Geo-FNO: Geometry-Aware Fourier Neural Operator

• The paper demonstrates that Geo-FNO integrates geometric information by learning a domain deformation, enabling FFT-based spectral methods on non-rectangular domains.
• The methodology leverages a small MLP to approximate the inverse mapping of coordinates, significantly reducing test errors and computational costs across various PDE benchmarks.
• Geometry conditioning via channel concatenation effectively differentiates spectral resonance structures in periodic problems, as seen in the cubic defocusing nonlinear Schrödinger equation.

Geometry-Aware Fourier Neural Operator (Geo-FNO) denotes a class of Fourier neural operator architectures in which geometric information is encoded explicitly into the learned operator rather than treated as an external preprocessing step. In its canonical formulation, Geo-FNO extends the FFT-based Fourier Neural Operator to PDEs on arbitrary geometries by learning a smooth deformation from the physical domain to a uniform computational domain, thereby retaining FFT efficiency while handling point clouds, meshes, and design parameters (Li et al., 2022). A later geometry-conditioned formulation applies the same principle on periodic domains by concatenating the torus aspect-ratio parameter $\omega^2$ as an input channel, so that a single model can distinguish geometries with different Fourier resonance structures in the cubic defocusing nonlinear Schrödinger equation (Oguadimma et al., 25 Jun 2026).

1. Geometric motivation and scope

The original motivation for Geo-FNO is the geometric restriction of the standard Fourier Neural Operator. Because the FNO uses the Fast Fourier transform, it is limited to rectangular domains with uniform grids. Geo-FNO addresses this restriction by introducing geometry directly into the operator architecture, rather than relying only on interpolation or coordinate padding (Li et al., 2022).

In the general-domain setting, the relevant geometry may be an irregular physical domain $\Omega=D_a\subset\mathbb R^d$, an unstructured mesh, or a shape parameter vector $a\in\mathcal A$. In the periodic-domain setting, the relevant geometry may be a global parameter that changes the spectral structure of the PDE. For the cubic nonlinear Schrödinger equation on $\mathbb T_\omega^2$, the aspect ratio $\omega^2$ governs the Fourier resonance structure, so rational and irrational geometries can exhibit different high-frequency cascade behaviors (Oguadimma et al., 25 Jun 2026).

This yields two closely related meanings of “geometry-aware” in the literature. One concerns learned coordinate deformation for arbitrary domains; the other concerns explicit conditioning on a geometry parameter that changes the underlying operator. The shared idea is architectural exposure of geometry to the neural operator.

2. Learned-deformation Geo-FNO on arbitrary domains

The defining construction in Geo-FNO is a diffeomorphism between a uniform computational domain and the physical domain. Let $\Omega=D_a\subset\mathbb R^d$ be the physical domain and let $\Omega'=D^c=[0,1]^d$ be the computational domain equipped with a uniform grid $T^c$ and standard Fourier basis $\psi_k^c(\xi)=e^{2\pi i\langle \xi,k\rangle}$. Geo-FNO introduces

$\phi_a:D^c\to D_a,\qquad \xi\mapsto x=\phi_a(\xi),$

with inverse $\Omega=D_a\subset\mathbb R^d$0 (Li et al., 2022).

When no analytic deformation is available, Geo-FNO learns $\Omega=D_a\subset\mathbb R^d$1 by a small MLP:

$\Omega=D_a\subset\mathbb R^d$2

where $\Omega=D_a\subset\mathbb R^d$3 is a 3-layer feed-forward network of width $\Omega=D_a\subset\mathbb R^d$4 with periodic feature encoding on $\Omega=D_a\subset\mathbb R^d$5,

$\Omega=D_a\subset\mathbb R^d$6

The “identity + MLP” form initializes $\Omega=D_a\subset\mathbb R^d$7 near identity. No explicit Jacobian-determinant regularization is used; diffeomorphism is encouraged by end-to-end training with the operator network (Li et al., 2022).

The full operator is implemented by deforming inputs to $\Omega=D_a\subset\mathbb R^d$8, applying an FFT-based FNO there, and mapping back. With channel lift $\Omega=D_a\subset\mathbb R^d$9, projection $a\in\mathcal A$0, pointwise linear maps $a\in\mathcal A$1, geometry-aware Fourier integral operators $a\in\mathcal A$2, and activation $a\in\mathcal A$3, the solution operator is written as

$a\in\mathcal A$4

Its Fourier layers use a deformed Fourier transform

$a\in\mathcal A$5

and inverse

$a\in\mathcal A$6

In practice, sampling on the pushed-forward mesh $a\in\mathcal A$7 absorbs the Jacobian weight into nonuniform mesh sampling (Li et al., 2022).

