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arxiv: 2605.00863 · v1 · submitted 2026-04-21 · 💻 cs.CE · physics.comp-ph

Recognition: unknown

Physics-informed neural networks for form-finding of unilateral membrane structures

Luigi Sibille , Sigrid Adriaenssens , Carlo Olivieri
Authors on Pith no claims yet

Pith reviewed 2026-05-10 00:31 UTC · model grok-4.3

classification 💻 cs.CE physics.comp-ph
keywords physics-informed neural networksmembrane equilibrium analysisform-findingunilateral membranesboundary conditionsfinite element methodspartial differential equations
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The pith

Physics-informed neural networks can generate equilibrium shapes for unilateral membranes by minimizing PDE residuals at collocation points, matching traditional finite element solutions.

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

The paper investigates physics-informed neural networks as a mesh-free way to solve the second-order elliptic PDE that governs the form-finding of unilateral membranes under the membrane equilibrium analysis approach. Two formulations are developed and tested: a soft-boundary version that adds a penalty term for boundary mismatches during training, and a hard-boundary version that builds the Dirichlet conditions exactly into the network using distance and lift functions. Both are applied to cases involving self-weight, concentrated loads, and horizontal actions in tension-only and compression-only states. The results show close agreement with FEM solutions, with the hard-boundary version producing smaller errors and smoother residuals especially near the edges.

Core claim

Two PINN formulations are proposed to enforce the equilibrium PDE residual at collocation points rather than through mesh-based discretization. The soft-BC version imposes boundary conditions via a penalty term, while the hard-BC version satisfies them exactly through distance and lift functions. On three case studies of varying geometrical complexity, both produce membrane surfaces in close agreement with an FEM-based PDE solver. The hard-BC formulation yields smaller errors and a smoother residual distribution, particularly near the boundary, while the soft-BC version still gives structurally meaningful results when simpler implementation is preferred.

What carries the argument

Physics-informed neural networks that minimize the residual of the second-order elliptic PDE for membrane equilibrium analysis at collocation points, with boundary conditions handled either through a penalty term (soft-BC) or exactly via distance and lift functions (hard-BC).

Load-bearing premise

That minimizing the PDE residual at collocation points during training, together with the chosen boundary-condition treatment, will converge to the unique solution of the second-order elliptic boundary-value problem.

What would settle it

A new test case where the trained PINN surface deviates substantially from the known FEM solution in regions away from training artifacts, even after refining collocation points or network capacity, would show the approach fails to recover the governing PDE solution.

Figures

Figures reproduced from arXiv: 2605.00863 by Carlo Olivieri, Luigi Sibille, Sigrid Adriaenssens.

