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

2026-08-09

AI Summary

This work addresses the challenge that Fourier Neural Operators (FNOs) struggle to generalize across variable geometries and independent discretizations by introducing a deterministic framework. The physical domain is embedded into a fixed hypercube, with geometry represented via a signed distance function. Input and solution fields are extended into an ambient domain and processed by an FNO on a unified, non-uniform Cartesian latent grid, after which they are interpolated back to the target mesh and restricted to the physical domain. This approach requires no trainable graph networks, point clouds, or geometric decoding modules, fully decoupling geometry handling from the optimization pipeline and enabling unified modeling of arbitrary domains. Evaluated on 2D/3D nonlinear Poisson and convection–reaction–diffusion problems, the method achieves relative L² errors of 0.32%–0.77%; when used as an initial field in CFD simulations, it reduces pseudo-time iterations by 44% on average, accelerates URANS physical time marching by 18.52%–27.51%, and shortens cross-condition DNS guidance intervals by 23.47%–48.21%.

Abstract

Fourier neural operators (FNOs) provide efficient nonlocal spectral learning, but varying geometries and independently chosen discretizations remain difficult to accommodate. We introduce the ambient-domain extension Fourier neural operator (ADEx-FNO), a deterministic framework that incorporates geometry without modifying the defining Fourier-operator layers. Each physical domain is embedded in a fixed ambient hypercube and represented by a signed distance function. Inputs and solution fields are deterministically extended to the ambient domain, transferred to a common, potentially nonuniform rectilinear latent grid, processed by the FNO, then interpolated to an independently chosen target discretization and restricted to the physical domain. All geometry-transfer operations lie outside the optimization procedure and require no trainable graph, point-cloud, deformation, or geometry-decoding modules. ADEx-FNO achieves relative l2 errors of 0.32%-0.77% on held-out smooth-domain nonlinear Poisson and advection-reaction-diffusion problems in 2D and 3D, and is also evaluated on unseen nonsmooth geometries. A single ADEx-FNO inference is then used to initialize conventional CFD solvers. For all 29 converged 2D and 3D RANS cases, pseudo-time iterations decrease, with mean reductions of 44.17% and 43.03%, respectively, with comparable gains across three mesh resolutions. URANS cases reduce post-window physical-time advances by 18.52%-27.51%. In transfer from 2D URANS training data to DNS at different Mach and Reynolds numbers, the bootstrap interval decreases by 23.47%-48.21%, depending on the target statistic. In all CFD tests, ADEx-FNO provides only the initial field; the governing-equation solver controls the subsequent solution, while physical or statistical consistency is assessed separately from computational savings.

Problem

• Fourier neural operators
• varying geometries
• discretization
• nonlocal spectral learning
• geometry representation

Innovation

• Fourier Neural Operator
• ambient-domain extension
• geometry-invariant learning
• signed distance function
• CFD acceleration