Convergence analysis of wPINNs on manifolds with boundary

Develop a rigorous convergence analysis for the proposed weak physics-informed neural network framework when applied to geometry-compatible hyperbolic conservation laws on Riemannian manifolds with boundary under Dirichlet boundary conditions, including precise error bounds and conditions ensuring well-posedness.

Background

The theoretical results in the paper focus on boundaryless manifolds. Although the authors demonstrate that their algorithm can be extended to manifolds with boundaries and present numerical experiments for Dirichlet boundary conditions, they explicitly state that the convergence theory for the boundary case is not yet developed.

Establishing convergence guarantees for manifolds with boundary is important for practical applications (e.g., geophysical models or image processing on surfaces) where boundary conditions are common and may affect stability and error accumulation in weak solution approximations.