Grid-based Neural Operators

• Grid-based neural operators are operator-learning models that lift discretized input fields to functions, transform them via spectral or convolutional layers, and project them back to outputs.
• They employ techniques such as Fourier transforms, multigrid parameterizations, and tensorized variants to effectively manage global and local scale features while addressing aliasing challenges.
• Advanced variants integrate direct spectral evaluation, graph-based methods, and kernel explainability to enhance simulation accuracy and support zero-shot super-resolution across varying resolutions.

Grid-based neural operators are operator-learning models that are trained on discretized fields on a spatial grid, but are intended to represent a continuous mapping between functions rather than a fixed-size vector-to-vector map. In this setting, the grid is a computational representation of the domain, not the object being learned: the target is a map between function spaces, typically arising from PDE solution operators, dynamical-system evolution operators, or related scientific surrogates. The family is closely associated with Fourier Neural Operators and related spectral architectures, but the literature now also includes multigrid parameterizations, tensorized and hierarchical variants, geometry-aware latent-grid models, and hybrid schemes that restore unresolved scales or couple grid representations to graphs and point clouds (Yang et al., 8 Apr 2026, He et al., 2023, Zhong et al., 7 Oct 2025).

1. Conceptual basis and operator-learning formulation

A standard formulation seeks an operator

$$\mathcal{G}^\dagger:\mathcal{A}\to\mathcal{U},\qquad a\mapsto u,$$

with a learned approximation written as a composition of lifting, operator, and projection stages,

$$\mathcal{G}_\theta = \mathcal{Q}\circ \mathcal{K}^{(L)}_\theta\circ \cdots \circ \mathcal{K}^{(1)}_\theta\circ \mathcal{P}.$$

This notation captures the common grid-based neural-operator view: sampled inputs are lifted to feature fields on a grid, transformed by learned integral or convolutional operators, and then projected back to output fields (Yang et al., 8 Apr 2026).

The defining distinction from FCNN-, CNN-, or RNN-based surrogates is that neural operators are intended to learn mappings between function spaces rather than finite-dimensional maps tied to one discretization. In the multiscale-closure setting, for example, the contrast is made explicit as finite-dimensional surrogates versus

$$G_\Theta:\;H_{\mathrm{in}}(D_x;\mathbb{R}^{d_{\mathrm{in}}}) \rightarrow H_{\mathrm{out}}(D_x;\mathbb{R}^{d_{\mathrm{out}}}),$$

which motivates claims of grid independence and evaluation on different mesh sizes or resolutions (Lütjens et al., 2022).

Some work has generalized this viewpoint beyond the usual lift–process–project template. MgNO defines an operator-valued neuron by

$O_i(u)=\sigma\left(\sum_j W_{ij}u+B_{ij}\right),$

where $W_{ij}$ are bounded linear operators and $B_{ij}$ are functional biases rather than scalars. In that formulation, the network remains function-to-function throughout, and the main design problem becomes how to parameterize the linear operators efficiently on a grid (He et al., 2023).

This function-space viewpoint is also the basis for zero-shot super-resolution and resolution transfer. In downscaling and surrogate modeling, a model may be trained on one discretization and evaluated on another because the learned object is a continuous operator rather than a fixed image-to-image map. Multiple papers, however, stress that this is a theoretical property whose practical realization depends strongly on architecture and task (Sinha et al., 2024).

2. Fourier layers, spectral efficiency, and the limits of regular-grid formulations

The canonical grid-based neural operator is the Fourier Neural Operator. In the benchmark literature, a Fourier layer is written as

$$G^{l+1}=\mathcal{U}_{\mathrm{FL}}(G^l)=\mathcal{F}^{-1}(R_{\phi}\cdot \mathcal{F}(G^l)),$$

where $\mathcal{F}$ and $\mathcal{F}^{-1}$ are Fourier transforms and $R_\phi$ is a learned spectral filter. This yields efficient global mixing on regular lattices and explains why FNO and related models became central in grid-based operator learning (Zhong et al., 7 Oct 2025).

The same spectral mechanism also imposes a structural limit. Fourier-based neural operators keep only a limited set of Fourier modes, so the learned kernel has a hard bandwidth limit. Recent work links this spectral truncation directly to the loss of small-scale spatial structure and degraded predictions at high resolution when models are trained on low-resolution data. In multiscale dynamics, the missing content is not merely visual detail; it affects power spectra, thin interfaces, vortical structures, and other high-frequency patterns that matter for the physics (Yang et al., 8 Apr 2026).