3. Input representations, optimization, and implementation regime

Geo-FNO was designed to accommodate three geometry representations. For point-cloud or unstructured-mesh input, the deformation network maps each spatial point $a\in\mathcal A$8 to $a\in\mathcal A$9 via $\mathbb T_\omega^2$0, and no explicit mesh connectivity is required. For structured meshes such as O- or C-grids, one may define

$\mathbb T_\omega^2$1

in which case Geo-FNO reduces exactly to FNO using the standard FFT. For design-parameter input, the geometry parameter $\mathbb T_\omega^2$2 is embedded jointly with each coordinate because $\mathbb T_\omega^2$3 takes $\mathbb T_\omega^2$4 as input (Li et al., 2022).

Training is end-to-end with the relative $\mathbb T_\omega^2$5 operator loss over a dataset $\mathbb T_\omega^2$6:

$\mathbb T_\omega^2$7

No additional Jacobian-determinant or invertibility regularizer is used. Reported optimization hyperparameters are Adam with initial learning rate $\mathbb T_\omega^2$8, decay by $\mathbb T_\omega^2$9 every 100 epochs, 500 training epochs, four Fourier layers, width $\omega^2$0, and maximum mode number $\omega^2$1, domain-dependent (Li et al., 2022).

The benchmark suite spans elasticity, plastic forging, advection on the sphere, Euler airfoil flow, and Navier–Stokes pipe flow. The reported datasets are: Elasticity, $\omega^2$2 train and $\omega^2$3 test; Plastic Forging, $\omega^2$4; Advection on Sphere, $\omega^2$5; Airfoil, $\omega^2$6; Pipe, $\omega^2$7 (Li et al., 2022).

4. Empirical performance on general geometries

On irregular and structured geometries, Geo-FNO was evaluated against FNO with interpolation, UNet baselines, GNO, and DeepONet. In elasticity with unstructured input, the learned-$\omega^2$8 Geo-FNO achieved $\omega^2$9 test error, compared with $\Omega=D_a\subset\mathbb R^d$0 for FNO+uniform interpolation, $\Omega=D_a\subset\mathbb R^d$1 for UNet+uniform interpolation, $\Omega=D_a\subset\mathbb R^d$2 for GNO, and $\Omega=D_a\subset\mathbb R^d$3 for DeepONet. Heuristic R-mesh and O-mesh variants reported $\Omega=D_a\subset\mathbb R^d$4 and $\Omega=D_a\subset\mathbb R^d$5 test error, respectively (Li et al., 2022).

On structured-mesh problems, the reported Geo-FNO test errors are $\Omega=D_a\subset\mathbb R^d$6 for Airfoil, $\Omega=D_a\subset\mathbb R^d$7 for Pipe flow, and $\Omega=D_a\subset\mathbb R^d$8 for Plasticity. On sphere advection, Geo-FNO2D reported $\Omega=D_a\subset\mathbb R^d$9 test error, compared with $\Omega'=D^c=[0,1]^d$0 for FNO2D, while FNO3D exceeded $\Omega'=D^c=[0,1]^d$1 and UNet2D reported $\Omega'=D^c=[0,1]^d$2 (Li et al., 2022).

The computational advantage is a central empirical result. Geo-FNO inference is reported as $\Omega'=D^c=[0,1]^d$3 s per sample on GPU, whereas the traditional Euler solver requires $\Omega'=D^c=[0,1]^d$4 s per sample on CPU, implying $\Omega'=D^c=[0,1]^d$5 speedup. The same report states that Geo-FNO requires $\Omega'=D^c=[0,1]^d$6 less compute for the same $\Omega'=D^c=[0,1]^d$7 error as an implicit solver. All experiments were run on a single NVIDIA 3090 GPU (Li et al., 2022).

These results establish the practical meaning of geometry awareness in the original Geo-FNO sense: the method preserves FFT-based global receptive fields while removing the requirement that the physical domain itself be a uniform Cartesian grid.

5. Geometry conditioning on periodic domains

A specialized geometry-aware formulation appears in operator learning for the cubic defocusing nonlinear Schrödinger equation on two-dimensional flat tori with varying aspect ratios. The domain is

$\Omega'=D^c=[0,1]^d$8

with anisotropic Laplacian

$\Omega'=D^c=[0,1]^d$9

and PDE

$T^c$0

In Fourier series $T^c$1, the anisotropic eigenvalues are $T^c$2 (Oguadimma et al., 25 Jun 2026).

