Shape optimization involves the minimization of a cost function defined over a shape, often governed by a partial differential equation (PDE). Since analytical solutions are typically unavailable, we need to rely on numerical method to find an approximate solution. The level set method, when coupled with finite element analysis, is one of the most versatile numerical shape optimization approach. However, its reliance on meshing introduces limitations inherent to mesh-based methods.

In this talk, we present a fully meshless level set framework that leverages neural networks to parameterize the level set function and employs the graph Laplacian to solve the underlying PDE. This approach enables precise computation of geometric quantities such as normals and curvature. Furthermore, we exploit the flexibility of neural networks to address optimization problems within the class of convex shapes.