A Physics-Informed Neural Network framework for solving PDEs on point clouds via surface reconstruction.

Abstract

We propose a novel Physics-Informed Neural Network (PINN) framework for solving Partial Differential Equations (PDEs) on manifolds represented by raw point clouds, without requiring any geometric priors such as level set functions or explicit surface parametrization. Unlike prior methods that assume the availability of normal vectors or a predefined representation of the surface, our approach reconstructs an implicit surface representation using normalizing flows, enabling accurate PDE solutions without requiring labeled training data. To the best of our knowledge, this is the first PINN framework that does not rely on predefined surface characteristics or supervision. Experimental results demonstrate that our method achieves high accuracy even with non-uniformly distributed and noisy point clouds, where traditional numerical approaches often fail. Furthermore, our approach exhibits significantly faster convergence compared to existing PINN-based methods that require explicit surface knowledge. This work highlights the potential of learning-based geometric methods in automating PDE simulations on arbitrary 3D surfaces.