A different critique concerns aliasing. The Spectral Neural Operator paper argues that in FNO the input is sampled on a grid and the output is parametrized by a neural network, which makes the output opaque and introduces systematic bias from aliasing errors. Because nonlinear activations generate higher harmonics on a fixed grid, unresolved frequencies are folded into lower ones. For the specific example

$$\mathcal{G}_\theta = \mathcal{Q}\circ \mathcal{K}^{(L)}_\theta\circ \cdots \circ \mathcal{K}^{(1)}_\theta\circ \mathcal{P}.$$0

the paper exhibits a nontrivial aliasing level even for a simple activation (Fanaskov et al., 2022).

These critiques produced two distinct responses. One response keeps the regular-grid spectral paradigm but tries to recover lost scales through multiscale refinement, generative enhancement, or multigrid correction. The other replaces the sampled-value output representation altogether with explicit spectral coefficient maps, as in SNO, which uses Chebyshev and Fourier series for both domain and codomain and is presented as having transparent output and never suffering from aliasing (Fanaskov et al., 2022).

A recurrent misconception is that resolution invariance by itself guarantees robust high-resolution prediction. Several papers reject that conclusion. Theoretical discretization transfer does not prevent spectral under-resolution, aliasing, or architecture-dependent degradation once the test regime demands frequencies not learned or retained during training (Yang et al., 8 Apr 2026).

3. Multiscale, multigrid, and resolution-restoring architectures

A major development within grid-based neural operators is the use of explicit multiscale structure. In Multiscale Neural Operator, the known coarse dynamics are kept in the simulator and the neural operator learns only the closure term

$$\mathcal{G}_\theta = \mathcal{Q}\circ \mathcal{K}^{(L)}_\theta\circ \cdots \circ \mathcal{K}^{(1)}_\theta\circ \mathcal{P}.$$1

so that

$$\mathcal{G}_\theta = \mathcal{Q}\circ \mathcal{K}^{(L)}_\theta\circ \cdots \circ \mathcal{K}^{(1)}_\theta\circ \mathcal{P}.$$2

This hybrid formulation is presented as the first neural-operator-based grid-independent non-local parametrization, with quasilinear $$\mathcal{G}_\theta = \mathcal{Q}\circ \mathcal{K}^{(L)}_\theta\circ \cdots \circ \mathcal{K}^{(1)}_\theta\circ \mathcal{P}.$$3 runtime and a reported $$\mathcal{G}_\theta = \mathcal{Q}\circ \mathcal{K}^{(L)}_\theta\circ \cdots \circ \mathcal{K}^{(1)}_\theta\circ \mathcal{P}.$$4 speedup over DNS on large-grid multiscale Lorenz96 (Lütjens et al., 2022).

MgFNO brings multigrid ideas directly into FNO training. Its three-level hierarchy trains a coarse network first, then an intermediate network on prolonged residuals, then a fine network for high-resolution correction. The paper frames this as a reconciliation between the frequency principle of neural networks, which fit low frequencies before high frequencies, and multigrid methods, which reduce high-frequency error first. Reported relative errors are $$\mathcal{G}_\theta = \mathcal{Q}\circ \mathcal{K}^{(L)}_\theta\circ \cdots \circ \mathcal{K}^{(1)}_\theta\circ \mathcal{P}.$$5, $$\mathcal{G}_\theta = \mathcal{Q}\circ \mathcal{K}^{(L)}_\theta\circ \cdots \circ \mathcal{K}^{(1)}_\theta\circ \mathcal{P}.$$6, and $$\mathcal{G}_\theta = \mathcal{Q}\circ \mathcal{K}^{(L)}_\theta\circ \cdots \circ \mathcal{K}^{(1)}_\theta\circ \mathcal{P}.$$7 on Burgers’ equation, Darcy flow, and Navier–Stokes, with reductions of $$\mathcal{G}_\theta = \mathcal{Q}\circ \mathcal{K}^{(L)}_\theta\circ \cdots \circ \mathcal{K}^{(1)}_\theta\circ \mathcal{P}.$$8, $$\mathcal{G}_\theta = \mathcal{Q}\circ \mathcal{K}^{(L)}_\theta\circ \cdots \circ \mathcal{K}^{(1)}_\theta\circ \mathcal{P}.$$9, and $$G_\Theta:\;H_{\mathrm{in}}(D_x;\mathbb{R}^{d_{\mathrm{in}}}) \rightarrow H_{\mathrm{out}}(D_x;\mathbb{R}^{d_{\mathrm{out}}}),$$0 compared to conventional FNO, and the model is also described as supporting zero-shot super-resolution prediction (Guo et al., 2024).

