FL-TFC method that differ by more than a few percent from independent Galerkin or finite-element results for the same perforation ratio, load amplitude, and beam geometry.

Figures

Figures reproduced from arXiv: 2604.24768 by Iswari Sahu, Ramanath Garai, S. Chakraverty.

**Figure 2.** Figure 2: Static and dynamic deflection relationship for a simply supported (SS) perforated nanobeam with varying filling ratio [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Figure 3: Comparison of static and dynamic deflection for different numbers of holes [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

**Figure 4.** Figure 4: Relationship between static and dynamic deflection of an SS perforated nanobeam with variation in non-local [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

read the original 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. 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.

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.

Desk Editor's Note private letter to a colleague

The paper applies FL-TFC with domain mapping to perforated nanobeam bending and pairs it with Galerkin for dynamic deflection, but supplies no numerical results or validation to support its accuracy claims.

read the letter

The paper applies a functional link neural network inside the theory of functional connections framework, with domain mapping to orthogonal polynomials, to compute static bending of simply-supported perforated nanobeams under sinusoidal load. It then compares those results to dynamic deflection obtained by the Galerkin method across different perforation cases. The constrained expression is built to satisfy boundary conditions exactly, and the free function is represented by the FLNN whose parameters are trained by minimizing the mean-square residual of the governing differential equation. This avoids deep architectures and penalty terms for the boundaries, which is a practical feature for this class of problems. The specific combination for perforated nanobeams appears to be new relative to the cited prior work on FL-TFC and standard PINNs. The method description is clear on how the constraints are embedded and how the domain is remapped, and the static-dynamic relationship is a reasonable quantity to examine for nanobeam design. Those elements are done competently on paper. The main weakness is the complete absence of any quantitative support. The abstract states that the approach ensures accuracy, efficiency, and superiority over standard PINN, yet it reports no error norms, no deflection plots, no convergence rates, and no side-by-side comparisons against finite-element or analytical solutions for the same perforated geometries. Without those checks it is impossible to know whether the low residual actually produces the physically correct deflection when perforations introduce local stiffness changes. The stress-test observation that residual minimization can coexist with large pointwise errors when the basis is insufficiently expressive is directly relevant here, and the abstract gives no indication that the FLNN basis was tested for that expressivity on the perforation cases. The governing equation modification for each perforation pattern is also not described in enough detail to judge whether it is handled correctly or simply approximated by a uniform beam model. This work is aimed at researchers already working on physics-informed networks for structural nanomechanics. A reader in that narrow area might extract the method outline and the static-dynamic comparison idea, but only if the full manuscript adds the missing benchmarks and implementation specifics. For anyone outside that subfield the paper offers little usable content. It deserves peer review because the framework is formally set out and the application is concrete; a referee can require the quantitative validation that is currently missing and assess whether the perforation modeling is handled rigorously.

Referee Report

3 major / 2 minor

Summary. The manuscript investigates the static bending of simply-supported perforated nanobeams under sinusoidal loading via a Physics-Informed Functional Link Constrained Framework with Domain Mapping (DFL-TFC). Static deflection is obtained by embedding boundary conditions exactly through the Theory of Functional Connections, mapping the domain to orthogonal polynomials, and representing the free function with a functional link neural network (FLNN) whose parameters are trained by minimizing the mean-square residual of the governing differential equation. Dynamic deflection is computed separately with the Galerkin method. The central claim is that this DFL-TFC approach yields accurate solutions, satisfies boundary conditions strictly, and is more efficient than standard PINNs without requiring deep architectures, for various perforation cases.

Significance. If the numerical validation holds, the method could provide a lightweight, constraint-satisfying alternative to deep PINNs or traditional FEM for nanobeam problems with geometric discontinuities, potentially lowering computational cost while preserving physical fidelity. The explicit comparison between static and dynamic responses for multiple perforation geometries would be a useful contribution if supported by quantitative benchmarks.

major comments (3)

[Abstract] Abstract: The claims of 'accuracy, efficiency, and strict satisfaction of boundary conditions as compared to standard PINN' are asserted without any reported error norms (e.g., L2 or pointwise deflection errors), residual values, convergence rates, or direct numerical comparisons against FEM/Galerkin or analytical solutions for the same perforated geometries. This absence prevents assessment of whether residual minimization actually yields the physically correct deflection when perforations induce local stiffness variations.
[Method] Method description (FL-TFC with domain mapping): No explicit statement is given on how the governing differential equation is altered for each perforation case (e.g., position-dependent moment of inertia or effective bending stiffness). Without this, it is unclear whether a single uniform beam equation is used or whether perforation-specific modifications are incorporated into the residual; the latter is required for the central claim to hold.
[FL-TFC formulation] FLNN expressivity: The manuscript provides no analysis or numerical test showing that the chosen functional-link basis (after TFC embedding) can represent the solution features induced by perforations, such as localized curvature changes or near-singularities. Standard PINN literature indicates that low mean-square residuals can coexist with large pointwise errors when the ansatz lacks sufficient expressivity; a concrete demonstration (e.g., comparison of FLNN vs. deeper networks on a known perforated benchmark) is needed.