The model learns the one-step solution operator. If $T^c$3, then the exact propagator $T^c$4 sends $T^c$5 to $T^c$6, and a first-order Duhamel or splitting update may be written as

$T^c$7

The learned-operator viewpoint is

$T^c$8

A single Fourier layer updates a latent field $T^c$9 by

$\psi_k^c(\xi)=e^{2\pi i\langle \xi,k\rangle}$0

with trainable spectral filter $\psi_k^c(\xi)=e^{2\pi i\langle \xi,k\rangle}$1, pointwise map $\psi_k^c(\xi)=e^{2\pi i\langle \xi,k\rangle}$2, and activation $\psi_k^c(\xi)=e^{2\pi i\langle \xi,k\rangle}$3 (Oguadimma et al., 25 Jun 2026).

The geometry-aware step is explicit concatenation of the aspect ratio as a constant third input channel at each spatial location. The lifting map acts on

$\psi_k^c(\xi)=e^{2\pi i\langle \xi,k\rangle}$4

to produce $\psi_k^c(\xi)=e^{2\pi i\langle \xi,k\rangle}$5. The reported architecture uses input channels $\psi_k^c(\xi)=e^{2\pi i\langle \xi,k\rangle}$6, lift $\psi_k^c(\xi)=e^{2\pi i\langle \xi,k\rangle}$7, $\psi_k^c(\xi)=e^{2\pi i\langle \xi,k\rangle}$8 Fourier layers of width $\psi_k^c(\xi)=e^{2\pi i\langle \xi,k\rangle}$9, retained modes $\phi_a:D^c\to D_a,\qquad \xi\mapsto x=\phi_a(\xi),$0 in each direction, GELU activation, and projection $\phi_a:D^c\to D_a,\qquad \xi\mapsto x=\phi_a(\xi),$1 to produce $\phi_a:D^c\to D_a,\qquad \xi\mapsto x=\phi_a(\xi),$2. Sigmoid is also reported as giving the best one-step error (Oguadimma et al., 25 Jun 2026).

The dataset comprises two geometries, rational $\phi_a:D^c\to D_a,\qquad \xi\mapsto x=\phi_a(\xi),$3 and irrational $\phi_a:D^c\to D_a,\qquad \xi\mapsto x=\phi_a(\xi),$4. Initial data are random-phase Fourier series supported on

$\phi_a:D^c\to D_a,\qquad \xi\mapsto x=\phi_a(\xi),$5

with phases $\phi_a:D^c\to D_a,\qquad \xi\mapsto x=\phi_a(\xi),$6, scaled so that $\phi_a:D^c\to D_a,\qquad \xi\mapsto x=\phi_a(\xi),$7. The reference solver is Fourier pseudo-spectral plus integrating-factor RK4 on a $\phi_a:D^c\to D_a,\qquad \xi\mapsto x=\phi_a(\xi),$8 grid with $\phi_a:D^c\to D_a,\qquad \xi\mapsto x=\phi_a(\xi),$9. Solution snapshots are downsampled to $\Omega=D_a\subset\mathbb R^d$00 every $\Omega=D_a\subset\mathbb R^d$01 with $\Omega=D_a\subset\mathbb R^d$02. The training, validation, and test splits are $\Omega=D_a\subset\mathbb R^d$03, $\Omega=D_a\subset\mathbb R^d$04, and $\Omega=D_a\subset\mathbb R^d$05 trajectories per geometry. Optimization uses mean-squared error on $\Omega=D_a\subset\mathbb R^d$06, Adam with initial learning rate $\Omega=D_a\subset\mathbb R^d$07, batch size $\Omega=D_a\subset\mathbb R^d$08, $\Omega=D_a\subset\mathbb R^d$09 epochs, halving every $\Omega=D_a\subset\mathbb R^d$10 epochs, early stopping on validation, and no additional weight decay or dropout (Oguadimma et al., 25 Jun 2026).

Reported one-step relative $\Omega=D_a\subset\mathbb R^d$11 test errors are approximately $\Omega=D_a\subset\mathbb R^d$12 on the rational torus and $\Omega=D_a\subset\mathbb R^d$13 on the irrational torus. Multi-step rollout up to $\Omega=D_a\subset\mathbb R^d$14 over $\Omega=D_a\subset\mathbb R^d$15 steps remains stable with errors $\Omega=D_a\subset\mathbb R^d$16. The Sobolev-norm evolution reproduces the qualitative dichotomy predicted by the resonance theory: on the rational torus the $\Omega=D_a\subset\mathbb R^d$17 norm grows more rapidly, up to approximately $\Omega=D_a\subset\mathbb R^d$18 at $\Omega=D_a\subset\mathbb R^d$19, whereas on the irrational torus it stays near approximately $\Omega=D_a\subset\mathbb R^d$20 (Oguadimma et al., 25 Jun 2026).

