DIF-FNO: Diffeomorphic Implicit Fourier Neural Operators for Solving PDEs on Irregular Domains

Abstract

Fourier Neural Operators (FNO) have emerged as a powerful paradigm for learning mappings between infinite-dimensional function spaces, significantly accelerating Partial Differential Equation (PDE) solutions. However, conventional FNOs rely on Fast Fourier Transforms (FFT) over regular Cartesian grids, limiting their applicability to complex, non-convex physical domains. In this work, we propose the Diffeomorphic Implicit Fourier Neural Operator (DIF-FNO), a novel architecture that learns a continuous diffeomorphic mapping from an arbitrary irregular domain to a canonical torus while enforcing a Jacobian determinant distortion penalty. Evaluated on 2D Poisson equations over non-convex star-shaped domains, DIF-FNO achieves a Relative L2 test error of 13.51% with O(N log N) computational complexity.