---
title: 'Geo-FNO: Geometry-Aware Fourier Neural Operator'
url: https://www.emergentmind.com/topics/geometry-aware-fourier-neural-operator-geo-fno
type: topic
---

# Geo-FNO: Geometry-Aware Fourier Neural Operator

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 [2207.05209]. 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 [2606.27459].

## 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 [2207.05209].

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 [2606.27459].

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 $\phi_a^{-1}:D_a\to D^c$ [2207.05209].

When no analytic deformation is available, Geo-FNO learns $\phi_a^{-1}$ by a small MLP:
$$
\phi_a^{-1}(x,a)=x+f_\theta(x,a),
$$
where $f_\theta:\mathbb R^d\times\mathbb R^{d_a}\to\mathbb R^d$ is a 3-layer feed-forward network of width $\sim 32$ with periodic feature encoding on $x$,
$$
x\mapsto [x,\sin(2^k\pi x),\cos(2^k\pi x)]_{k=0\ldots K}.
$$
The “identity + MLP” form initializes $\phi_a^{-1}$ near identity. No explicit Jacobian-determinant regularization is used; diffeomorphism is encouraged by end-to-end training with the operator network [2207.05209].

The full operator is implemented by deforming inputs to $D^c$, applying an FFT-based FNO there, and mapping back. With channel lift $P$, projection $Q$, pointwise linear maps $W_\ell$, geometry-aware Fourier integral operators $K_\ell(\phi_a)$, and activation $\sigma$, the solution operator is written as
$$
G_\theta(a)(x)
=Q\circ[W_L+K_L(\phi_a)+b_L]\circ \sigma\circ\cdots\circ \sigma\circ[W_1+K_1(\phi_a)+b_1]\circ P(a)(\phi_a^{-1}(x)).
$$
Its Fourier layers use a deformed Fourier transform
$$
F_a[v](k)=\int_{D_a} v(x)\,e^{-2\pi i\langle \phi_a^{-1}(x),k\rangle}\,|\det \nabla\phi_a^{-1}(x)|\,dx,
$$
and inverse
$$
F_a^{-1}[\hat w](x)=\sum_{|k|\le \kappa_{\max}} \hat w(k)\,e^{2\pi i\langle \phi_a^{-1}(x),k\rangle}.
$$
In practice, sampling on the pushed-forward mesh $T_a=\phi_a(T^c)$ absorbs the Jacobian weight into nonuniform mesh sampling [2207.05209].

## 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 $x_i$ to $D^c$ via $\phi_a^{-1}(x_i,a)$, and no explicit mesh connectivity is required. For structured meshes such as O- or C-grids, one may define
$$
\phi_a^{-1}(i_1,\dots,i_d)=(i_1/s_1,\dots,i_d/s_d),
$$
in which case Geo-FNO reduces exactly to FNO using the standard FFT. For design-parameter input, the geometry parameter $a\in\mathbb R^{d_a}$ is embedded jointly with each coordinate because $\phi_a^{-1}$ takes $(x,a)$ as input [2207.05209].

Training is end-to-end with the relative $L^2$ operator loss over a dataset $\{(a_j,u_j)\}$:
$$
\mathcal L_{\mathrm{data}}(G_\theta)\approx \frac1N\sum_j \frac{\|u_j-G_\theta(a_j)\|_{L^2(D_{a_j})}}{\|u_j\|_{L^2(D_{a_j})}}.
$$
No additional Jacobian-determinant or invertibility regularizer is used. Reported optimization hyperparameters are Adam with initial learning rate $10^{-3}$, decay by $1/2$ every 100 epochs, 500 training epochs, four Fourier layers, width $d_v=32$, and maximum mode number $\sim 12$, domain-dependent [2207.05209].

The benchmark suite spans elasticity, plastic forging, advection on the sphere, Euler airfoil flow, and Navier–Stokes pipe flow. The reported datasets are: Elasticity, $1\,000$ train and $200$ test; Plastic Forging, $900/80$; Advection on Sphere, $1\,000/200$; Airfoil, $1\,000/200$; Pipe, $1\,000/200$ [2207.05209].

## 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-$\phi$ Geo-FNO achieved $2.29\%$ test error, compared with $5.08\%$ for FNO+uniform interpolation, $5.31\%$ for UNet+uniform interpolation, $12.60\%$ for GNO, and $9.65\%$ for DeepONet. Heuristic R-mesh and O-mesh variants reported $5.36\%$ and $3.63\%$ test error, respectively [2207.05209].

On structured-mesh problems, the reported Geo-FNO test errors are $1.38\%$ for Airfoil, $0.67\%$ for Pipe flow, and $0.74\%$ for Plasticity. On sphere advection, Geo-FNO2D reported $3.32\%$ test error, compared with $3.81\%$ for FNO2D, while FNO3D exceeded $100\%$ and UNet2D reported $31.32\%$ [2207.05209].

The computational advantage is a central empirical result. Geo-FNO inference is reported as $\lesssim 0.01$ s per sample on GPU, whereas the traditional Euler solver requires $\sim 3\,600$ s per sample on CPU, implying $\gtrsim 10^5\times$ speedup. The same report states that Geo-FNO requires $\sim 10^4\times$ less compute for the same $1.4\%$ error as an implicit solver. All experiments were run on a single NVIDIA 3090 GPU [2207.05209].

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
$$
\mathbb T^2_{\,\omega}=\mathbb R/2\pi\mathbb Z\times \mathbb R/2\pi\mathbb Z
$$
with anisotropic Laplacian
$$
\Delta_\omega=\partial_x^2+\omega^2\partial_y^2,
$$
and PDE
$$
i\,\partial_t u+\Delta_\omega u=|u|^2u,\qquad u(0,\cdot)=u_0\in H^s(\mathbb T^2).
$$
In Fourier series $u(t,x,y)=\sum_{k=(m,\ell)\in\mathbb Z^2}\hat u_k(t)e^{i(mx+\ell y)}$, the anisotropic eigenvalues are $\lambda_k=m^2+\omega^2\ell^2$ [2606.27459].

