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

Source: arXiv - 2601.01756v1

Overview

The paper introduces a new way to enforce Dirichlet boundary conditions exactly in Physics‑Informed Neural Networks (PINNs) when the computational domain is a convex polygon. By borrowing the classic Wachspress coordinates from geometric modeling, the authors build a transfinite interpolation that lifts boundary data into the interior, guaranteeing that the neural‑network trial solution always satisfies the prescribed boundary values.

Key Contributions

• Wachspress‑based transfinite interpolant for convex polygons that generalizes the Coons bilinear interpolation used on rectangles.
• Exact Dirichlet enforcement in the deep Ritz formulation, eliminating the need for penalty terms or approximate distance functions.
• Geometric feature map: the vector of Wachspress coordinates (\boldsymbol\lambda(\mathbf{x})) is fed to the network, encoding the domain’s edge information directly.
• Bounded Laplacian of the trial function, which resolves stability issues observed in earlier exact‑enforcement schemes.
• Comprehensive evaluation on forward, inverse, and parametrized Poisson problems, demonstrating accuracy comparable to or better than standard PINN approaches.

Methodology

1. Domain representation – For an (n)-sided convex polygon (P), compute the Wachspress barycentric coordinates
   [ \boldsymbol\lambda(\mathbf{x}) = (\lambda_1,\dots,\lambda_n). ]
   These rational functions are 1 on a given edge and 0 on all others, providing a smooth “edge‑indicator” for any interior point.

2. Transfinite extension of boundary data – Given a Dirichlet boundary function (\mathcal{B}) defined on (\partial P), construct a continuous extension
   [ g(\mathbf{x}) = \sum_{i=1}^{n} \lambda_i(\mathbf{x}),\mathcal{B}_i(\mathbf{x}), ]
   where (\mathcal{B}_i) is the restriction of (\mathcal{B}) to edge (i). This classic transfinite interpolation exactly matches (\mathcal{B}) on each edge and varies smoothly inside the polygon.

3. Neural‑network trial function – Let (N_\theta(\mathbf{x})) be a standard feed‑forward network (e.g., fully‑connected with tanh or ReLU activations). The admissible trial solution is defined as
   [ u_\theta(\mathbf{x}) = g(\mathbf{x}) + \bigl(N_\theta(\mathbf{x}) - N_\theta|{\partial P}(\mathbf{x})\bigr), ]
   where (N\theta|{\partial P}) denotes the network evaluated only on the boundary and then lifted by the same transfinite map. By construction, (u\theta = \mathcal{B}) on (\partial P) for any (\theta).

4. Deep Ritz loss – The loss functional is the energy
   [ \mathcal{L}(\theta)=\int_P \left(\tfrac12|\nabla u_\theta|^2 - f,u_\theta\right),d\mathbf{x}, ]
   evaluated with automatic differentiation and quadrature points sampled inside (P). No penalty term for the boundary is required.

5. Training pipeline – The Wachspress coordinates are pre‑computed (or evaluated analytically) and supplied as additional inputs to the network, allowing the model to learn geometry‑aware features. Standard optimizers (Adam → L‑BFGS) are used.

Results & Findings

The bounded Laplacian of the trial function leads to smoother loss landscapes, which translates into faster convergence (≈30 % fewer epochs) and more stable training across different network depths.

Practical Implications

• Plug‑and‑play BC handling – Developers can drop the Wachspress transfinite layer into existing PINN pipelines to guarantee Dirichlet compliance without fiddling with penalty weights.
• Geometry‑aware networks – By feeding (\boldsymbol\lambda(\mathbf{x})) as input, a single model can solve PDEs on a whole family of convex polygons, enabling rapid design‑space exploration (e.g., shape optimization, parametric CAD).
• Reduced training cost – Exact BCs shrink the solution space, often requiring smaller networks and fewer training iterations—valuable when GPU time is at a premium.
• Compatibility with existing frameworks – The method relies only on standard automatic‑diff tools (TensorFlow, PyTorch) and simple rational functions, so integration into current scientific‑ML codebases is straightforward.

Limitations & Future Work

• Convexity requirement – Wachspress coordinates are defined only for convex polygons; extending the approach to non‑convex or curved domains will need alternative barycentric schemes (e.g., mean‑value or harmonic coordinates).
• Scalability to 3‑D – The paper focuses on 2‑D polygons; a 3‑D analogue would involve Wachspress polyhedral coordinates, which are more expensive to evaluate.
• Complex boundary conditions – The current formulation handles scalar Dirichlet data. Extending to vector‑valued or mixed (Dirichlet/Neumann) conditions is an open question.
• Adaptive sampling – While uniform random sampling works for the presented benchmarks, coupling the method with adaptive quadrature or error‑driven point selection could further improve efficiency.

Bottom line: By marrying classical geometric interpolation with modern deep learning, this work offers a clean, mathematically exact solution to one of the most cumbersome aspects of PINNs—enforcing Dirichlet boundaries—while keeping the implementation lightweight and developer‑friendly.

Authors

• N. Sukumar
• Ritwick Roy

Paper Information

• arXiv ID: 2601.01756v1
• Categories: math.NA, cs.NE
• Published: January 5, 2026
• PDF: Download PDF