The ablation results isolate the role of geometry conditioning. With the $\Omega=D_a\subset\mathbb R^d$21 channel, the mean errors are $\Omega=D_a\subset\mathbb R^d$22; without $\Omega=D_a\subset\mathbb R^d$23, they rise to $\Omega=D_a\subset\mathbb R^d$24. The study also reports that $\Omega=D_a\subset\mathbb R^d$25 is the best retained-mode choice among $\Omega=D_a\subset\mathbb R^d$26, that Sigmoid gives the lowest mean error, and that $\Omega=D_a\subset\mathbb R^d$27 Fourier layers give the best mean error while $\Omega=D_a\subset\mathbb R^d$28 to $\Omega=D_a\subset\mathbb R^d$29 layers do not improve performance and make optimization harder (Oguadimma et al., 25 Jun 2026). This shows that, in periodic dispersive PDEs, geometry awareness can be implemented by global parameter conditioning rather than by learned deformation.

6. Related operators, misconceptions, and open problems

Geo-FNO sits within a broader class of geometry-aware spectral operators, but its mechanism is specific. DAFNO introduces a smoothed characteristic function $\Omega=D_a\subset\mathbb R^d$30 on an enclosing rectangular box $\Omega=D_a\subset\mathbb R^d$31 and masks the Fourier integral layer by $\Omega=D_a\subset\mathbb R^d$32, preserving FFT acceleration while handling irregular geometries and topology changes (Liu et al., 2023). EqGINO addresses a different failure mode in 3D PDE surrogates by enforcing isotropy in the spectral domain through orbit-based weight sharing and combining this with E(3)-equivariant encoder and decoder blocks, yielding exact equivariance to the 24-element octahedral symmetry group of the cubic grid (Kim et al., 2 Jun 2026).

A concise comparison is given below.

Framework	Geometry mechanism	Representative setting
Geo-FNO	learned deformation $\Omega=D_a\subset\mathbb R^d$33	arbitrary geometries, point clouds, meshes, design parameters
geometry-conditioned FNO	concatenation of $\Omega=D_a\subset\mathbb R^d$34 as a constant input channel	cubic defocusing NLS on rational and irrational tori
DAFNO	smoothed characteristic mask $\Omega=D_a\subset\mathbb R^d$35 on an enclosing box	irregular geometries and topology changes
EqGINO	orbit-based isotropic spectral weights plus E(3)-equivariant encoder/decoder	irregular 3D geometries with rotations

Several misconceptions are clarified by the literature. Geo-FNO is not merely FNO plus interpolation; in the original framework, the learned deformation is part of the operator itself and is trained end-to-end (Li et al., 2022). Nor does “geometry-aware” imply a single architectural recipe: the NLS study shows that explicit conditioning on a scalar geometry parameter can be sufficient when the geometry acts primarily through the operator spectrum (Oguadimma et al., 25 Jun 2026). Conversely, DAFNO demonstrates that masking-based geometry encoding can outperform Geo-FNO on some benchmarks, reporting, for example, $\Omega=D_a\subset\mathbb R^d$36 versus $\Omega=D_a\subset\mathbb R^d$37 on hyperelastic material modeling at $\Omega=D_a\subset\mathbb R^d$38 samples and $\Omega=D_a\subset\mathbb R^d$39 versus $\Omega=D_a\subset\mathbb R^d$40 on the airfoil benchmark (Liu et al., 2023). This suggests that geometry awareness is a design principle rather than a unique implementation.

Open problems are stated explicitly in the Geo-FNO literature. For learned-deformation Geo-FNO, these include non-diffeomorphic topologies, possible mesh-regularization or barrier losses on $\Omega=D_a\subset\mathbb R^d$41, extension of FNO universal approximation and discretization-convergence theorems to the deformed setting, and scaling to 3D and time-dependent kernels (Li et al., 2022). For geometry-conditioned periodic operators, the reported gains from explicit $\Omega=D_a\subset\mathbb R^d$42 conditioning suggest that lightweight geometry channels may be effective whenever domain shape dictates resonance structure, but the paper limits its empirical study to two geometries, one-step operator learning, and the cubic defocusing NLS equation (Oguadimma et al., 25 Jun 2026).