MeshONet: A Generalizable and Efficient Operator Learning Method for Structured Mesh Generation

TL;DR

MeshONet reframes structured mesh generation as a multivariable operator-learning problem to overcome generalization limits of physics-informed approaches. It introduces a dual-branch, shared-trunk architecture with a Lift-Layer to jointly map boundary functions to physical coordinates, trained with an interior/boundary loss $L( heta)=\alpha L_{ ext{interior}}+\beta L_{ ext{boundary}}$ and data-fidelity terms. Empirical results show up to four orders of magnitude speedup over traditional methods while maintaining high mesh quality and enabling generalization across unseen geometries without retraining. The method also demonstrates robust mesh refinement performance, suggesting strong practical impact for real-time and large-scale mesh generation tasks, with future work extending to 3D and optimizing boundary sampling to control parameter size.

Abstract

Mesh generation plays a crucial role in scientific computing. Traditional mesh generation methods, such as TFI and PDE-based methods, often struggle to achieve a balance between efficiency and mesh quality. To address this challenge, physics-informed intelligent learning methods have recently emerged, significantly improving generation efficiency while maintaining high mesh quality. However, physics-informed methods fail to generalize when applied to previously unseen geometries, as even small changes in the boundary shape necessitate burdensome retraining to adapt to new geometric variations. In this paper, we introduce MeshONet, the first generalizable intelligent learning method for structured mesh generation. The method transforms the mesh generation task into an operator learning problem with multiple input and solution functions. To effectively overcome the multivariable mapping restriction of operator learning methods, we propose a dual-branch, shared-trunk architecture to approximate the mapping between function spaces based on input-output pairs. Experimental results show that MeshONet achieves a speedup of up to four orders of magnitude in generation efficiency over traditional methods. It also enables generalization to different geometries without retraining, greatly enhancing the practicality of intelligent methods.

Figures (17)

• Figure 1: Schematic of the structured mesh generation process, showing the mapping of the computational mesh $(\xi, \eta)$ to the physical mesh $(x, y)$ through the function $f$.
• Figure 2: Illustration of univariable and multivariable mapping. The input set $\{x_1, x_2, \dots, x_m\}$ represents the sensor locations, where each $x_i \in \mathbb{R}^k$. The variable $y$ represents the evaluation points, with $d$ points, each having $k$ dimensions. $G$ is the operator that maps the input functions to their corresponding solution values. (a) Univariable Mapping: The input is a single function $u$, evaluated at $m$ sensors, and the output is a $d$-dimensional vector $G(u)(y)$, representing the solution values at $d$ evaluation points. (b) Multivariable Mapping: The input consists of $k$ functions $u_1, u_2, \ldots, u_k$, evaluated at $m$ sensors. The output is a $d \times k$ matrix $G(u_1, \ldots, u_k)(y)$, where each row represents the solution values at $d$ evaluation points $y$, and the $j$-th column, generated by the sub-operator $G_j$, represents the $j$-th component of the solution, where $j = 1, \dots, k$.
• Figure 3: Architecture of MeshONet for mesh generation. The model consists of three main components: Branch-x Network, Trunk Network, and Branch-y Network. The Branch-x and Branch-y networks process sub-sampled boundary data, extracting features specific to the $x$ and $y$ coordinates, respectively. The Trunk Network operates on the computational mesh, applying a Lift-Layer to transform input coordinates $(\xi, \eta)$ into a higher-dimensional space, followed by feature extraction in the Trunk-Layer. Outputs from the Branch and Trunk networks are combined using a dot product operation in the Output Layer, producing physical mesh coordinates $(x, y)$ as functions $\hat{G}_x(u_1)(\xi, \eta)$ and $\hat{G}_y(u_2)(\xi, \eta)$, effectively mapping the computational domain to the physical domain.
• Figure 4: The figure presents the test cases, where (a) illustrates the test cases involving variations in the outer boundary, and (b) depicts the test cases with modifications to the inner boundary. The arrows indicate the direction of the corresponding changes.
• Figure 5: Loss comparison across different operator learning-based methods over 100,000 iterations, displayed on a logarithmic scale.