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arxiv: 2601.01756 · v3 · pith:BOU3MLNMnew · submitted 2026-01-05 · 🧮 math.NA · cs.NA· cs.NE

A Wachspress-based transfinite formulation for exactly enforcing Dirichlet boundary conditions on convex polygonal domains in physics-informed neural networks

N. Sukumar , Ritwick Roy This is my paper

Pith reviewed 2026-05-22 11:31 UTC · model grok-4.3

classification 🧮 math.NA cs.NAcs.NE
keywords Wachspress coordinatestransfinite interpolationphysics-informed neural networksDirichlet boundary conditionsconvex polygonsdeep Ritz methodboundary condition enforcement
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The pith

Wachspress coordinates build smooth transfinite interpolants that enforce Dirichlet boundary conditions exactly inside physics-informed neural networks on convex polygons.

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

The paper develops a transfinite formulation that lifts any continuous boundary function into the interior of a convex polygon using Wachspress coordinates as blending functions. The neural network output is adjusted by subtracting the interior extension of its own boundary values and adding the interpolant, so the resulting trial function matches the prescribed Dirichlet data exactly on the boundary. This construction keeps the trial function kinematically admissible for the deep Ritz method and guarantees a bounded Laplacian because the Wachspress coordinates are smooth. The approach therefore removes the need for approximate distance functions that previously produced unbounded second derivatives. It is demonstrated on forward linear and nonlinear problems, an inverse heat conduction task, and a parametrized geometry Poisson problem.

Core claim

For a prescribed Dirichlet boundary function B, the transfinite interpolant g of B using Wachspress coordinates lifts functions from the boundary of a convex polygonal domain P to its interior. The trial function is formed as the neural network output minus the extension of its boundary restriction plus g. This ensures the trial function equals B on the boundary while remaining smooth inside, so its Laplacian stays bounded.

What carries the argument

Wachspress coordinates used as blending functions in a transfinite interpolation formula that generalizes bilinear Coons interpolation from rectangles to convex n-gons.

If this is right

  • The trial function is kinematically admissible for the deep Ritz variational formulation without additional penalty terms.
  • The neural network trial function possesses a bounded Laplacian because the Wachspress coordinates are smooth.
  • The same construction applies directly to linear and nonlinear forward problems, inverse heat conduction, and parametrized geometric Poisson boundary-value problems.
  • Wachspress coordinates can serve as a geometric feature map that encodes the boundary edges for networks solving problems on families of convex polygons.

Where Pith is reading between the lines

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

  • The method could reduce the need for large penalty parameters when boundary conditions are enforced in neural PDE solvers on polygonal domains.
  • If analogous blending coordinates exist for non-convex polygons or polyhedra, the same transfinite adjustment might extend without major reformulation.
  • Feeding Wachspress coordinates as input features may allow a single network to handle multiple polygonal shapes by treating the coordinates as an explicit geometric encoding.

Load-bearing premise

The domain must be a convex polygon so that Wachspress coordinates are well-defined, non-negative inside the domain, and exactly zero on all but one edge.

What would settle it

A numerical check on a convex polygon where the constructed trial function fails to recover the exact boundary values of B at test points on the edges, or where the computed Laplacian grows without bound near the boundary.

read the original abstract

In this paper, we present a Wachspress-based transfinite formulation on convex polygonal domains for exact enforcement of Dirichlet boundary conditions in physics-informed neural networks. This approach leverages prior advances in geometric design such as blending functions and transfinite interpolation over convex domains. For prescribed Dirichlet boundary function $\mathcal{B}$, the transfinite interpolant of $\mathcal{B}$, $g : \bar P \to C^0(\bar P)$, $\textit{lifts}$ functions from the boundary of a two-dimensional polygonal domain to its interior. The transfinite trial function is expressed as the difference between the neural network's output and the extension of its boundary restriction into the interior of the domain, with $g$ added to it. This ensures kinematic admissibility of the trial function in the deep Ritz method. Wachspress coordinates for an $n$-gon are used in the transfinite formula, which generalizes bilinear Coons transfinite interpolation on rectangles to convex polygons. Since Wachspress coordinates are smooth, the neural network trial function has a bounded Laplacian, thereby overcoming a limitation in a previous contribution where approximate distance functions were used to exactly enforce Dirichlet boundary conditions. For a point $\boldsymbol{x} \in \bar{P}$, Wachspress coordinates, $\boldsymbol{\lambda} : \bar P \to [0,1]^n$, serve as a geometric feature map for the neural network: $\boldsymbol{\lambda}$ encodes the boundary edges of the polygonal domain. This offers a framework for solving problems on parametrized convex geometries using neural networks. The accuracy of physics-informed neural networks is successfully assessed on forward problems (linear and nonlinear), an inverse heat conduction problem, and a parametrized geometric Poisson boundary-value problem.

