Neural Representation of Minimal Surfaces

2026-07-25

AI Summary

This work addresses the approximation errors inherent in conventional approaches to minimal surface representation and generation—errors arising either from discretization or from the use of physics-informed neural networks (PINNs). To overcome these limitations, the paper proposes a neural representation grounded in the Weierstrass–Enneper parameterization, which leverages an analytic formulation of minimal surfaces. By training the model with a variational objective derived from the Plateau problem, the method eliminates reliance on mesh-based representations or numerical solutions of differential equations. This framework constitutes the first neural approach to achieve nearly integration-error-free representation of minimal surfaces, simultaneously enforcing exact boundary conditions and preserving high geometric fidelity. Empirical results demonstrate its significant superiority over existing PINN-based and discrete mesh methods.

Abstract

We propose a neural representation for minimal surfaces. Unlike prior approaches based on discretization or Physics-Informed Neural Networks (PINNs), where meshes or neural fields are optimized to approximate the governing equations, our method builds on an exact representation, similar to the classical Weierstrass--Enneper parameterization, yielding minimal surfaces up to negligible quadrature error in evaluation. We formulate a training objective for the Plateau problem that optimizes over this representation.

Problem

• minimal surfaces
• neural representation
• Weierstrass–Enneper parameterization
• Plateau problem

Innovation

• neural representation
• minimal surfaces
• Weierstrass–Enneper parameterization
• Plateau problem
• exact representation