PINNsur: Physics-Informed Neural Networks for PDEs on Curved Surfaces

2026-05-26

AI Summary

This work addresses the limitations of traditional finite element methods for solving partial differential equations (PDEs) on surfaces—namely, sensitivity to mesh quality and geometric discretization errors—as well as the lack of convergence guarantees and systematic validation in existing physics-informed neural networks (PINNs). The authors propose PINNsur, a unified mesh-free framework capable of solving PDEs on arbitrary orientable surfaces. By leveraging neural implicit fields to approximate surface normals and projecting ambient-space differential operators onto the surface, PINNsur enables accurate intrinsic computations. Furthermore, the framework introduces a simple yet effective empirical convergence test that simultaneously validates both functional and geometric convergence—a gap unaddressed by prior approaches. Numerical experiments demonstrate that PINNsur reliably solves PDEs across surfaces with diverse topologies and curvatures while exhibiting robust convergence behavior.

Abstract

Partial differential equations (PDEs) on surfaces are fundamental to scientific computing and geometry processing. A popular approach to solving PDEs on surfaces is the finite element method (FEM), where the surface is divided into discrete geometric elements (usually triangles). Recently, physics-informed neural networks (PINNs) have emerged as a continuous, mesh-free alternative that does not suffer from FEM's sensitivity to mesh quality or geometric discretization errors. We present PINNSur, a simple framework for using PINNs on curved surfaces: we train a neural field to approximate the surface's normals, and then we express surface differential operators using their projection from $\mathbb{R}^3$ onto the surface. Since every orientable manifold has well-defined normals, our method is suitable for all such surfaces, regardless of curvature or topology, enabling many geometry processing applications. Moreover, despite their empirical success in solving PDEs in flat Euclidean domains, PINNs lack convergence guarantees to the true solution of the underlying PDE, and there is limited systematic experimental evidence demonstrating such convergence. This gap restricts their adoption as reliable solvers compared to established methods like FEM, where convergence to the true solution is well understood and theoretically grounded. These surface PDEs are particularly challenging to solve convergently, as one must not only deal with the convergence of the function approximation, but also with the convergence of the geometric approximation of the surface itself. In this work, we empirically investigate the convergence behavior of PINNs for solving surface PDEs by introducing a simple empirical convergence test.

Problem

• Physics-Informed Neural Networks
• Surface PDEs
• Convergence
• Curved Surfaces
• Geometric Approximation

Innovation

• Physics-Informed Neural Networks
• Surface PDEs
• Mesh-free Methods
• Convergence Analysis
• Neural Fields