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 ($\det J \le 0$). 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 ($\mathcal{L}_{\text{barrier}}$) that explicitly penalizes non-positive determinants, ensuring a bijective mapping without metric collapse. Furthermore, by evaluating physical derivatives via exact metric transformation $J^{-T}\nabla_\xi u$, 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 $2.3\times$ lower relative $L^2$ error ($1.82\%$) and a $4.0\times$ improvement in Sobolev $H^1$ accuracy over state-of-the-art baselines. Official repository: https://github.com/GiovanniDagnese-paper/DIF-FNO.