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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
FFT-based FNOs are restricted to rectangular domains with uniform meshes, limiting their use on irregular geometries. Geo-FNO learns to deform irregular physical domains into uniform latent meshes where FFT-based FNO computation remains available. Across varied PDEs, geometries, and forward or inverse-design tasks, it reports up to 10^5 acceleration over numerical solvers and half the error rate of interpolation-based methods.
Problem
FFT-based FNOs require rectangular domains with uniform meshes, while irregular-domain embeddings and interpolation can be inefficient or inaccurate.
Method
Geo-FNO learns a deformation from an irregular physical domain to a uniform latent mesh, applies FNO operations there, and supports multiple geometry input formats.
Results
Up to 10^5 acceleration over numerical solvers and half the error rate of previous interpolation-based methods are reported across the experiments.
Takeaways & Limitations
Geo-FNO combines FFT computational efficiency with learned-deformation flexibility for PDE modeling on irregular geometries.
Takeaways & Limitations
The paper leaves theoretical approximation bounds for Geo-FNO on general geometries as future work.
Abstract
from arXiv · showhide
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.
1. Introduction
Existing PDE surrogate methods are fast but struggle to represent full solutions on irregular geometries, while FFT-based FNOs require rectangular uniform meshes. Geo-FNO addresses this by learning deformations into a uniform latent mesh, preserving FFT efficiency across geometries and input formats.
- Motivation: FFT-based FNOs are limited to rectangular domains with uniform meshes, while interpolation on irregular meshes can introduce large errors.Irregular-domain embeddings can also waste computation, especially for highly irregular geometries.
- Contribution: Geo-FNO supports arbitrary geometries and inputs including point clouds, non-uniform meshes, and design parameters.The framework is designed for both forward modeling and inverse design across several PDE families.
- Method: Geo-FNO deforms an irregular physical domain into a uniform latent mesh, applies FNO and FFT operations there, then maps the solution back.The deformation may be fixed or learned end-to-end with the FNO architecture.
- Results: 10^5 acceleration compared to the numerical solver and half the error rate compared to previous interpolation-based methods are reported across the experiments.Geo-FNO also maintains discretization convergence and can evaluate high-resolution cases after training on low-resolution data.
- Method: The deformation induces an adaptive mesh and generalized Fourier basis on the physical domain while retaining the computational efficiency of FFT.Unlike traditional deformed-mesh spectral solvers, Geo-FNO approximates the solution operator data-dependently rather than solving PDE equations in deformed Fourier space.
- Extensions: The framework is presented as extensible to general topologies through domain decomposition and to physics-informed neural operators through differentiable deformations.These extensions are described as possibilities rather than evaluated results in the supplied passage.
2. Problem Settings and Preliminaries
The paper formulates parametric PDE solution operators over varied domains and introduces neural operators as function-space mappings learned from data. It then describes Fourier integral operators whose FFT implementation depends on periodic square domains and uniform meshes.
- Problem settings: The problem considers stationary and time-dependent parametric PDEs on bounded, orientable manifolds embedded in Euclidean space.The domain is parameterized by design parameters, while initial and boundary conditions are treated as fixed in the stated setting.
- Input formats: Domain inputs may be represented as meshes or point clouds, functions, and design parameters.These representations describe the geometry supplied to the solution operator.
- Operator learning: A neural operator Gθ is trained as a surrogate for the PDE solution operator using paired input-output data.The supplied formulation optimizes the operator with a relative empirical data loss over sampled data pairs.
- Neural operators: Neural operators compose linear integral operators with pointwise nonlinear activations to approximate nonlinear operators between function spaces.Their layers include pointwise linear maps, integral kernels, biases, and fixed activation functions.
- Fourier operators: The Fourier integral operator uses a Fourier transform and basis functions, with the standard FFT available for periodic square domains and uniform meshes.The paper extends this setting toward non-uniform meshes and irregular domains.
3. Geometry-Aware Fourier Neural Operator
Geo-FNO deforms physical domains into a uniform computational space, enabling Fourier transforms while adapting to irregular geometries and meshes. It supports fixed or learned coordinate maps and reduces to standard FNO for structured inputs.
