Technical Summary: PINNsur

Problem Statement

Partial differential equations (PDEs) on curved two-dimensional surfaces embedded in $\mathbb{R}^3$ are fundamental to geometry processing, physical simulation, and scientific computing. The standard approach, the Finite Element Method (FEM), discretizes the surface into geometric elements (typically triangles). While FEM offers theoretical convergence guarantees, it is sensitive to mesh quality, requires careful refinement, and struggles with complex topologies or poor-quality triangulations.

Physics-Informed Neural Networks (PINNs) have emerged as a mesh-free alternative that approximates solutions via differentiable neural networks. However, existing PINN frameworks generally operate on flat Euclidean domains ($\mathbb{R}^d$) and lack robust methods for curved surfaces. Recent attempts to extend PINNs to surfaces often rely on parametric mappings (e.g., mapping a sphere to the surface), which limits applicability to specific topologies (e.g., genus-0) and introduces discretization errors. Furthermore, while PINNs show empirical success, there is a lack of systematic evidence and theoretical understanding regarding their convergence to the true solution, particularly when both the function approximation and the geometric approximation of the surface must converge simultaneously.

Methodology: PINNsur

The authors propose PINNsur, a framework for solving PDEs on arbitrary curved manifolds without requiring a parametric domain or explicit mesh discretization. The core innovation is the use of a normal-only neural field to construct surface differential operators.

1. Surface Operator Construction

Instead of parameterizing the surface $\Omega$, PINNsur defines surface differential operators by projecting Euclidean operators onto the tangent plane using the surface normal vector $\mathbf{n}$.

• Surface Gradient: $\nabla_\Omega u = \nabla_{\mathbb{R}^3} u - (\nabla_{\mathbb{R}^3} u \cdot \mathbf{n})\mathbf{n}$
• Surface Divergence: $\nabla_\Omega \cdot \mathbf{F} = \nabla_{\mathbb{R}^3} \cdot \mathbf{F} - \mathbf{n}^T J(\mathbf{F}) \mathbf{n}$
• Laplace-Beltrami Operator: $\Delta_\Omega u = \nabla_{\mathbb{R}^3} \cdot (\nabla_\Omega u) - \mathbf{n}^T J(\nabla_\Omega u) \mathbf{n}$

This formulation allows the Laplace-Beltrami operator to be computed via automatic differentiation on a standard coordinate network $u_\theta: \mathbb{R}^3 \to \mathbb{R}$, provided the normal field $\mathbf{n}$ is known.

2. Network Architecture

The framework employs two distinct neural networks:

• Normal Network ($\hat{n}_\theta$): A Siren network (MLP with sine activations) trained to approximate the surface normal field $\mathbf{n}$ at any point on $\Omega$. For surfaces given as meshes, this network learns to smooth the piecewise-constant mesh normals. For Neumann boundary conditions, a separate network learns the boundary normals $\mathbf{\nu}$.
• PDE Network ($u_\theta$): A coordinate network that solves the target PDE (e.g., Poisson or Helmholtz). It utilizes the pre-trained normal network to compute the Laplace-Beltrami operator via the projection formulas above.

3. Training and Loss

The PDE is solved by minimizing a loss function composed of:

• PDE Residual: The squared $L_2$ norm of the equation residual (e.g., $\|\Delta_\Omega u_\theta - f\|_2^2$).
• Soft Boundary Constraints: Penalty terms for Dirichlet ($\|u_\theta - g\|_2^2$) and Neumann ($\|\nabla_\Omega u_\theta \cdot \mathbf{\nu} - h\|_2^2$) conditions.

The method supports arbitrary manifold topologies, including closed surfaces (no boundary) and surfaces with complex boundaries, provided the normal field is continuous.

Key Contributions

1. PINNsur Framework: A simple, parameterization-free method for solving surface PDEs using a normals-only neural field. It eliminates the need for parametric maps, thereby supporting arbitrary topologies.
2. Empirical Convergence Criterion: The authors introduce a systematic test for PINN convergence. Analogous to FEM where error decreases with the number of elements, this criterion posits that PINN error should decrease as the number of tunable weights (#W) increases. Convergence is empirically defined by a negative slope in the log-log plot of error vs. #W.
3. Systematic Convergence Study: The paper provides extensive empirical evidence demonstrating that PINNsur converges to the true solution across various PDEs (Poisson, Helmholtz, Heat), boundary conditions (Dirichlet, Neumann, no-boundary), and geometries.

Results

The authors validate PINNsur against FEM and other PINN baselines:

• Convergence Behavior: Experiments show that as the network width (and thus #W) increases, the relative $L_2$ error systematically decreases. The convergence rates (slopes of -0.3 to -1.4 depending on the problem) are comparable to or better than FEM and significantly better than standard PINNs using tanh or ReLU activations.
• Comparison with Baselines:
  • vs. FEM: PINNsur achieves similar convergence rates to FEM for comparable degrees of freedom but is less sensitive to mesh quality.
  • vs. Parametric PINNs: Compared to the work of Williamson and Mitra [2025], PINNsur achieves lower errors and applies to arbitrary topologies, whereas the parametric approach is limited to surfaces homeomorphic to a sphere.
  • vs. Standard PINNs: Standard PINNs (e.g., Kharazmi et al., Yu et al.) fail to converge under the same refinement strategy, attributed to the superior high-frequency approximation capabilities of the Siren architecture.
• Applications: The method successfully solves:
  • Mean Curvature Flow: Simulating surface evolution.
  • Heat Geodesics: Computing geodesic distances via the heat equation.
  • Minimal Surfaces: Recovering catenoids from cylinders.
  • Harmonic Interpolation: Filling holes in surfaces with Dirichlet boundary conditions.

Significance and Limitations

The paper claims that PINNsur bridges a critical gap in geometric computing by providing a mesh-free, topology-agnostic solver with demonstrated convergence properties. It establishes that PINNs can be reliable solvers for surface PDEs if the geometric approximation (normals) is handled correctly and the network capacity is sufficiently refined.

Limitations:

• The method fails if the surface has a discontinuous normal field (e.g., sharp corners with Neumann boundaries).
• Convergence is not guaranteed if the solution $u$ contains extremely high frequencies or if the function range is too large.
• While mesh-free, the training computational cost can be higher than FEM, and the method does not yet offer theoretical convergence bounds, relying instead on empirical evidence.

The authors conclude by noting that while their current work is empirical, future directions include deriving theoretical convergence bounds based on neural network expressiveness and scaling laws.