minor comments (2)

[Abstract] The abstract refers to 'this chapter' and 'various perforation cases' without defining the specific geometries or loading parameters; these should be stated explicitly with a table or figure.
[Method] Notation for the constrained expression (CE) and the domain-mapping transformation should be introduced with a clear equation reference rather than inline description.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each of the major comments below and have made revisions to enhance the manuscript's clarity and validation.

read point-by-point responses

Referee: [Abstract] Abstract: The claims of 'accuracy, efficiency, and strict satisfaction of boundary conditions as compared to standard PINN' are asserted without any reported error norms (e.g., L2 or pointwise deflection errors), residual values, convergence rates, or direct numerical comparisons against FEM/Galerkin or analytical solutions for the same perforated geometries. This absence prevents assessment of whether residual minimization actually yields the physically correct deflection when perforations induce local stiffness variations.

Authors: We agree with the referee that quantitative metrics are necessary to support the claims in the abstract. In the revised manuscript, we have added specific L2 error norms for the deflection fields, mean-square residual values, and convergence behavior with respect to the number of basis functions in the FLNN. Additionally, we now include direct comparisons of the DFL-TFC results with the Galerkin method for the dynamic case and with FEM for validation on perforated geometries. revision: yes
Referee: [Method] Method description (FL-TFC with domain mapping): No explicit statement is given on how the governing differential equation is altered for each perforation case (e.g., position-dependent moment of inertia or effective bending stiffness). Without this, it is unclear whether a single uniform beam equation is used or whether perforation-specific modifications are incorporated into the residual; the latter is required for the central claim to hold.

Authors: The governing equation is indeed modified for each perforation case by incorporating a position-dependent moment of inertia I(x), which reflects the reduced cross-section at perforation locations. This leads to a variable-coefficient differential equation that is embedded in the residual. We have revised the Method section to explicitly present the modified Euler-Bernoulli equation with I(x) for the different perforation configurations and explain how it is used in the domain-mapped TFC framework. revision: yes
Referee: [FL-TFC formulation] FLNN expressivity: The manuscript provides no analysis or numerical test showing that the chosen functional-link basis (after TFC embedding) can represent the solution features induced by perforations, such as localized curvature changes or near-singularities. Standard PINN literature indicates that low mean-square residuals can coexist with large pointwise errors when the ansatz lacks sufficient expressivity; a concrete demonstration (e.g., comparison of FLNN vs. deeper networks on a known perforated benchmark) is needed.

Authors: We recognize the importance of demonstrating the expressivity of the FLNN ansatz for capturing perforation-induced features. To address this, we have included in the revised paper a dedicated subsection with numerical benchmarks on a perforated beam problem, comparing the FL-TFC approach to both analytical/FEM solutions and a standard deep PINN. The results show that the domain-mapped FLNN accurately captures localized curvature variations with lower computational cost, and we provide error distributions to confirm no large pointwise discrepancies despite low residuals. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's chain applies TFC to embed ICs/BCs exactly into a constrained expression, maps the domain to orthogonal polynomials, represents the free function via FLNN, and minimizes the mean-square residual of the governing DE to obtain the static deflection. This is then compared to an independent Galerkin computation of dynamic deflection. None of these steps reduce to self-definition (e.g., no output defined in terms of itself), fitted parameters renamed as predictions, or load-bearing self-citations that import uniqueness or ansatzes. The method is a constructive solver whose accuracy claims rest on residual minimization and exact constraint satisfaction rather than tautological equivalence to inputs. No evidence of the enumerated circular patterns appears in the abstract or described framework.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on standard beam bending differential equations from classical theory and the theory of functional connections; no free parameters or invented entities are described in the abstract.

axioms (1)

domain assumption The governing differential equation for static bending of a simply-supported perforated nanobeam under sinusoidal load is known and can be used to form a residual.
Invoked to define the cost function for training the network.

pith-pipeline@v0.9.0 · 5546 in / 1287 out tokens · 37023 ms · 2026-05-10T14:24:42.267408+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · 1 internal anchor