MG-TFNO addresses a different bottleneck: high-resolution domains and large spectral parameter tensors. It decomposes both the domain and the operator parameters through multi-grid domain decomposition and tensor factorization of Fourier weights. On turbulent Navier–Stokes, the paper reports less than half the error with over $$G_\Theta:\;H_{\mathrm{in}}(D_x;\mathbb{R}^{d_{\mathrm{in}}}) \rightarrow H_{\mathrm{out}}(D_x;\mathbb{R}^{d_{\mathrm{out}}}),$$1 compression, and states that tensorization plus domain decomposition yields over $$G_\Theta:\;H_{\mathrm{in}}(D_x;\mathbb{R}^{d_{\mathrm{in}}}) \rightarrow H_{\mathrm{out}}(D_x;\mathbb{R}^{d_{\mathrm{out}}}),$$2 reduction in the number of parameters and $$G_\Theta:\;H_{\mathrm{in}}(D_x;\mathbb{R}^{d_{\mathrm{in}}}) \rightarrow H_{\mathrm{out}}(D_x;\mathbb{R}^{d_{\mathrm{out}}}),$$3 reduction in domain size without losses in accuracy (Kossaifi et al., 2023).

MENO is a recent example of multiscale restoration layered on top of conventional grid-based backbones. It couples FNO or UNO with a one-step generative decoder based on improved MeanFlow. The neural operator predicts coarse low-resolution evolution; the decoder upsamples and refines each predicted frame through

$$G_\Theta:\;H_{\mathrm{in}}(D_x;\mathbb{R}^{d_{\mathrm{in}}}) \rightarrow H_{\mathrm{out}}(D_x;\mathbb{R}^{d_{\mathrm{out}}}),$$4

Across phase-field dynamics, 2D Kolmogorov flow, and active matter at resolutions up to $$G_\Theta:\;H_{\mathrm{in}}(D_x;\mathbb{R}^{d_{\mathrm{in}}}) \rightarrow H_{\mathrm{out}}(D_x;\mathbb{R}^{d_{\mathrm{out}}}),$$5, MENO improves PSDD by up to a factor of 2 relative to baseline neural operators while achieving $$G_\Theta:\;H_{\mathrm{in}}(D_x;\mathbb{R}^{d_{\mathrm{in}}}) \rightarrow H_{\mathrm{out}}(D_x;\mathbb{R}^{d_{\mathrm{out}}}),$$6 faster inference than DDIM-enhanced counterparts (Yang et al., 8 Apr 2026).

A related but distinct multilevel direction learns grid operators for solvers rather than PDE solution maps themselves. The work on neural construction of multigrid operators learns the coupling operator $$G_\Theta:\;H_{\mathrm{in}}(D_x;\mathbb{R}^{d_{\mathrm{in}}}) \rightarrow H_{\mathrm{out}}(D_x;\mathbb{R}^{d_{\mathrm{out}}}),$$7 and recovers

$$G_\Theta:\;H_{\mathrm{in}}(D_x;\mathbb{R}^{d_{\mathrm{in}}}) \rightarrow H_{\mathrm{out}}(D_x;\mathbb{R}^{d_{\mathrm{out}}}),$$8

which is then used recursively to build coarser levels through

$$G_\Theta:\;H_{\mathrm{in}}(D_x;\mathbb{R}^{d_{\mathrm{in}}}) \rightarrow H_{\mathrm{out}}(D_x;\mathbb{R}^{d_{\mathrm{out}}}),$$9

The method is explicitly presented as operator-learning-for-numerical-linear-algebra rather than a general end-to-end neural operator (Tomasi et al., 2021).

