Structure-Preserving Learning Improves Geometry Generalization in Neural PDEs

The authors introduce General-Geometry Neural Whitney Forms (Geo-NeW), a data-driven finite element method designed to solve Partial Differential Equations (PDEs) while preserving physical conservation laws through Finite Element Exterior Calculus. The model integrates geometry via transformer-based encoding and learned finite element spaces, enhancing generalization to unseen geometries and demonstrating state-of-the-art performance on steady-state PDE benchmarks. This approach is significant for practitioners as it offers a robust framework for developing physics-informed models that maintain accuracy across diverse geometrical configurations.