Geometry-Aware Neural Operators

• Geometry-aware neural operators are advanced methods that integrate explicit geometric descriptors like latent charts and boundary frames to model non-Euclidean domains.
• They employ diverse encoding techniques—including graph structures, point-cloud attention, and deformation-based transforms—to capture complex geometric features.
• Empirical studies demonstrate significant error reduction and improved efficiency in simulations across CFD, elasticity, and aerodynamics applications.

Geometry-aware neural operators are operator-learning methods in which geometry enters the approximation not merely as raw coordinates but as an explicit argument, latent chart, symmetry prior, boundary frame, deformation field, manifold model, or solver-side structural constraint. In this literature, the operator may be written as $\mathcal{G}[w,\Phi]\to u$ with geometric descriptor $\Phi$ (Quackenbush et al., 2024), as $\mathcal{G}^\dagger:(a,D)\mapsto u$ on variable domains $D$ (Han et al., 2 Feb 2026), or as latent dynamics constrained to Lie-group actions rather than unconstrained Euclidean residual updates (Zhang et al., 18 Feb 2026). The shared aim is to make operator learning more faithful to domain shape, mesh topology, manifold structure, boundary anisotropy, coordinate choice, and physical symmetries than standard geometry-agnostic formulations.

1. Problem setting and conceptual scope

Neural operators are typically introduced as surrogates for PDE solution operators, but many practically important operators depend on geometry as strongly as they depend on coefficients, forcing, or boundary data. The relevant setting is therefore not only $a \mapsto u$, but $(a,\mu)\mapsto u$ or $(a,D)\mapsto u$, where $\mu$ parameterizes geometry or $D$ denotes a variable domain. This is explicit in formulations for parametric PDEs on $\mathcal{D}(\mu)$ (Sarkar et al., 13 Aug 2025), arbitrary geometries learned directly from point clouds (Chen et al., 12 Feb 2026), and kernel-integral operators defined on variable boundaries or domains (Han et al., 2 Feb 2026).

A recurring diagnosis is that standard neural operators are geometry-misaligned. Fixed-grid spectral models are naturally strong on uniform Cartesian domains but less natural on irregular meshes, point clouds, curved boundaries, or non-periodic geometries (Koh et al., 18 Apr 2025). Fixed Eulerian coordinates force the network to represent transport, moving interfaces, vortices, and shocks as unnecessarily non-local mappings, even when the underlying physics is simpler in adapted coordinates (Liu et al., 7 May 2026). On arbitrary shapes, direct geometry-to-solution learning can also be data-inefficient because the space of possible geometries is large, combinatorial, and expensive to sample with high-fidelity simulations (Cheng et al., 2024).

Geometry-awareness is therefore broader than irregular-mesh compatibility. In the cited literature it includes manifold-aware operator learning for Laplace–Beltrami problems and geometric estimation (Quackenbush et al., 2024), geometry-conditioned attention on arbitrary domains (Chen et al., 12 Feb 2026), tangent-normal boundary corrections for aerodynamics (Zhang et al., 8 Jun 2026), equivariance to discrete 3D symmetries in spectral space (Kim et al., 2 Jun 2026), and even post-hoc uncertainty quantification built in the operator’s geometry-sensitive feature space (Vendrell-Gallart et al., 16 Jun 2026). A related operator-centric lineage also appears in shape analysis: OperatorNet reconstructs 3D shapes from intrinsic and extrinsic functional operators, treating operator-valued geometric descriptors as the primary representation rather than raw geometry samples (Huang et al., 2019).

2. Geometric representations and encoding mechanisms

One major design axis is the representation of geometry itself. In Geometric Neural Operators, geometry may be supplied as coordinate charts, ambient embeddings, or explicit geometric features such as curvature, and the learned operator takes the form $\Phi$0 on non-Euclidean domains (Quackenbush et al., 2024). The point-cloud-native GNP foundation-model variant constructs local neighborhoods $\Phi$1, aligns them by PCA, represents them in Monge gauge, and predicts a local surface patch $\Phi$2 from which normals, inverse metric tensors, the Weingarten map, Gaussian curvature, and mean curvature are recovered (Quackenbush et al., 6 Mar 2025). In that formulation, geometry is not an auxiliary side channel; it is the target latent structure.