4. From regular lattices to arbitrary domains, meshes, and latent grids

Although “grid-based” often suggests equispaced Cartesian meshes, the literature increasingly treats the grid as a computational scaffold rather than a restriction to regular geometry. A central contribution in this direction is direct spectral evaluation on arbitrary point sets. Instead of FFTs, which require regular sampling, the spectral transform is written as a matrix multiplication

$O_i(u)=\sigma\left(\sum_j W_{ij}u+B_{ij}\right),$0

where $O_i(u)=\sigma\left(\sum_j W_{ij}u+B_{ij}\right),$1 contains truncated basis functions evaluated at the sample locations. Because neural operators usually retain only a small number of modes, the cost is $O_i(u)=\sigma\left(\sum_j W_{ij}u+B_{ij}\right),$2, and the paper argues that direct evaluation is competitive in the usual FNO regime of roughly 12–32 modes and remains efficient even up to about 64 modes (Lingsch et al., 2023).

This direct-evaluation strategy is presented as a drop-in replacement inside FNO, UFNO, FFNO, and SFNO. It eliminates interpolation to a regular grid and extends Fourier- or spherical-harmonic layers to arbitrary non-equispaced distributions of points. On Burgers’ equation with a contracting-expanding irregular distribution, for example, FNO-DSE is reported at $O_i(u)=\sigma\left(\sum_j W_{ij}u+B_{ij}\right),$3 test error and $O_i(u)=\sigma\left(\sum_j W_{ij}u+B_{ij}\right),$4 s/epoch versus $O_i(u)=\sigma\left(\sum_j W_{ij}u+B_{ij}\right),$5 and $O_i(u)=\sigma\left(\sum_j W_{ij}u+B_{ij}\right),$6 s/epoch for FFT plus cubic interpolation (Lingsch et al., 2023).

For unstructured grids and irregular domains, another strategy is to build graphs and blend local and global operators. Sp$O_i(u)=\sigma\left(\sum_j W_{ij}u+B_{ij}\right),$7GNO constructs a kNN graph on the point cloud, applies a truncated spectral graph branch for long-range dependencies and a gated spatial branch for local aggregation, and fuses both inside each operator block. The method is reported to achieve the best performance in 3 out of 4 tests, second-best in the remaining case, and about $O_i(u)=\sigma\left(\sum_j W_{ij}u+B_{ij}\right),$8 gain over the nearest competitor on the airfoil problem (Sarkar et al., 2024).

Industry-scale engineering work makes a related distinction between native grid-based models and geometry-aware adaptations. In that benchmark, the grid-based family includes FNO, F-FNO, U-NO, Geo-FNO, GINO, and FigConv-UNet. The paper emphasizes that curved boundaries and unstructured FEM meshes are challenging for vanilla grid methods, so latent regular-grid projection, point-grid coupling, or factorized implicit grids become necessary. Among the compared grid-based models, FigConvUNet is reported as the strongest overall, but point-based methods remain better on freeform vehicle geometries such as DrivAer and DrivAer++ (Zhong et al., 7 Oct 2025).

The same irregular-domain literature also contains approaches that are explicitly not grid-based in the usual sense. GOLA is described as a graph-based alternative to grid-based neural operators, combining a learnable Fourier encoder with attention-enhanced message passing on graphs constructed from irregular spatial samples. Its relevance is chiefly comparative: it addresses the shortcomings of regular-grid assumptions rather than extending the grid paradigm itself (Li et al., 25 May 2025).

5. Kernel viewpoints, operator parameterization, and explainability

A broader theoretical account of grid-based models treats them as weighted averages over grid nodes,

$O_i(u)=\sigma\left(\sum_j W_{ij}u+B_{ij}\right),$9

Within this framework, the Grid Tangent Kernel is defined by

$W_{ij}$0

and the training dynamics under gradient flow become

$W_{ij}$1

A notable claim is that for the grid-model abstraction the GTK is stationary,

$W_{ij}$2

so training is a fixed-kernel linear dynamics problem once the interpolation kernel is fixed (Zhao et al., 2024).

That theory also motivates model design through the quantity

$W_{ij}$3

which appears in the generalization bound. MulFAGrid is proposed as a grid-based model with a wider kernel spectrum and lower generalization bound than several predecessors, and is reported to achieve state-of-the-art results in 2D image fitting, 3D SDF reconstruction, and novel view synthesis. This is not a neural-operator paper in the PDE-solver sense, but it is directly relevant to the analysis of grid-based architectures because it treats kernel geometry as the central architectural object (Zhao et al., 2024).

