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

2026-05-22 · Source: cs.NE updates on arXiv.org · Field: Science & Research — Mathematics & Computational Sciences, Engineering & Applied Sciences · Depth: Expert, extended

Summary

A new Wachspress-based transfinite formulation is introduced for exactly enforcing Dirichlet boundary conditions in Physics-Informed Neural Networks (PINNs) on convex polygonal domains. This method utilizes Wachspress coordinates as a geometric feature map, addressing a critical limitation of prior Approximate Distance Function (ADF) methods where the Laplacian of the trial function became unbounded at polygon vertices, causing training instability. The proposed approach ensures a bounded Laplacian, significantly improving solution accuracy, particularly when collocation points are near boundaries. The formulation is validated on forward, inverse, and parametrized geometric Poisson boundary-value problems, demonstrating superior accuracy with training losses as low as O(10^-9) for harmonic problems and O(10^-12) for Poisson problems on a quadrilateral, and robustness across different activation functions like tanh and SIREN.

Key takeaway

For Machine Learning Engineers developing PINNs for PDEs on complex polygonal geometries, you should adopt the Wachspress-based transfinite formulation. This method ensures exact Dirichlet boundary condition enforcement and a bounded Laplacian, significantly improving solution accuracy and training stability, especially near domain vertices. You can achieve O(10^-9) to O(10^-12) error rates, overcoming limitations of older ADF methods.

Key insights

Wachspress-based transfinite interpolation enables exact Dirichlet boundary condition enforcement in PINNs on complex geometries.

Principles

• Exact boundary condition enforcement simplifies PINN optimization.
• Bounded Laplacian of trial functions improves training stability.
• Geometric feature maps enhance neural network performance on complex shapes.

Method

Construct a trial function by adding a Wachspress-based transfinite interpolant of Dirichlet conditions to the neural network output, then subtracting the network's boundary restriction extension.

In practice

• Use Wachspress coordinates as geometric features for polygonal domains.
• Employ Adam + L-BFGS optimizers for robust PINN training.
• Apply to forward, inverse, and parametrized Poisson problems.

Topics

• Physics-Informed Neural Networks
• Dirichlet Boundary Conditions
• Wachspress Coordinates
• Transfinite Interpolation
• Polygonal Domains
• Deep Ritz Method

Best for: AI Scientist, Machine Learning Engineer, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by cs.NE updates on arXiv.org.