Neural Geometry for PDEs: Regularity, Stability, and Convergence Guarantees

Implicit Neural Representations (INRs) have emerged as a powerful tool for geometric representation, yet their suitability for physics-based simulation remains underexplored. While metrics like Hausdorff distance quantify surface reconstruction quality, they fail to capture the geometric regularity required for provable numerical performance. This work establishes a unified theoretical framework connecting INR training errors to Partial Differential Equation (PDE) (specifically, linear elliptic equation) solution accuracy. We define the minimal geometric regularity required for INRs to support well-posed boundary value problems and derive \emph{a priori} error estimates linking the neural network's function approximation error to the finite element discretization error. Our analysis reveals that to match the convergence rate of linear finite elements, the INR training loss must scale quadratically relative to the mesh size.