Convolutional Neural Operator

• Convolutional Neural Operator is a CNN-based framework that learns mappings between function spaces to solve PDEs, using alias-free multiresolution processing.
• It enforces continuous-discrete equivalence by redesigning convolution and activation methods to preserve high-frequency details and ensure resolution invariance.
• CNOs have been applied in physics-informed simulations and seismic inversion, demonstrating enhanced accuracy and reduced computational costs over traditional methods.

Convolutional neural operator (CNO) denotes a CNN-based neural-operator paradigm for learning mappings between function spaces, particularly PDE solution operators, rather than fixed-size vector-to-vector maps. In its strict formulation, a CNO is a modified U-Net-style neural operator acting between bandlimited function spaces, designed to preserve a correspondence between a continuous operator and its discrete implementation through continuous-discrete equivalence and alias-free multiresolution processing (Raonić et al., 2023). Subsequent literature uses the same label more broadly for convolutional realizations and extensions of operator learning, including physics-informed surrogates for the Helmholtz equation (Ma et al., 22 Jul 2025), dynamic local propagators for Maxwell simulation (Ma et al., 2024), multiscale Fourier-convolution hybrids (Xu et al., 2024), and transfer-learning adaptations for shifted PDE regimes (Fan et al., 19 Dec 2025).

1. Conceptual definition and scope

A neural operator learns a map between function spaces. In PDE settings, the input can be a coefficient field, forcing, geometry, source condition, or initial condition, and the output is a function such as a solution field, wavefield, or evolved state. The CNO program arose from the observation that convolution-based architectures, although highly successful in conventional machine learning, had been “largely ignored” for PDE operator learning because standard CNNs were regarded as inconsistent in function space (Raonić et al., 2023).

The central claim of the original CNO formulation is that convolutional architectures are not intrinsically unsuitable for operator learning; rather, naive CNN implementations fail when the discrete network does not correspond to a well-defined operator on the underlying continuous function space. The resulting pathologies include grid dependence, aliasing, and lack of resolution invariance. CNO addresses this by constraining the architecture to operate on bandlimited function spaces and by redesigning convolution, activation, and multiresolution transfers so that discretization and model application commute (Raonić et al., 2023).

In a narrower usage, “CNO” refers specifically to the architecture introduced in “Convolutional Neural Operators for robust and accurate learning of PDEs” (Raonić et al., 2023). In a broader usage, the term also covers convolutional neural-operator backbones embedded in application-specific pipelines. This broader usage includes a physics-informed CNO for frequency-domain seismic wavefield prediction (Ma et al., 22 Jul 2025), a causality-aware dynamic convolutional neural operator for photonic FDTD rollout (Ma et al., 2024), and a velocity-to-image seismic forward surrogate embedded in an inversion loop (Ma et al., 24 Sep 2025). This suggests that “convolutional neural operator” functions both as the name of a specific alias-free architecture and as a wider design family centered on local convolutional operator approximation.

2. Continuous-discrete architecture

The original CNO is written as an operator

$\mathcal{G}: \mathcal{B}_w(D)\to \mathcal{B}_w(D),$

where $\mathcal{B}_w(D)$ denotes a bandlimited function space. Abstractly, the network is composed as

$$\mathcal{G}: u \mapsto P(u)=v_0 \mapsto v_1 \mapsto \cdots \mapsto v_L \mapsto Q(v_L)=\bar u,$$

with layers of the form

$$v_{l+1}=\mathcal{P}_l\circ \Sigma_l\circ \mathcal{K}_l(v_l).$$

Here, $\mathcal{K}_l$ is a convolution operator in physical space, $\mathcal{P}_l$ is an upsampling or downsampling operator, $\Sigma_l$ is a modified activation, $P$ is a lifting map into latent channels, and $Q$ is a projection back to the output function space (Raonić et al., 2023).

The architecture is implemented as an operator U-Net with encoder-decoder structure, skip connections, and four named block types: downsampling blocks $D$, upsampling blocks $\mathcal{B}_w(D)$0, ResNet blocks $\mathcal{B}_w(D)$1, and invariant blocks $\mathcal{B}_w(D)$2 (Raonić et al., 2023). The multiscale design is not merely architectural convenience. In the original formulation, skip connections transfer high-frequency information across scales, while the encoder progressively filters higher frequencies, yielding a genuinely multiscale operator (Raonić et al., 2023).

