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

Area of Science:

• Computational Geometry
• Numerical Analysis
• Machine Learning

Background:

• Solving Partial Differential Equations (PDEs) on complex 3D surfaces is crucial in various scientific and engineering fields.
• Existing methods often require explicit surface representations or geometric priors, limiting their applicability to raw, unstructured data.
• Physics-Informed Neural Networks (PINNs) offer a promising data-driven approach but typically rely on well-defined surface geometries.

Purpose of the Study:

• To develop a novel Physics-Informed Neural Network (PINN) framework capable of solving PDEs directly on manifolds represented by raw point clouds.
• To eliminate the necessity for geometric priors, such as level set functions or explicit surface parametrizations, in PDE simulations.
• To establish a supervision-free PINN framework for automated PDE simulations on arbitrary 3D surfaces.

Main Methods:

• Reconstruction of an implicit surface representation from raw point clouds using normalizing flows.
• Integration of the implicit surface representation within a PINN framework to enforce physical laws.
• Training the PINN without requiring labeled data or predefined surface characteristics like normal vectors.

Main Results:

• The proposed PINN framework accurately solves PDEs on manifolds represented by non-uniformly distributed and noisy point clouds.
• Achieved high accuracy in scenarios where traditional numerical methods often fail.
• Demonstrated significantly faster convergence rates compared to existing PINN methods that require explicit surface knowledge.

Conclusions:

• The novel PINN framework successfully addresses the challenge of solving PDEs on complex 3D surfaces from raw point cloud data.
• This approach represents a significant advancement, being the first PINN framework to operate without predefined surface characteristics or supervision.
• The work underscores the potential of learning-based geometric methods for automating and enhancing PDE simulations on arbitrary 3D manifolds.