3DMeshNet: A Three-Dimensional Differential Neural Network for Structured Mesh Generation

TL;DR

The paper addresses the challenge of generating fast, high-quality 3D structured meshes for numerical simulations. It introduces 3DMeshNet, a physics-informed neural network that embeds elliptic PDE residuals in an unsupervised loss to learn the mapping between a parametric domain and a computational mesh, with a Finite Difference layer enabling efficient derivatives. Key contributions include loss function reweighting via task uncertainties, a surface-fitting boundary loss, and a gradient projection mechanism to mitigate conflicting gradients, enabling rapid, CPU-friendly mesh generation after offline training. Empirical results show that 3DMeshNet achieves superior mesh quality and significantly lower meshing overhead than traditional methods and other neural approaches, with training time reduced by up to 85%. This work demonstrates a scalable, data-efficient pathway for 3D mesh generation and lays groundwork for extending PINN-based meshing to more complex geometries.

Abstract

Mesh generation is a crucial step in numerical simulations, significantly impacting simulation accuracy and efficiency. However, generating meshes remains time-consuming and requires expensive computational resources. In this paper, we propose a novel method, 3DMeshNet, for three-dimensional structured mesh generation. The method embeds the meshing-related differential equations into the loss function of neural networks, formulating the meshing task as an unsupervised optimization problem. It takes geometric points as input to learn the potential mapping between parametric and computational domains. After suitable offline training, 3DMeshNet can efficiently output a three-dimensional structured mesh with a user-defined number of quadrilateral/hexahedral cells through the feed-forward neural prediction. To enhance training stability and accelerate convergence, we integrate loss function reweighting through weight adjustments and gradient projection alongside applying finite difference methods to streamline derivative computations in the loss. Experiments on different cases show that 3DMeshNet is robust and fast. It outperforms neural network-based methods and yields superior meshes compared to traditional mesh partitioning methods. 3DMeshNet significantly reduces training times by up to 85% compared to other neural network-based approaches and lowers meshing overhead by 4 to 8 times relative to traditional meshing methods.

Figures (12)

• Figure 1: The architecture of the proposed 3DMeshNet. It fuses a Neural network, which learns mesh partitioning from 3D points, with a Physics-informed learning part that enforces physical laws via elliptic differential equations. The Neural network extracts features and predicts mesh coordinates, while the Physics-informed learning part uses a finite difference layer for efficient derivatives calculation, guiding the NN's training.
• Figure 2: Coordinate points of the three-dimensional finite difference.
• Figure 3: Conflicting gradients and gradient projections.
• Figure 4: Cell Non-Orthogonality; The concept of cell non-orthogonality, where the black dots represent the centroids of the cells, and $\boldsymbol{f}$ is the normal of the shared face. The angle $\boldsymbol{\theta }$ is defined by the vector normal to the shared face and the line connecting the two centroids.
• Figure 5: Comparison of structured mesh generation results for 2D-case1 geometry using TFI, PDE, PINN, MGNet, and 3DMeshNet methods. The colors in the subfigures below indicate mesh quality: red represents higher orthogonality, and blue denotes lower.