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D. Dan, X. Liao, L. Ge, F. Han, Physics-informed neural networks for solving free vibration response of cables considering bending stiﬀness, Journal of Vibration Engineering & Technologies 13 (8) (2025) 1–21

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A. Martínez-Esteban, P. Calvo-Barlés, L. Martín-Moreno, S. G. Rodrigo, Physics-informed neural net- works with dynamical boundary constraints, Communications in Nonlinear Science and Numerical Sim- ulation (2026) 109866

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Fernández-Sáez, R

J. Fernández-Sáez, R. Zaera, Vibrations of bernoulli-euler beams using the two-phase nonlocal elasticity theory, International Journal of Engineering Science 119 (2017) 232–248

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Garai, A

R. Garai, A. K. Gartia, S. Chakraverty, Eﬀect of axial functional gradation and periodic square per- forations on the vibration of nanobeams with elastic foundation, International Journal of Structural Stability and Dynamics (2026) 2750188

work page 2026
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I. Sahu, R. Garai, S. Chakraverty, Physics-informed functional link constrained framework with do- main mapping for solving bending analysis of an exponentially loaded perforated beam, arXiv preprint arXiv:2604.07025 (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026
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Mortari, H

D. Mortari, H. Johnston, L. Smith, High accuracy least-squares solutions of nonlinear diﬀerential equa- tions, Journal of computational and applied mathematics 352 (2019) 293–307

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S. Mall, S. Chakraverty, Single layer chebyshev neural network model for solving elliptic partial diﬀer- ential equations, Neural Processing Letters 45 (2017) 825–840. 15

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Mortari, The theory of connections: Connecting points, Mathematics 5 (4) (2017) 57

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Mortari, C

D. Mortari, C. Leake, The multivariate theory of connections, Mathematics 7 (3) (2019) 296

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Leake, H

C. Leake, H. Johnston, D. Mortari, The multivariate theory of functional connections: Theory, proofs, and application in partial diﬀerential equations, Mathematics 8 (8) (2020) 1303

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Paszke, S

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, A. Lerer, Automatic diﬀerentiation in pytorch, NIPS 2017 Autodiﬀ Workshop (2017)

work page 2017
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I. Sahu, S. Kumar, S. Chakraverty, Physics-informed functional link with theory of functional connec- tions technique for solving diﬀerential equations, Neurocomputing (2026) 132795. 16

work page 2026

[1] [1]

Luschi, F

L. Luschi, F. Pieri, A simple analytical model for the resonance frequency of perforated beams, Procedia Engineering 47 (2012) 1093–1096

work page 2012

[2] [2]

Luschi, F

L. Luschi, F. Pieri, An analytical model for the determination of resonance frequencies of perforated beams, Journal of Micromechanics and Microengineering 24 (5) (2014) 055004

work page 2014

[3] [3]

A. A. Abdelrahman, N. A. Mohamed, M. A. Eltaher, Static bending of perforated nanobeams including surface energy and microstructure eﬀects, Engineering with Computers 38 (Suppl 1) (2022) 415–435

work page 2022

[4] [4]

Vadivelu, V

C. Vadivelu, V. Prabaharan, P. K. Akhas, Flexural behavior of perforated rectangular hollow section cold-formed steel beams: An experimental and numerical study, Results in Engineering 26 (2025) 104821

work page 2025

[5] [5]

Raissi, P

M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning frame- work for solving forward and inverse problems involving nonlinear partial diﬀerential equations, Journal of Computational physics 378 (2019) 686–707

work page 2019

[6] [6]

Leake, D

C. Leake, D. Mortari, Deep theory of functional connections: A new method for estimating the solutions of partial diﬀerential equations, Machine learning and knowledge extraction 2 (1) (2020) 37–55

work page 2020

[7] [7]

Schiassi, C

E. Schiassi, C. Leake, M. De Florio, H. Johnston, R. Furfaro, D. Mortari, Extreme theory of functional connections: A physics-informed neural network method for solving parametric diﬀerential equations, arXiv preprint arXiv:2005.10632 (2020)

work page arXiv 2005

[8] [8]

Kumar, A

S. Kumar, A. K. Sahoo, S. Chakraverty, Physics-informed machine learning framework for approximat- ing the modiﬁed degasperis-procesi equation, in: 2023 International Conference on Ambient Intelligence, Knowledge Informatics and Industrial Electronics (AIKIIE), 2023, pp. 1–6

work page 2023

[9] [9]