Figure 1
Figure 1. Figure 1: Geometry and loads in the MEA formulation. The membrane surface 𝑆 is represented above its plan projection Ω. The right panel shows the projected membrane stresses and the external loads acting on an infinitesimal region of the planform. where 𝑓 ∶ Ω → ℝ represents the height of the membrane above the planform. The membrane is subjected to external loads acting on its surface, including self-weight and poss… view at source ↗
Figure 2
Figure 2. Figure 2: Overview of the proposed PINN architectures. The weights 𝑤pde and 𝑤bc in eq. (16) are adapted during training using the ReLoBRaLo algorithm [26]. At each epoch, ReLoBRaLo computes tentative weights propor￾tional to the exponential of the relative loss improvement over a randomly selected past epoch. The result is a con￾vex combination of the previous weights and the tentative ones, controlled by a balance … view at source ↗
Figure 3
Figure 3. Figure 3: Considered case studies together with their boundary conditions and load configurations. From left to right: rectangular domain, the three-legged domain, and the four-legged domain. The supports are highlighted in black, while the self-weight 𝑞 load, the applied vertical point load 𝑃 , and the horizontal action 𝐻 are indicated in red [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Vertical load 𝑞, cumulative horizontal load ℎ1 , and cumulative horizontal load ℎ2 entering the governing PDE. From left to right, the columns correspond to the rectangular domain, the three-legged domain, and the four-legged domain [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The top row compares the membrane surfaces obtained with the Soft-BC PINN solution (red) and with FEniCSx (gray). The bottom row shows the pointwise difference between the two solutions, computed as 𝑓PINN − 𝑓FEniCSx. From left to right, the columns correspond to the rectangular domain, the three-legged domain, and the four-legged domain. for the four-legged domain. The relative 𝐿2 errors, i.e. the RMSE nor… view at source ↗
Figure 6
Figure 6. Figure 6: Distance and lifting functions adopted in the hard-BC formulation for the three case studies. The rows correspond, from top to bottom, to the rectangular domain, the three-legged domain, and the four-legged domain. The columns show, from left to right, the reference membrane geometry, the distance function 𝐷(𝒙), and the lifting function 𝐺(𝒙) [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The top row compares the membrane surfaces obtained with the Hard-BC PINN solution (red) and with FEniCSx (gray). The bottom row shows the pointwise difference between the two solutions, computed as 𝑓PINN − 𝑓FEniCSx. From left to right, the columns correspond to the rectangular domain, the three-legged domain, and the four-legged domain. Page 9 of 16 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Convergence of the RMSE of the PDE residual on the training collocation point set and on the fixed validation point set for the three case studies. The rows correspond, from top to bottom, to the rectangular domain, the three-legged domain, and the four-legged domain. The soft-BC and hard-BC formulations are shown in the left and right panels, respectively. In each panel, solid lines denote the training PD… view at source ↗
Figure 9
Figure 9. Figure 9: RMSE convergence of the PINN solutions relative to the FEniCSx reference for the three case studies. The soft-BC and hard-BC formulations are shown in the left and right panels, respectively. The dashed vertical line marks the transition from Adam training (30 000 epochs) to L-BFGS refinement (10 000 steps). The curves correspond to the rectangular, three-legged, and four-legged domains [PITH_FULL_IMAGE:f… view at source ↗
Figure 10
Figure 10. Figure 10: ASF surfaces associated with the three case studies. From left to right, the columns correspond to the rectangular domain, the three-legged domain, and the four-legged domain [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Maximum principal stress 𝜎1 for the three case studies, shown on the three-dimensional membrane surface (top row) and in plan view (bottom row). From left to right, the columns correspond to the rectangular domain, the three-legged domain, and the four-legged domain Let the prescribed boundary profile on the outer circle 𝑟 = 𝑅 be represented by the truncated Fourier expansion 𝑔(𝜃) = 𝑎0 + ∑ 𝐾 𝑘=1 ( 𝑎𝑘 cos(… view at source ↗
Figure 12
Figure 12. Figure 12: Principal membrane stress directions shown on the three-dimensional membrane surfaces (top row) and in plan view (bottom row). At each sampled point, the blue and red segments indicate the directions associated with the two principal stresses, 𝜎1 and 𝜎2 , respectively. From left to right, the columns correspond to the rectangular domain, the three-legged domain, and the four-legged domain. The coefficient… view at source ↗
read the original abstract

Form-finding of unilateral membrane structures is commonly addressed by solving equilibrium equations with Finite Element Methods (FEMs). This paper investigates Physics-Informed Neural Networks (PINNs) as an alternative, where the equilibrium equation is enforced by minimizing its residual at collocation points during neural-network training rather than by solving a mesh-based discretized system. This approach is well suited to form-finding problems based on Membrane Equilibrium Analysis (MEA), in which the unknown membrane surface is governed by a second-order elliptic Partial Differential Equation (PDE) with Dirichlet boundary conditions. Two PINN formulations are proposed and compared: a soft-Boundary Condition (soft-BC) approach, where the boundary conditions are imposed through a penalty term, and a hard-BC approach, where they are satisfied exactly by construction through distance and lift functions. The methods are assessed on three case studies with different geometrical complexity, including compression-only and tension-only stress states, and combined self-weight, concentrated vertical loads, and horizontal actions. Both formulations produce membrane surfaces in close agreement with solutions obtained using an FEM-based PDE solver. The hard-BC formulation gives smaller errors and a smoother residual distribution, especially near the boundary, showing that exact enforcement of the Dirichlet conditions improves overall accuracy. The soft-BC formulation still provides structurally meaningful solutions and remains attractive when simpler implementation is preferred and limited relaxation of the boundary data is acceptable. Overall, the results show that PINNs are a viable alternative for MEA-based form-finding.

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 / 1 minor

Summary. The manuscript proposes Physics-Informed Neural Networks (PINNs) as an alternative to FEM for form-finding of unilateral membrane structures under Membrane Equilibrium Analysis (MEA). It introduces two formulations for enforcing the governing second-order elliptic PDE with Dirichlet boundary conditions: a soft-BC approach using a penalty term in the loss and a hard-BC approach that exactly satisfies the boundary conditions via distance and lift functions. The methods are evaluated on three case studies with varying geometry, stress states (compression-only, tension-only), and loads (self-weight, concentrated, horizontal), claiming close agreement with FEM solutions and superior performance for the hard-BC variant.

