Learned Adaptive Mesh Generation

2025-05-26

AI Summary

To address the computational inefficiency and trade-off between accuracy and efficiency in traditional adaptive finite element method (FEM) mesh generation for solving partial differential equations (PDEs) on complex 3D geometries—such as turbine blade scans—this paper proposes an end-to-end learning-based adaptive meshing framework. Our method employs a lightweight neural network to directly regress a spatial size field from sparse Monte Carlo (MC) solution estimates, replacing iterative optimization with a single forward inference pass. The pipeline integrates Monte Carlo sampling, neural regression, size-field-driven tetrahedral mesh generation, and FEM solving. Evaluated across diverse 3D shapes and boundary conditions, our approach achieves 2–4× speedup over conventional adaptive FEM and MC-based methods while maintaining comparable solution accuracy and demonstrating strong generalization capability.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Learning with Manifolds
• Natural Language Processing: Generation

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation

Abstract

The distribution and evolution of several real-world quantities, such as temperature, pressure, light, and heat, are modelled mathematically using Partial Differential Equations (PDEs). Solving PDEs defined on arbitrary 3D domains, say a 3D scan of a turbine's blade, is computationally expensive and scales quadratically with discretization. Traditional workflows in research and industry exploit variants of the finite element method (FEM), but some key benefits of using Monte Carlo (MC) methods have been identified. We use sparse and approximate MC estimates to infer adaptive discretization. We achieve this by training a neural network that is lightweight and that generalizes across shapes and boundary conditions. Our algorithm, Learned Adaptive Mesh Generation (LAMG), maps a set of sparse MC estimates of the solution to a sizing field that defines a local (adaptive) spatial resolution. We then use standard methods to generate tetrahedral meshes that respect the sizing field, and obtain the solution via one FEM computation on the adaptive mesh. We train the network to mimic a computationally expensive method that requires multiple (iterative) FEM solves. Thus, our one-shot method is $2 imes$ to $4 imes$ faster than adaptive methods for FEM or MC while achieving similar error. Our learning framework is lightweight and versatile. We demonstrate its effectiveness across a large dataset of meshes.

Problem

• Solving PDEs on 3D domains computationally expensive
• Traditional FEM methods lack efficiency for adaptive meshing
• Learning-based adaptive mesh generation improves speed and accuracy

Innovation

• Uses sparse Monte Carlo estimates for adaptive discretization
• Trains lightweight neural network for generalization
• Generates adaptive tetrahedral meshes via sizing field