[Paper Review] Fourier Neural Operator with Learned Deformations for PDEs on General Geometries

Zongyi Li, Daniel Zhengyu Huang+2 more|arXiv (Cornell University)|Jul 11, 2022

Model Reduction and Neural Networks147 citations

TL;DR

Geo-FNO extends Fourier Neural Operator to arbitrary geometries by learning a deformation from irregular physical domains to a uniform latent space, enabling FFT-based solving on general meshes with end-to-end training.

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.

Motivation & Objective

• Address the challenge of applying FFT-based neural operators to irregular geometries and non-uniform meshes.
• Develop a geometry-aware framework that deforms physical domains to a uniform latent space for spectral methods.
• Enable end-to-end learning of the domain deformation jointly with the operator for improved accuracy.
• Demonstrate applicability to forward problems and inverse design across multiple PDEs (Elasticity, Plasticity, Euler, Navier–Stokes).

Proposed method

• Introduce geo-FNO by learning a (diffeomorphic) deformation from the physical domain to a unit torus so FFTs can be applied in latent space.
• Define forward and inverse geometric Fourier transforms that incorporate the deformation (F_a and F_a^{-1}) with appropriate Jacobian weighting.
• Parameterize the deformation map with a neural network and train it end-to-end with the Geo-FNO operator.
• Allow input formats including point clouds, non-uniform meshes, and design parameters; special cases reduce to standard FNO on structured meshes.
• Use Fourier-based global convolutions (K_l) in spectral space with activation layers to approximate the solution operator; extend to Fourier continuation when topology is irregular.
• Optionally embed complex domains into regular supersets and apply Fourier continuation to enable spectral solving on non-homeomorphic domains.

Experimental results

Research questions

• RQ1 Can Geo-FNO accurately approximate solution operators on arbitrary geometries with irregular meshes?
• RQ2 Does learning the domain deformation improve accuracy and speed compared to interpolation-based or fixed-deformation approaches?
• RQ3 How does Geo-FNO perform across forward PDE problems (Elasticity, Plasticity, Euler, Navier–Stokes) and in inverse design tasks?
• RQ4 What are the computational gains relative to traditional solvers and other ML-based PDE surrogates?
• RQ5 Can the deformation be learned jointly with the operator in an end-to-end framework to enable end-to-end differentiable design optimization?

Key findings

• Geo-FNO achieves substantial speedups, up to 1e5 times faster than traditional solvers on airfoil problems.
• Geo-FNO with learned deformation yields lower error than interpolation-based ML solvers and outperforms mesh-free methods like GraphNO and DeepONet in reported tests.
• End-to-end learned deformation (vs fixed R- or O-mesh) improves accuracy across tested Elasticity, Pipe flow, and Plasticity problems.
• When inputs are structured meshes, Geo-FNO reduces to standard FNO and retains FFT efficiency; learned deformation mainly helps irregular geometries.
• Inference time remains around 0.01 seconds per instance, preserving FNO’s speed while enabling geometry flexibility.
• Geo-FNO enables inverse design by optimizing geometry parameters (e.g., airfoil spline nodes) to achieve desired flow characteristics, with forward predictions guiding the optimization.