Other methods encode geometry through graph or token structure. Geo-DeepONet extracts an adjacency matrix $\Phi$3 from finite-element connectivity, uses a graph-trunk network $\Phi$4, and combines it with a CNN branch operating on a structured-grid projection $\Phi$5, yielding

$\Phi$6

This makes unstructured mesh topology part of the operator basis itself (Lee et al., 16 Dec 2025). ArGEnT instead treats geometry as a point cloud and uses self-attention, cross-attention, or hybrid-attention to encode geometric context directly in the DeepONet trunk, with the cross-attention variant decoupling geometry sampling from query evaluation (Chen et al., 12 Feb 2026).

A distinct representation strategy is the geometry-grounded latent point cloud of enf2enf. Its latent state is

$\Phi$7

where $\Phi$8 are latent point locations in physical space and $\Phi$9 are features attached to those points. The decoder uses relative offsets $\mathcal{G}^\dagger:(a,D)\mapsto u$0 through Fourier features and a locality-biased attention rule, so the latent code remains spatially interpretable and discretization-invariant (Catalani et al., 24 Apr 2025). GAOT likewise adds explicit geometry embeddings computed from local neighborhood statistics—neighbor count, average distance, distance variance, centroid offset, and PCA/covariance eigenvalues—and fuses them with multiscale attentional graph-neural-operator features in its MAGNO encoder/decoder (Wen et al., 24 May 2025).

Across these formulations, geometry-awareness is not reducible to appending coordinates. It may instead mean learning from manifold patches, graph topology, latent point clouds, or point-cloud attention, each chosen to preserve a different aspect of geometric structure.

3. Deformations, charts, and learned coordinate systems

A second major paradigm changes the coordinate system in which the operator is learned. Reference Neural Operators recast geometry variation as smooth deformation around a known reference geometry. Given a reference solution $\mathcal{G}^\dagger:(a,D)\mapsto u$1, reference geometry $\mathcal{G}^\dagger:(a,D)\mapsto u$2, and a smooth map $\mathcal{G}^\dagger:(a,D)\mapsto u$3, RNO learns the residual

$\mathcal{G}^\dagger:(a,D)\mapsto u$4

and predicts

$\mathcal{G}^\dagger:(a,D)\mapsto u$5

The paper connects this directly to the material derivative of the solution operator under shape perturbation, so the model learns the local dependence of the solution on geometric deformation rather than a global geometry-to-solution map (Cheng et al., 2024).

Adaptive Coordinate Transforms generalize that idea from input preprocessing to layer-wise latent realignment. Each ACT head predicts a residual coordinate transform

$\mathcal{G}^\dagger:(a,D)\mapsto u$6

then resamples the feature field at transformed coordinates,

$\mathcal{G}^\dagger:(a,D)\mapsto u$7

Because the transform is differentiable and identity-preserving at initialization, ACT can be inserted after Fourier blocks, transformer blocks, or ConvNeXt-UNet stages to make latent dynamics follow moving structures more naturally (Liu et al., 7 May 2026).

CATO makes the coordinate change itself the core architectural object. It learns a continuous chart

$\mathcal{G}^\dagger:(a,D)\mapsto u$8

maps physical coordinates into a latent chart space, and applies row-wise and column-wise attention in that space. The theoretical motivation is that, under a favorable chart, the PDE operator becomes approximately axial low-rank; the empirical motivation is that raw mesh coordinates may obscure the intrinsic directions along which physical interactions are simpler (Cheng et al., 9 May 2026).

These methods differ in mechanism, but they share a strong claim: geometry-awareness can be achieved by changing the learning objective or latent coordinate system, not only by changing the geometry encoder.

4. Locality, anisotropy, symmetry, and latent geometric structure

Another branch of the literature makes geometry-aware inductive bias explicit at the level of neighborhoods, interfaces, boundaries, and symmetries. LA2Former is exemplary: it uses instant KNN patchifying on arbitrary meshes and a global-local attention block,

$\mathcal{G}^\dagger:(a,D)\mapsto u$9

combining linear global attention with full pairwise attention restricted to local KNN neighborhoods. The local neighborhoods are themselves softened by a learned sigmoid mask over the $D$0 nearest neighbors, so the architecture treats localized geometric interaction as a first-class component rather than an afterthought (Koh et al., 18 Apr 2025).

