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

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.