Significance. If the central claims hold under quantitative scrutiny, the work would demonstrate a viable mesh-free alternative for solving the elliptic BVP in MEA form-finding and provide concrete evidence that exact boundary-condition enforcement improves accuracy near boundaries in structural PINN applications. The direct numerical comparison to an independent FEM solver is a positive feature, as is the explicit contrast between soft and hard BC treatments.

major comments (2)
  1. [Numerical results] Numerical results section: The claims of 'close agreement' and that 'the hard-BC formulation gives smaller errors and a smoother residual distribution' rest on qualitative visual comparisons of surfaces and residual plots only. No quantitative error tables, L2 or maximum-norm differences between PINN and FEM solutions, or pointwise error statistics are supplied, which is load-bearing for the assertion that PINNs are a viable alternative and that hard-BC is demonstrably superior.
  2. [Method] Method and training description: No convergence studies (collocation-point density, network size, or training epochs), residual-decay rates, optimization diagnostics, or multi-seed statistics are reported. This leaves unverified the key assumption that gradient descent on the collocation residual (plus soft penalty or hard enforcement) reaches the unique solution of the second-order elliptic BVP rather than a spurious stationary point, especially for the soft-BC formulation.
minor comments (1)
  1. [Figures] Figure captions and residual plots would be clearer with explicit color-bar scales and quantitative annotations indicating the magnitude of the reported 'smoother' residual distribution for hard-BC.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the major comments below and will revise the manuscript to incorporate quantitative metrics and convergence analyses.

read point-by-point responses

  1. Referee: [Numerical results] Numerical results section: The claims of 'close agreement' and that 'the hard-BC formulation gives smaller errors and a smoother residual distribution' rest on qualitative visual comparisons of surfaces and residual plots only. No quantitative error tables, L2 or maximum-norm differences between PINN and FEM solutions, or pointwise error statistics are supplied, which is load-bearing for the assertion that PINNs are a viable alternative and that hard-BC is demonstrably superior.

    Authors: We agree with this observation. The revised manuscript will include quantitative error tables reporting L2 and maximum-norm differences between the PINN and FEM solutions for each of the three case studies. Pointwise error statistics will also be provided to substantiate the claims of close agreement and the improved performance of the hard-BC formulation, particularly near the boundaries. revision: yes

  2. Referee: [Method] Method and training description: No convergence studies (collocation-point density, network size, or training epochs), residual-decay rates, optimization diagnostics, or multi-seed statistics are reported. This leaves unverified the key assumption that gradient descent on the collocation residual (plus soft penalty or hard enforcement) reaches the unique solution of the second-order elliptic BVP rather than a spurious stationary point, especially for the soft-BC formulation.

    Authors: We acknowledge the need for more detailed training analysis. In the revision, we will add convergence studies with respect to collocation point density and network size. Residual decay plots and optimization diagnostics will be included. Multi-seed statistics from several training runs with different initializations will be reported to demonstrate consistency. We will also discuss how the hard-BC approach directly enforces the BVP, while the soft-BC provides a good approximation as evidenced by the FEM comparison. revision: yes

Circularity Check

0 steps flagged

No circularity; PINN residual minimization is validated against independent FEM solver

full rationale

The derivation proceeds by encoding the second-order elliptic MEA equilibrium PDE as a loss minimized at collocation points (with soft or hard BC enforcement) and directly comparing the resulting surfaces to those from a separate FEM-based PDE solver on three case studies. No equation reduces to its own inputs by construction, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests on self-citation chains or uniqueness theorems imported from the authors' prior work. The central claim is externally falsifiable via the FEM benchmark and remains independent of the training procedure itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard assumptions of membrane equilibrium theory and the ability of neural networks to approximate solutions to elliptic PDEs when residuals are minimized; no new entities are postulated.

axioms (2)
  • domain assumption Unilateral membrane equilibrium is governed by a second-order elliptic PDE with Dirichlet boundary conditions.
    This is the governing equation stated in the abstract for MEA-based form-finding.
  • domain assumption A neural network can represent the solution surface sufficiently well when the PDE residual is driven to zero at collocation points.
    Core premise of the PINN training procedure described.

pith-pipeline@v0.9.0 · 5564 in / 1270 out tokens · 42204 ms · 2026-05-10T00:31:24.929584+00:00 · methodology

discussion (0)

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