Boundary-aware and interface-aware models specialize this locality further. GeoABC constructs a local tangent-normal frame

$D$1

and uses separate tangential, normal, and mixed correction branches inside a mid-layer residual update near trusted wall regions. This reflects the paper’s premise that near-wall aerodynamics is anisotropic: flow propagates along the wall, while normal variation is tightly constrained by the wall (Zhang et al., 8 Jun 2026). IANO similarly elevates interface geometry to a structural prior in multiphase flow, using interface-aware multiple-function encoding and geometry-aware positional encoding so that each spatial query depends explicitly on interface location and coupled field–interface interactions (Wang et al., 9 Nov 2025).

Symmetry-aware methods encode geometric structure at a more global level. EqGINO constrains Fourier mixing weights by symmetry orbits in frequency space, producing exact equivariance to the discrete grid-preserving subgroup of 3D rotations and reflections represented on the discretized domain (Kim et al., 2 Jun 2026). MCL addresses a different symmetry problem: latent evolution in operator layers. Instead of unconstrained Euclidean residual updates,

$D$2

it uses a Lie-group-constrained update on $D$3, parameterized in the Lie algebra by low-rank skew-symmetric matrices and linearized as

$D$4

The stated effect is near-isometric latent evolution, reduced latent drift, and improved long-horizon stability (Zhang et al., 18 Feb 2026).

Taken together, these works show that geometry-awareness may refer to local neighborhood geometry, interface geometry, anisotropic boundary frames, or symmetry structure in either physical or latent space.

5. Physics-informed, kernel-integral, solver-aware, and probabilistic extensions

Geometry-aware neural operators are not confined to supervised surrogate regression. PI-GANO combines a DCON backbone with a geometry encoder trained only through PDE residuals and boundary losses, targeting the simultaneous variation of PDE parameters and domain geometry without FEM-generated supervision (Zhong et al., 2024). The related $D$5G-Sp$D$6GNO extends a spatio-spectral graph neural operator with two geometry-awareness mechanisms—interpolation-based enrichment and an explicit geometry encoder—and uses a simulation-free physics loss. For time-dependent PDEs it couples a Crank–Nicolson-style time-marching residual with a stochastic projection estimator for spatial derivatives, avoiding direct reliance on higher-order automatic differentiation (Sarkar et al., 13 Aug 2025).

A more structural reformulation is given by the kernel-integral perspective. Here variable-geometry operator learning is cast as approximation of geometry-dependent kernel operators, often singular, motivated by classical boundary integral formulations. The proposed multiscale point-cloud neural operator decomposes the kernel as

$D$7

with a Fourier-truncated long-range component and a locally approximated short-range component inspired by Ewald splitting. In this view, geometric generalization arises because the model learns the kernel mechanism itself—displacement, normals, and singular local structure—rather than a mesh-bound surrogate on a fixed domain (Han et al., 2 Feb 2026).

Geometry-aware operators have also been incorporated into numerical solvers. Geo-DeepONet is used as a neural preconditioner in hybrid relaxation and Krylov schemes; its trunk-basis construction yields a coarse operator $D$8, while flexible Krylov variants can use the nonlinear neural prediction directly (Lee et al., 16 Dec 2025). REEF-GP moves in a probabilistic direction: it freezes a geometry-aware operator, extracts internal embeddings $D$9, and fits a GP residual model with kernel

$a \mapsto u$0

supplemented by spectral-normalized projections and heteroscedastic geometry-aware noise (Vendrell-Gallart et al., 16 Jun 2026).

These extensions suggest that geometry-awareness is increasingly treated as a reusable substrate for training without labels, accelerating linear algebra, and quantifying uncertainty, not only for direct solution prediction.

6. Empirical behavior, applications, and limitations

The application range is broad: Laplace–Beltrami and geometric PDEs on manifolds (Quackenbush et al., 2024, Quackenbush et al., 6 Mar 2025), Darcy flow and elasticity on variable domains (Zhong et al., 2024, Sarkar et al., 13 Aug 2025), Navier–Stokes, Burgers, shallow water, and diffusion-type systems (Zhang et al., 18 Feb 2026, Liu et al., 7 May 2026), arbitrary-domain steady-state PDEs such as airfoils, pipes, elasticity, and plasticity (Koh et al., 18 Apr 2025, Cheng et al., 9 May 2026), aerodynamic CFD around airfoils and cars (Zhang et al., 8 Jun 2026), industrial 3D CFD on DrivAerNet++ (Wen et al., 24 May 2025), and solver acceleration on unstructured FEM meshes for Poisson and linear elasticity (Lee et al., 16 Dec 2025). This breadth indicates that “geometry-aware neural operator” is now a methodological class rather than a single architecture.