Z. Li, J. Bai, H. Ouyang, S. Martelli, M. Tang, Y. Yang, H. Wei, P. Liu, R. Wei, Y. Gu, Physics-informed neural networks for friction-involved nonsmooth dynamics problems, Nonlinear Dynamics 112 (9) (2024) 7159–7183. 14

work page 2024

[10] [10]

Fallah, M

A. Fallah, M. M. Aghdam, Physics-informed neural network for bending and free vibration analysis of three-dimensional functionally graded porous beam resting on elastic foundation, Engineering with Computers 40 (1) (2024) 437–454

work page 2024

[11] [11]

A. K. Sahoo, S. Kumar, S. Chakraverty, Physics-informed neural network for vibration analysis of large membranes, Journal of Nonlinear, Complex and Data Science 25 (7-8) (2025) 505–521

work page 2025

[12] [12]

D. Dan, X. Liao, L. Ge, F. Han, Physics-informed neural networks for solving free vibration response of cables considering bending stiﬀness, Journal of Vibration Engineering & Technologies 13 (8) (2025) 1–21

work page 2025

[13] [13]

Martínez-Esteban, P

A. Martínez-Esteban, P. Calvo-Barlés, L. Martín-Moreno, S. G. Rodrigo, Physics-informed neural net- works with dynamical boundary constraints, Communications in Nonlinear Science and Numerical Sim- ulation (2026) 109866

work page 2026

[14] [14]

Eltaher, A

M. Eltaher, A. Abdraboh, K. Almitani, Resonance frequencies of size dependent perforated nonlocal nanobeam, Microsystem Technologies 24 (9) (2018) 3925–3937

work page 2018

[15] [15]

Reddy, Nonlocal theories for bending, buckling and vibration of beams, International journal of engineering science 45 (2-8) (2007) 288–307

J. Reddy, Nonlocal theories for bending, buckling and vibration of beams, International journal of engineering science 45 (2-8) (2007) 288–307

work page 2007

[16] [16]

Fernández-Sáez, R

J. Fernández-Sáez, R. Zaera, Vibrations of bernoulli-euler beams using the two-phase nonlocal elasticity theory, International Journal of Engineering Science 119 (2017) 232–248

work page 2017

[17] [17]

A. C. Eringen, On diﬀerential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54 (9) (1983) 47034710

work page 1983

[18] [18]

Kafkas, Size-dependent bending behavior of perforated nanobeams on winkler-pasternak foundation., International Journal of Engineering & Applied Sciences (1309-0267) 17 (1) (2025)

U. Kafkas, Size-dependent bending behavior of perforated nanobeams on winkler-pasternak foundation., International Journal of Engineering & Applied Sciences (1309-0267) 17 (1) (2025)

work page 2025

[19] [19]

Garai, A

R. Garai, A. K. Gartia, S. Chakraverty, Eﬀect of axial functional gradation and periodic square per- forations on the vibration of nanobeams with elastic foundation, International Journal of Structural Stability and Dynamics (2026) 2750188

work page 2026

[20] [20]

I. Sahu, R. Garai, S. Chakraverty, Physics-informed functional link constrained framework with do- main mapping for solving bending analysis of an exponentially loaded perforated beam, arXiv preprint arXiv:2604.07025 (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026

[21] [21]

Mortari, H

D. Mortari, H. Johnston, L. Smith, High accuracy least-squares solutions of nonlinear diﬀerential equa- tions, Journal of computational and applied mathematics 352 (2019) 293–307

work page 2019

[22] [22]

S. Mall, S. Chakraverty, Single layer chebyshev neural network model for solving elliptic partial diﬀer- ential equations, Neural Processing Letters 45 (2017) 825–840. 15

work page 2017

[23] [23]

Mortari, The theory of connections: Connecting points, Mathematics 5 (4) (2017) 57

D. Mortari, The theory of connections: Connecting points, Mathematics 5 (4) (2017) 57

work page 2017

[24] [24]

Mortari, C

D. Mortari, C. Leake, The multivariate theory of connections, Mathematics 7 (3) (2019) 296

work page 2019

[25] [25]

Leake, H

C. Leake, H. Johnston, D. Mortari, The multivariate theory of functional connections: Theory, proofs, and application in partial diﬀerential equations, Mathematics 8 (8) (2020) 1303

work page 2020

[26] [26]

Paszke, S

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, A. Lerer, Automatic diﬀerentiation in pytorch, NIPS 2017 Autodiﬀ Workshop (2017)

work page 2017

[27] [27]

I. Sahu, S. Kumar, S. Chakraverty, Physics-informed functional link with theory of functional connec- tions technique for solving diﬀerential equations, Neurocomputing (2026) 132795. 16

work page 2026