The model learns the one-step solution operator. If $u_n(x)=u(n\Delta t,x)$, then the exact propagator $S^{\Delta t}$ sends $\{\hat u_k(t)\}$ to $\{\hat u_k(t+\Delta t)\}$, and a first-order Duhamel or splitting update may be written as
$$
\hat u_{n+1}(k)
= e^{-i\lambda_k\Delta t}\hat u_n(k)
-i\Delta t\,e^{-i\lambda_k\Delta t}\widehat{|u_n|^2u_n}(k)
+\mathcal O(\Delta t^2).
$$
The learned-operator viewpoint is
$$
u_{n+1}(x)=\mathcal G_\theta(u_n(x),\omega^2).
$$
A single Fourier layer updates a latent field $v^{(t)}:\mathbb T^2\to\mathbb R^d$ by
$$
v^{(t+1)}(x)=\sigma\Bigl(Wv^{(t)}(x)+\mathcal F^{-1}[R\,\mathcal F(v^{(t)})](x)\Bigr),
$$
with trainable spectral filter $R$, pointwise map $W$, and activation $\sigma$ [2606.27459].

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
$$
(\Re u,\Im u,\omega^2)\in\mathbb R^3
$$
to produce $v^{(0)}\in\mathbb R^{d_v}$. The reported architecture uses input channels $3$, lift $P:\mathbb R^3\to\mathbb R^{64}$, $T=4$ Fourier layers of width $64$, retained modes $K=12$ in each direction, GELU activation, and projection $Q:\mathbb R^{64}\to\mathbb R^2$ to produce $(\Re u_{n+1},\Im u_{n+1})$. Sigmoid is also reported as giving the best one-step error [2606.27459].

The dataset comprises two geometries, rational $\omega^2=1$ and irrational $\omega^2=\sqrt2$. Initial data are random-phase Fourier series supported on
$$
K_0=\{(m,\ell):-2\le m,\ell\le 2,\ (m,\ell)\neq (0,0)\},
$$
with phases $\phi_{m,\ell}\sim\mathrm{Uniform}[0,2\pi]$, scaled so that $\|u_0\|_{H^2}=R=1.8263$. The reference solver is Fourier pseudo-spectral plus integrating-factor RK4 on a $256^2$ grid with $\Delta t_{\mathrm{ref}}=2\times 10^{-3}$. Solution snapshots are downsampled to $64^2$ every $\Delta t_{\mathrm{data}}=T_f/40$ with $T_f=2\pi$. The training, validation, and test splits are $800$, $200$, and $200$ trajectories per geometry. Optimization uses mean-squared error on $(\Re,\Im)$, Adam with initial learning rate $10^{-3}$, batch size $16$, $100$ epochs, halving every $25$ epochs, early stopping on validation, and no additional weight decay or dropout [2606.27459].

Reported one-step relative $L^2$ test errors are approximately $6.25\times 10^{-2}$ on the rational torus and $3.66\times 10^{-2}$ on the irrational torus. Multi-step rollout up to $t=10T_f$ over $400$ steps remains stable with errors $\mathcal O(10^{-2})$. The Sobolev-norm evolution reproduces the qualitative dichotomy predicted by the resonance theory: on the rational torus the $H^2$ norm grows more rapidly, up to approximately $2.1$ at $t=10T_f$, whereas on the irrational torus it stays near approximately $1.9$ [2606.27459].

The ablation results isolate the role of geometry conditioning. With the $\omega^2$ channel, the mean errors are $(3.66,\,6.25)\times 10^{-2}$; without $\omega^2$, they rise to $(4.67,\,10.34)\times 10^{-2}$. The study also reports that $K=12$ is the best retained-mode choice among $\{8,12,16,20\}$, that Sigmoid gives the lowest mean error, and that $2$ Fourier layers give the best mean error while $6$ to $10$ layers do not improve performance and make optimization harder [2606.27459]. 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 $\chi_\epsilon$ on an enclosing rectangular box $\mathbb T$ and masks the Fourier integral layer by $\chi_\epsilon(x)\chi_\epsilon(y)$, preserving FFT acceleration while handling irregular geometries and topology changes [2305.00478]. 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 [2606.03260].

A concise comparison is given below.

| Framework | Geometry mechanism | Representative setting |
|---|---|---|
| Geo-FNO | learned deformation $\phi_a^{-1}(x,a)=x+f_\theta(x,a)$ | arbitrary geometries, point clouds, meshes, design parameters |
| geometry-conditioned FNO | concatenation of $\omega^2$ as a constant input channel | cubic defocusing NLS on rational and irrational tori |
| DAFNO | smoothed characteristic mask $\chi_\epsilon$ 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 [2207.05209]. 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 [2606.27459]. Conversely, DAFNO demonstrates that masking-based geometry encoding can outperform Geo-FNO on some benchmarks, reporting, for example, $1.09\%$ versus $2.32\%$ on hyperelastic material modeling at $1000$ samples and $0.596\%$ versus $1.65\%$ on the airfoil benchmark [2305.00478]. 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 $\det\nabla\phi_a$, extension of FNO universal approximation and discretization-convergence theorems to the deformed setting, and scaling to 3D and time-dependent kernels [2207.05209]. For geometry-conditioned periodic operators, the reported gains from explicit $\omega^2$ 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 [2606.27459].

Source: https://www.emergentmind.com/topics/geometry-aware-fourier-neural-operator-geo-fno