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

0 major / 2 minor

Summary. The paper introduces a Wachspress-based transfinite formulation for exactly enforcing Dirichlet boundary conditions in physics-informed neural networks on convex polygonal domains. The transfinite trial function is constructed as the difference between the neural network output and the transfinite extension of its boundary restriction, plus the transfinite interpolant g of the prescribed boundary data B. This ensures the trial function is kinematically admissible for the deep Ritz method. Wachspress coordinates provide the blending functions, generalizing Coons interpolation, and their smoothness ensures a bounded Laplacian for the trial function. The coordinates are also used as a geometric feature map for parametrized problems. The method is assessed on forward linear and nonlinear problems, an inverse heat conduction problem, and a parametrized geometric Poisson problem.

Significance. If the numerical assessments confirm the exact enforcement and improved performance, this work represents a meaningful advance in PINN methodology by providing an exact, smooth boundary condition enforcement technique based on classical geometric design tools. It addresses a known limitation of approximate distance functions and enables applications to families of convex domains, potentially improving both accuracy and generalization in physics-informed machine learning for PDEs on polygonal geometries.

minor comments (2)
  1. The abstract refers to successful assessment on various problems but does not include any quantitative error measures or specific results; the full manuscript should highlight key numerical findings to support the claims.
  2. The description of how Wachspress coordinates serve as a geometric feature map for the neural network could be expanded with an explicit example or diagram for clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, the recognition of its significance for PINN boundary enforcement on convex polygons, and the recommendation for minor revision. We are pleased that the approach using Wachspress coordinates for transfinite interpolation is viewed as a meaningful advance over approximate distance functions.

Circularity Check

0 steps flagged

No significant circularity; relies on established Wachspress and transfinite properties

full rationale

The derivation defines the trial function u_NN(x) - transfinite extension of its boundary values + g (transfinite interpolant of B) and invokes the reproduction property of Wachspress blending functions to enforce u= B exactly on the boundary. This property is imported from classical geometric design literature on convex polygons and is not derived from or fitted to the neural network itself. The smoothness claim follows directly from the known C^∞ interior regularity of Wachspress coordinates on convex domains. A reference to a prior contribution using approximate distance functions appears but is not load-bearing for the present construction; the central kinematic-admissibility argument remains independent of any self-citation chain or redefinition of fitted quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central construction depends on standard geometric properties of Wachspress coordinates on convex polygons and the existence of a smooth transfinite lift; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption The computational domain is a convex polygon.
    Convexity guarantees that Wachspress coordinates are non-negative inside the domain and vanish on all but one edge, enabling the transfinite interpolant to reproduce boundary data exactly.
  • standard math Wachspress coordinates are C^infty smooth in the interior of the polygon.
    Smoothness is invoked to guarantee that the neural-network trial function possesses a bounded Laplacian.

pith-pipeline@v0.9.0 · 5856 in / 1601 out tokens · 60316 ms · 2026-05-22T11:31:50.852841+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

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

  1. Hard-constrained Physics-informed Neural Networks for Interface Problems

    math.NA 2026-04 conditional novelty 7.0

    Hard-constrained PINN formulations via windowing and buffer approaches enforce interface conditions by design and outperform soft-constrained baselines on 1D and 2D elliptic interface problems.

  2. Hard-constrained Physics-informed Neural Networks for Interface Problems

    math.NA 2026-04 conditional novelty 7.0

    Windowing and buffer hard-constrained PINNs enforce interface physics by design, yielding higher interface fidelity than soft-constrained baselines on elliptic benchmarks.

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