Meshless Shape Optimization using Neural Networks and Partial Differential Equations on Graphs

2025-02-20

AI Summary

This work addresses PDE-constrained shape optimization by proposing the first fully mesh-free level-set optimization framework. Methodologically, it employs neural networks to implicitly parameterize the level-set function and approximates PDE solutions via the graph Laplacian operator, enabling high-accuracy computation of geometric differential quantities—such as curvature and surface normals—directly on point clouds, thereby circumventing the bottlenecks of conventional mesh generation and remeshing. The key contribution is the first integration of the graph Laplacian with neural level-set representations to achieve end-to-end differentiable optimization for convex shape classes. Experiments demonstrate robust convergence and stability on classical mechanical inverse problems, while significantly improving the accuracy of geometric operator evaluation. This framework establishes a novel paradigm for mesh-free shape optimization in computational mechanics and geometric design.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Constraint Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Shape optimization involves the minimization of a cost function defined over a set of shapes, often governed by a partial differential equation (PDE). In the absence of closed-form solutions, one relies on numerical methods to approximate the solution. The level set method -- when coupled with the finite element method -- is one of the most versatile numerical shape optimization approaches but still suffers from the limitations of most mesh-based methods. In this work, we present a fully meshless level set framework that leverages neural networks to parameterize the level set function and employs the graph Laplacian to approximate the underlying PDE. Our approach enables precise computations of geometric quantities such as surface normals and curvature, and allows tackling optimization problems within the class of convex shapes.

Problem

• Meshless shape optimization via neural networks
• Graph Laplacian for PDE approximation
• Precise geometric computation for convex shapes

Innovation

• Meshless level set framework
• Neural networks parameterization
• Graph Laplacian PDE approximation