IKNO moves the operator parameterization problem in a different direction. It argues that existing neural operators rely mainly on first-order kernel integral approximations and therefore have limited expressivity. The proposed infinite-order operator replaces the usual one-hop kernel map by

$W_{ij}$4

with two realizations: IKNO-Vanilla on the full product grid and IKNO-TP as a tensor-product composition of per-axis resolvents. The paper reports that IKNO-TP achieves the best result on 10 of 14 benchmarks and second-best on the other 4, while remaining scalable to very large point clouds and large industrial geometries (Zhu et al., 21 May 2026).

Explainability has also been reformulated at operator level. OperatorSHAP observes that classical Shapley values are tied to a fixed set of discrete “players,” which is incompatible with neural operators that naturally transfer across resolutions or irregular meshes. It therefore learns a continuous attribution density in function space and connects the construction to Aumann–Shapley theory. The paper states that the resulting explanations are consistent with state-of-the-art discrete Shapley values across resolutions and transfer across grid sizes without retraining (Stiller et al., 26 Jun 2026).

6. Empirical regimes, limitations, and current debates

One of the central debates concerns whether practical zero-shot super-resolution follows from theoretical resolution invariance. The weather-downscaling study gives a negative answer for a particularly demanding setting. Across ERA5$W_{ij}$5ERA5 and ERA5$W_{ij}$6WTK experiments with challenging zero-shot factors such as $W_{ij}$7 and $W_{ij}$8, neural-operator-based downscaling models outperform interpolation and a simple convolutional baseline, but a SwinIR-based approach with interpolation mostly outperforms models with neural operator layers on average error metrics. The paper further notes that an ESRGAN-based approach is better than most models in terms of capturing the physics of the ground truth data, and recommends such methods as strong baselines (Sinha et al., 2024).

The same paper also complicates the interpretation of spectral fidelity. SwinIR most closely matches the energy spectrum at low to medium wavenumbers, DAFNO sometimes does best at the highest wavenumbers, and yet all methods underestimate energy at high wavenumbers, especially in zero-shot settings. Its conclusion is not that neural operators fail in general, but that resolution invariance alone is not enough for static weather downscaling, where fine-scale physics may be weakly inferable from the training-scale information (Sinha et al., 2024).

By contrast, several dynamical-system papers show that grid-based neural operators can be highly effective when the missing scales are tightly constrained by dynamics or by a hybrid formulation. MENO improves free-energy decay in phase-field dynamics, autocorrelation behavior over long horizons, and PSDD across three multiscale systems while remaining substantially faster than diffusion-based enhancement (Yang et al., 8 Apr 2026). MgFNO and MG-TFNO likewise report strong accuracy gains on Burgers, Darcy, Navier–Stokes, and high-resolution turbulence through hierarchical residual learning or decomposed spectral parameterization (Guo et al., 2024, Kossaifi et al., 2023).

Irregular-domain extensions introduce their own limitations. Direct spectral evaluation depends on truncation; if many modes are required, its efficiency advantage shrinks, and sample-dependent point clouds require sample-dependent transform matrices (Lingsch et al., 2023). Sp$W_{ij}$9GNO still depends on graph construction and spectral decomposition, with kNN graphs noted as not necessarily optimal and eigenvector computation adding overhead (Sarkar et al., 2024). OperatorSHAP’s clean theory assumes Fréchet $B_{ij}$0 regularity on Sobolev spaces and smooth masks, so the theoretical guarantees do not apply verbatim to nonsmooth activations such as ReLU (Stiller et al., 26 Jun 2026).

Taken together, these results support a narrower interpretation of the term “grid-based neural operator” than early Fourier-operator work sometimes implied. The grid remains an effective computational substrate for global spectral mixing, multilevel correction, and latent operator parameterization, but robustness depends on how the model handles unresolved frequencies, geometry, and the relation between training and deployment discretizations. This suggests that the contemporary field is less a single architecture class than a design space organized around three recurring questions: how to represent functions on grids without losing physically relevant scales, how to transfer those representations beyond regular lattices, and how to preserve the operator-level advantages of efficiency and discretization transfer while remaining faithful to multiscale scientific structure.