- Geometric deformation: Geo-FNO maps each physical domain to a shared unit-torus computational domain with a uniform mesh and standard Fourier basis.The deformation induces adaptive physical meshes and bases through pushforward and pullback operations.
- Geometric deformation: Chebyshev methods arise by applying a cosine deformation to cosine-series bases, concentrating grid points near non-periodic boundaries.The cosine deformation places more points near −1 and 1 and can be implemented with an extended-domain FFT.
- Geometric Fourier transform: The geometric Fourier transform maps physical-domain functions to computational spectral space, where standard Fourier transforms apply on uniform structured meshes.Its inverse transform evaluates the spectral representation back on the physical domain; mesh sampling and optional monitor weights enter the transform.
- Coordinate-map learning: Geo-FNO can use a given coordinate map or learn one jointly with the solution operator in an end-to-end manner.The learned inverse map is parameterized by a neural network taking spatial coordinates and geometry parameters as inputs, with sinusoidal features improving expressiveness.
- Architecture and special cases: For structured meshes, canonical indexing creates a uniform unit-cube mesh, so Geo-FNO reduces directly to standard FNO and its transforms reduce to FFT.The architecture lifts inputs, applies geometric Fourier convolutions, and projects outputs; structured inputs recover the standard FNO case.
- Domain decomposition: If the input topology is not homeomorphic to a disk or torus, no diffeomorphism to the computational domain exists, requiring embedding into a larger regular domain.For general 2-manifolds, decomposition into sphere- or torus-like subdomains uses multiple sub-models and deformation maps.
4. Numerical Examples
Geo-FNO is evaluated across PDEs, geometries, meshes, and tasks, including elasticity, plasticity, spherical advection, airfoils, and pipe flows. The experiments report improved accuracy, preserved discretization-convergence, and substantial acceleration relative to numerical and interpolation-based methods.
- Experimental scope: Geo-FNO is tested on varied geometries and PDEs, including elasticity, plasticity, spherical advection, airfoil, and pipe problems.The evaluation includes both forward modeling and inverse design tasks.
- Benchmark comparisons: Geo-FNO outperforms interpolation-based FNO and UNet, mesh-free GNO and DeepONet, while preserving standard FNO speed.These comparisons cover irregular domains and non-uniform meshes.
- Elasticity: Geo-FNO has lower elasticity error than interpolation-based FNO and UNet, and learned deformation is more accurate than fixed heuristic deformations.The interpolation baselines have test errors larger than 5%.
- Spherical advection: Geo-FNO supports spherical topology through domain decomposition and outperforms projected FNO and UNet baselines; embedded FNO3D has remarkably high test error.Splitting the sphere into northern and southern hemispheres enables FFT2D without raising dimensionality.
- Discretization convergence: Geo-FNO retains discretization-convergence: training on a 56 × 51 airfoil mesh yields test errors of 0.0147, 0.0329, and 0.0428 on three resolutions.The evaluated meshes are 56 × 51, 111 × 51, and 221 × 51.
5. Conclusion and future works
Geo-FNO combines learned or fixed domain deformations with FNO computation to handle arbitrary geometries and input formats. The paper outlines physics-informed, topology, and theoretical extensions as future directions.
- Conclusion: Geo-FNO deforms irregular physical domains into uniform latent meshes, where FFT-based FNO computation retains efficiency while improving geometric representation.The deformation may be fixed or learned end-to-end with the FNO architecture.
- Physics-informed settings: Physics-informed Geo-PINO can compute derivatives through the deformation map and optimize PDE residuals to represent solution functions.This extends the framework beyond data-driven surrogate learning when datasets are unavailable.
- General topologies: For challenging topologies without a diffeomorphism to uniform computational space, Fourier continuation or domain decomposition may be required.The authors leave coupled training across decomposed sub-domains as future work.
- Theoretical guarantees: Extending universal approximation bounds from FNO to Geo-FNO on general geometries remains an open theoretical direction.The proposed route would combine bounds for the deformed equation with approximation bounds for the inverse deformation network.