DIF-FNO: Diffeomorphic Fourier Neural Operators for Complex Geometries via Topological Barrier Loss

Learning solution operators of partial differential equations (PDEs) on arbitrary deformed geometries remains a fundamental challenge in scientific machine learning. Existing approaches like Geo-FNO map physical domains to regular latent grids but frequently suffer from spatial metric collapse and grid folding under high boundary curvature (J ≤ 0). In this work, we propose DIF-FNO (Diffeomorphic Fourier Neural Operator), a novel operator learning framework that enforces topological integrity and exact Sobolev gradient representations. DIF-FNO incorporates a smooth Jacobian Barrier Loss (Lbarrier) that explicitly penalizes non-positive determinants, ensuring a bijective mapping without metric collapse. Furthermore, by evaluating physical derivatives via exact metric transformation J-T∇_ξ u, DIF-FNO eliminates high-frequency discretization artifacts on irregular boundaries. Benchmark evaluations on Deformed Darcy Flow and Transonic Airfoil Flow (NACA 0012, Mach 0.8) demonstrate that DIF-FNO achieves up to 2.3× lower relative L² error ($1.82%$) and a 4.0× improvement in Sobolev H¹ accuracy over state-of-the-art baselines. Official repository: https://github.com/GiovanniDagnese-paper/DIF-FNO.