A distinguishing architectural choice is that convolution is performed directly in physical space rather than via truncated global Fourier modes. In the seismic Helmholtz formulation, the convolutional layer is written discretely as

$\mathcal{B}_w(D)$3

with trainable kernel weights $\mathcal{B}_w(D)$4. That paper emphasizes locality, interpretability, and non-periodic boundaries as reasons to prefer physical-space convolution to Fourier neural operators in that setting (Ma et al., 22 Jul 2025).

The activation mechanism is also specialized. In the original CNO, a naive pointwise nonlinearity is avoided because it generates frequencies outside the prescribed bandlimit. Instead, activation is implemented by upsampling to a larger bandlimit, applying the pointwise nonlinearity, and downsampling back: $\mathcal{B}_w(D)$5 Upsampling and downsampling are based on sinc interpolation and filtering, with

$\mathcal{B}_w(D)$6

so that multiresolution transfers remain consistent with the bandlimited function-space model (Raonić et al., 2023).

In application papers that use the CNO backbone more pragmatically, the same operator-level structure is retained but expressed in conventional deep-learning terms. The physics-informed Helmholtz work describes the operator as

$\mathcal{B}_w(D)$7

with layer update

$\mathcal{B}_w(D)$8

where $\mathcal{B}_w(D)$9 is a convolution operator, $$\mathcal{G}: u \mapsto P(u)=v_0 \mapsto v_1 \mapsto \cdots \mapsto v_L \mapsto Q(v_L)=\bar u,$$0 is Leaky ReLU, and $$\mathcal{G}: u \mapsto P(u)=v_0 \mapsto v_1 \mapsto \cdots \mapsto v_L \mapsto Q(v_L)=\bar u,$$1 changes resolution by upsampling or downsampling (Ma et al., 22 Jul 2025). The continuity of the operator viewpoint across these formulations is one of the defining features of the CNO literature.

3. Approximation theory and operator consistency

The most distinctive theoretical contribution of the original CNO paper is the insistence on continuous-discrete equivalence. The model is constructed so that, for bandlimited functions sampled at sufficiently fine resolution, the discrete implementation commutes with the continuous operator. This yields representation equivalence. The paper states explicitly: “Convolutional Neural Operator $$\mathcal{G}: u \mapsto P(u)=v_0 \mapsto v_1 \mapsto \cdots \mapsto v_L \mapsto Q(v_L)=\bar u,$$2 is a Representation equivalent neural operator or ReNO” (Raonić et al., 2023).

On the approximation side, the same paper proves a universality theorem for a broad class of PDE solution operators. Under regularity and stability assumptions, for every $$\mathcal{G}: u \mapsto P(u)=v_0 \mapsto v_1 \mapsto \cdots \mapsto v_L \mapsto Q(v_L)=\bar u,$$3 there exists a CNO $$\mathcal{G}: u \mapsto P(u)=v_0 \mapsto v_1 \mapsto \cdots \mapsto v_L \mapsto Q(v_L)=\bar u,$$4 such that

$$\mathcal{G}: u \mapsto P(u)=v_0 \mapsto v_1 \mapsto \cdots \mapsto v_L \mapsto Q(v_L)=\bar u,$$5

for all admissible coefficient fields $$\mathcal{G}: u \mapsto P(u)=v_0 \mapsto v_1 \mapsto \cdots \mapsto v_L \mapsto Q(v_L)=\bar u,$$6 in bounded subsets of the relevant Sobolev space (Raonić et al., 2023). The proof projects the PDE operator onto finite Fourier or trigonometric polynomial spaces, approximates the resulting finite-dimensional map by a shallow neural network, realizes that map by a CNN, and then converts it into CNO form with sinc-based interpolation (Raonić et al., 2023).

