Oriol Colomés* ([email protected]), Jan Modderman ([email protected])
Neural operators are gaining attention as powerful tools for solving PDEs. However, handling complex geometries remains a challenge, especially in the training phase. This is especially relevant when domain geometry is an input of the neural operator. Unfitted/Immersed or Embedded boundary methods have gained attention for allowing simulations in complex domains without requiring boundary-conforming meshes. Such approaches could facilitate the development of neural operators in complex geometries. Nonetheless, many existing methods require ad-hoc modifications to standard Finite Element structures, such as tessellating elements intersected by the embedded boundary, constructing special quadrature rules, or defining geometry-dependent surrogate boundaries. In this talk we present the Generalized Weighted Shifted Boundary Method (G-WSBM), a generalized formulation of the WSBM approach [1], which offers a generalized framework for handling complex geometries in a geometry-agnostic manner. By treating the geometry as a, possibly parametrized, field, the G-WSBM enables neural operators to efficiently process irregular domains without compromising accuracy. In the talk we will present the formulation and give some examples of its application to the solution of PDEs in complex domains.
[1] O. Colomés, A. Main, L. Nouveau and G. Scovazzi (2021). A weighted shifted boundary method for free surface flow problems. Journal of Computational Physics, 424, 109837.