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

A team led by N. Sukumar and Ritwick Roy has published a significant mathematical advance for physics-informed neural networks (PINNs). Their paper introduces a Wachspress-based transfinite formulation designed to exactly enforce Dirichlet boundary conditions—a fundamental requirement in solving physics equations—on convex polygonal domains. This addresses a persistent challenge where PINNs often struggle to satisfy boundary conditions precisely, leading to solution inaccuracies. The core innovation is using Wachspress coordinates, which are smooth functions defined for any convex polygon, to create a geometric feature map. This map, denoted λ, encodes the domain's boundary edges and allows for the exact 'lifting' of boundary data into the interior.

The new formulation constructs a trial function by blending the neural network's output with a transfinite interpolant built from Wachspress coordinates. This ensures the function is 'kinematically admissible,' meaning it automatically satisfies the prescribed boundary conditions everywhere. Critically, because Wachspress coordinates are smooth, the resulting trial function has a bounded Laplacian. This resolves a major drawback of previous methods that used approximate distance functions, which could cause unbounded second derivatives and numerical instability. The method generalizes the well-known Coons patch interpolation from rectangles to any convex polygon.

In testing, the researchers successfully applied their framework to a range of problems, demonstrating its practical utility. They validated accuracy on standard forward problems (both linear and nonlinear), a challenging inverse heat conduction problem, and a parametrized geometric Poisson problem. This shows the technique is not just a theoretical improvement but a robust tool for solving real engineering and physics simulations where domains are often complex polygons. The work provides a foundational framework for using neural networks to solve PDEs on parametrized convex geometries with guaranteed boundary condition enforcement.