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Fourier Neural Operator with Learned Deformations for PDEs on General Geometries

Zongyi Li, Daniel Zhengyu Huang, Burigede Liu, Anima Anandkumar

TL;DR

Geo-FNO extends Fourier neural operators to arbitrary geometries by learning a deformation that maps irregular physical domains to a uniform latent grid, allowing FFT-based computation on nonuniform meshes and point-cloud inputs. The framework supports multiple input formats, employs a geometry-aware Fourier transform, and can optionally learn the deformation end-to-end, preserving discretization convergence. Across elasticity, plasticity, advection on the sphere, Euler airfoils, and Navier–Stokes pipe problems, Geo-FNO achieves substantial speedups (up to ~10^5x) and improved accuracy over interpolation-based baselines while maintaining forward and inverse design capabilities. The work lays a foundation for physics-informed variants and extensions to broader topologies with potential theoretical guarantees.

Abstract

Deep learning surrogate models have shown promise in solving partial differential equations (PDEs). Among them, the Fourier neural operator (FNO) achieves good accuracy, and is significantly faster compared to numerical solvers, on a variety of PDEs, such as fluid flows. However, the FNO uses the Fast Fourier transform (FFT), which is limited to rectangular domains with uniform grids. In this work, we propose a new framework, viz., geo-FNO, to solve PDEs on arbitrary geometries. Geo-FNO learns to deform the input (physical) domain, which may be irregular, into a latent space with a uniform grid. The FNO model with the FFT is applied in the latent space. The resulting geo-FNO model has both the computation efficiency of FFT and the flexibility of handling arbitrary geometries. Our geo-FNO is also flexible in terms of its input formats, viz., point clouds, meshes, and design parameters are all valid inputs. We consider a variety of PDEs such as the Elasticity, Plasticity, Euler's, and Navier-Stokes equations, and both forward modeling and inverse design problems. Geo-FNO is $10^5$ times faster than the standard numerical solvers and twice more accurate compared to direct interpolation on existing ML-based PDE solvers such as the standard FNO.

Fourier Neural Operator with Learned Deformations for PDEs on General Geometries

TL;DR

Geo-FNO extends Fourier neural operators to arbitrary geometries by learning a deformation that maps irregular physical domains to a uniform latent grid, allowing FFT-based computation on nonuniform meshes and point-cloud inputs. The framework supports multiple input formats, employs a geometry-aware Fourier transform, and can optionally learn the deformation end-to-end, preserving discretization convergence. Across elasticity, plasticity, advection on the sphere, Euler airfoils, and Navier–Stokes pipe problems, Geo-FNO achieves substantial speedups (up to ~10^5x) and improved accuracy over interpolation-based baselines while maintaining forward and inverse design capabilities. The work lays a foundation for physics-informed variants and extensions to broader topologies with potential theoretical guarantees.

Abstract

Deep learning surrogate models have shown promise in solving partial differential equations (PDEs). Among them, the Fourier neural operator (FNO) achieves good accuracy, and is significantly faster compared to numerical solvers, on a variety of PDEs, such as fluid flows. However, the FNO uses the Fast Fourier transform (FFT), which is limited to rectangular domains with uniform grids. In this work, we propose a new framework, viz., geo-FNO, to solve PDEs on arbitrary geometries. Geo-FNO learns to deform the input (physical) domain, which may be irregular, into a latent space with a uniform grid. The FNO model with the FFT is applied in the latent space. The resulting geo-FNO model has both the computation efficiency of FFT and the flexibility of handling arbitrary geometries. Our geo-FNO is also flexible in terms of its input formats, viz., point clouds, meshes, and design parameters are all valid inputs. We consider a variety of PDEs such as the Elasticity, Plasticity, Euler's, and Navier-Stokes equations, and both forward modeling and inverse design problems. Geo-FNO is times faster than the standard numerical solvers and twice more accurate compared to direct interpolation on existing ML-based PDE solvers such as the standard FNO.
Paper Structure (30 sections, 27 equations, 9 figures, 4 tables)

This paper contains 30 sections, 27 equations, 9 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Deformable meshes.
  3. Neural operators.
  4. Limitations of FNO on irregular geometries.
  5. Our contributions.
  6. Problem Settings and Preliminaries
  7. Problem settings.
  8. Input formats.
  9. Learning the solution operator
  10. Neural operator
  11. Geometry-Aware Fourier Neural Operator
  12. Deformation from the physical space to the computational space.
  13. Examples: Chebyshev Basis.
  14. Geometric Fourier transform
  15. Architecture of Geo-FNO and the deformation neural network
  16. ...and 15 more sections

Figures (9)

  • Figure 1: Geometry-aware FNO
  • Figure 2: Elasticity (a) and plasticity problems (b) introduced in \ref{['ssec:solid-elastic', 'ssec:solid-plastic']}. The comparison is shown between the reference obtained using a traditional solver (left) and the Geo-FNO result (right).
  • Figure 3: The cost-accuracy trade-off for Geo-FNO/UNet and traditional numerical solver with implicit/explicit temporal integrators on the airfoil problem.
  • Figure 4: Interpolation into different meshes for the elasticity problem
  • Figure 5: Visualization of $\phi_{a}^{-1}$
  • ...and 4 more figures

Theorems & Definitions (3)

  • Definition 1: Neural operator $\mathcal{G}_\theta$
  • Definition 2: Fourier integral operator $\mathcal{K}$
  • Remark 3: Visualization of $\phi_{a}^{-1}$