Reported gains are substantial but method-specific. RNO reports up to $a \mapsto u$1 error reduction and emphasizes small-data efficiency on deforming geometries (Cheng et al., 2024). MCL lowers relative prediction error by $a \mapsto u$2–$a \mapsto u$3 on average with only $a \mapsto u$4 more parameters, and its ablation against a standard MLP augmentation is used to argue that the benefit comes from geometric constraint rather than extra capacity (Zhang et al., 18 Feb 2026). LA2Former reports over $a \mapsto u$5 relative improvement over existing linear-attention methods on several benchmarks and best results on Elasticity, Plasticity, Airfoil, and Darcy (Koh et al., 18 Apr 2025). ACT yields average relative error reductions of $a \mapsto u$6 for CNextU, $a \mapsto u$7 for Transolver, and $a \mapsto u$8 for FNO, while an ablation on Navier shows that layer-wise ACT outperforms simply doubling FNO parameters (Liu et al., 7 May 2026). CATO reports an average improvement of approximately $a \mapsto u$9 over the strongest competing baselines while reducing parameter count by $(a,\mu)\mapsto u$0 (Cheng et al., 9 May 2026). GeoABC reduces near-boundary relative $(a,\mu)\mapsto u$1 error by about $(a,\mu)\mapsto u$2 on average across 2D airfoil and 3D car tasks (Zhang et al., 8 Jun 2026). GAOT reports state-of-the-art performance on a large-scale three-dimensional industrial CFD dataset and, on the main benchmark suite, is described as about $(a,\mu)\mapsto u$3 more accurate than the second-best baseline on time-independent datasets while also being faster and more memory-efficient (Wen et al., 24 May 2025). In steady-state neural fields, enf2enf attains volume pressure MSE $(a,\mu)\mapsto u$4, surface pressure MSE $(a,\mu)\mapsto u$5, lift coefficient MSE $(a,\mu)\mapsto u$6, and Spearman correlation $(a,\mu)\mapsto u$7 on AirFRANS, together with zero-shot super-resolution and roughly $(a,\mu)\mapsto u$8–$(a,\mu)\mapsto u$9 orders of magnitude speedup relative to high-fidelity CFD (Catalani et al., 24 Apr 2025). For uncertainty quantification, REEF-GP preserves the base operator’s predictive accuracy while achieving calibration competitive with deep ensembles; on ShapeNet Car it reports NLL $(a,D)\mapsto u$0 versus $(a,D)\mapsto u$1 for deep ensembles (Vendrell-Gallart et al., 16 Jun 2026).

The literature is also explicit about limits. LA2Former’s best neighborhood size $(a,D)\mapsto u$2 remains problem-dependent (Koh et al., 18 Apr 2025). ACT is currently developed for structured grids, with extension to irregular geometries left open (Liu et al., 7 May 2026). CATO’s empirical study is centered on 2D benchmarks (Cheng et al., 9 May 2026). PI-GANO is demonstrated on steady-state PDEs and notes instability typical of physics-informed optimization (Zhong et al., 2024). The point-cloud GNP foundation-model approach depends on local charting, PCA alignment, and low-order polynomial patch models, and its training distribution is synthetic (Quackenbush et al., 6 Mar 2025). REEF-GP relies on a base operator whose hidden states preserve spatial correspondence; the reported experiments therefore use Transolver (Vendrell-Gallart et al., 16 Jun 2026). A common misconception—that gains come mainly from added parameters—is directly challenged by the MCL and ACT ablations, both of which report improvements not replicated by simple capacity increases (Zhang et al., 18 Feb 2026, Liu et al., 7 May 2026).

A plausible synthesis is that geometry-awareness in neural operators has moved from input-side geometry encoding toward operator-side structural design. In recent work, geometry may define what is encoded, which coordinates are used, how attention neighborhoods are formed, how latent dynamics evolve, what symmetries are enforced, where corrections are applied, how physics residuals are computed, and even how uncertainty kernels are built. That expansion is what now distinguishes geometry-aware neural operators from earlier formulations that treated geometry as little more than another feature channel.