ADEx-FNO: A Unified Ambient-Domain Framework for Fourier Neural Operators on Varying Geometries

Fourier Neural Operators (FNOs) are powerful tools for learning solution operators of partial differential equations (PDEs), but they are traditionally restricted to uniform, structured grids. This paper addresses the challenge of applying FNOs to complex, varying geometries and unstructured meshes, which are common in engineering and computational fluid dynamics (CFD).

The authors introduce the Ambient-Domain Extension Fourier Neural Operator (ADEx-FNO). Instead of using trainable modules to learn geometry-to-grid mappings, ADEx-FNO uses a deterministic approach:

1. Embedding: Each physical domain is embedded into a fixed ambient hypercube.
2. Representation: The geometry is represented via a signed distance function.
3. Transfer: Data is extended to the ambient domain and mapped to a nonuniform rectilinear latent grid.
4. Inference: The FNO processes the data on this latent grid, and the output is then interpolated back to the target physical discretization.

By using a nonuniform grid and a Nonuniform Discrete Fourier Transform (NDFT), the model concentrates computational resources where spatial gradients are highest, such as near solid boundaries in fluid flow, without increasing the number of trainable parameters.

ADEx-FNO demonstrates high accuracy, achieving relative L2 errors between 0.32% and 0.77% on nonlinear Poisson and advection-reaction-diffusion problems. In practical CFD applications, using ADEx-FNO to provide an initial field for conventional solvers significantly accelerates convergence. For RANS simulations, the method reduced pseudo-time iterations by approximately 43–44%. In more complex URANS and DNS transfer scenarios, the framework successfully reduced the time required for flow development and statistical accumulation, proving its utility as a robust initialization tool for high-fidelity simulations.