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

TL;DR

This paper develops a theoretical framework linking neural implicit representations to PDE solution accuracy, establishing conditions for geometric regularity and error bounds that ensure reliable physics-based simulations.

Contribution

It introduces a unified theory connecting INR training errors to PDE solution accuracy and defines the geometric regularity needed for stable boundary value problem solutions.

Findings

INRs can support well-posed PDE problems under certain regularity conditions.

Training loss must scale quadratically with mesh size to match finite element convergence.

Provides error estimates linking neural approximation to finite element discretization.

Abstract

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.