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

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.