A complementary line of theory analyzes CNN-based operator learning in a constructive, discretized, Fourier-based setting. “Approximation bounds for convolutional neural networks in operator learning” studies operators of the form

$$\mathcal{G}: u \mapsto P(u)=v_0 \mapsto v_1 \mapsto \cdots \mapsto v_L \mapsto Q(v_L)=\bar u,$$7

with the output discretized on a grid and approximated by a network $$\mathcal{G}: u \mapsto P(u)=v_0 \mapsto v_1 \mapsto \cdots \mapsto v_L \mapsto Q(v_L)=\bar u,$$8. The key decomposition is

$$\mathcal{G}: u \mapsto P(u)=v_0 \mapsto v_1 \mapsto \cdots \mapsto v_L \mapsto Q(v_L)=\bar u,$$9

where $$v_{l+1}=\mathcal{P}_l\circ \Sigma_l\circ \mathcal{K}_l(v_l).$$0 is a fully connected network that learns coefficient-like information from parameters, while $$v_{l+1}=\mathcal{P}_l\circ \Sigma_l\circ \mathcal{K}_l(v_l).$$1 is a convolutional network that reconstructs the spatial field (Franco et al., 2022). The main theorem gives explicit complexity scaling: $$v_{l+1}=\mathcal{P}_l\circ \Sigma_l\circ \mathcal{K}_l(v_l).$$2 with active weights bounded by

$$v_{l+1}=\mathcal{P}_l\circ \Sigma_l\circ \mathcal{K}_l(v_l).$$3

and channels bounded by $$v_{l+1}=\mathcal{P}_l\circ \Sigma_l\circ \mathcal{K}_l(v_l).$$4 (Franco et al., 2022). The same paper is explicit that this is not the full modern infinite-dimensional neural-operator theory; the result is discrete-output, 1D in the rigorous proofs, and inherits a curse-of-dimensionality factor $$v_{l+1}=\mathcal{P}_l\circ \Sigma_l\circ \mathcal{K}_l(v_l).$$5 in the parameter dimension (Franco et al., 2022).

A further theoretical refinement concerns locality. “Neural Operators with Localized Integral and Differential Kernels” shows that a standard finite-stencil convolution with fixed kernel values collapses to a pointwise linear operator under grid refinement. To recover a genuine differential operator, the kernel must be centered by subtracting its mean and rescaled by $$v_{l+1}=\mathcal{P}_l\circ \Sigma_l\circ \mathcal{K}_l(v_l).$$6; for localized integral operators, the kernel must instead be represented continuously and sampled on arbitrary meshes through DISCO-style discrete-continuous convolutions (Liu-Schiaffini et al., 2024). This is directly relevant to CNO-type models because it clarifies when a convolution layer is merely grid-bound and when it defines a stable function-space operator.

4. Architectural variants and domain-specific extensions

One major extension is physics-informed regularization. In “An effective physics-informed neural operator framework for predicting wavefields,” the convolutional backbone is kept intact but trained with an additional PDE-residual term derived from the scattered-field Helmholtz equation (Ma et al., 22 Jul 2025). The learned map is

$$v_{l+1}=\mathcal{P}_l\circ \Sigma_l\circ \mathcal{K}_l(v_l).$$7

where $$v_{l+1}=\mathcal{P}_l\circ \Sigma_l\circ \mathcal{K}_l(v_l).$$8 is the analytically computed background wavefield in a homogeneous medium and $$v_{l+1}=\mathcal{P}_l\circ \Sigma_l\circ \mathcal{K}_l(v_l).$$9 is the scattered wavefield (Ma et al., 22 Jul 2025). The total objective is

$\mathcal{K}_l$0

with the Laplacian evaluated on the grid by an eighth-order finite-difference stencil rather than automatic differentiation (Ma et al., 22 Jul 2025). The modification is therefore not a new operator architecture but a physics-informed training strategy layered onto a standard CNO.

A second extension localizes the operator even more aggressively in space-time. PIC2O-Sim formulates Maxwell simulation as a causality-constrained, local, permittivity-conditioned propagation problem and uses a causality-aware dynamic convolutional neural operator as backbone (Ma et al., 2024). Its position-adaptive convolution is

$\mathcal{K}_l$1

with a Gaussian similarity kernel

$\mathcal{K}_l$2

The receptive field is chosen to honor the light cone, with a radius estimate

$\mathcal{K}_l$3

and the paper recommends a receptive field about “30 pixels larger” than the theoretical value for best fidelity (Ma et al., 2024). This is a CNO in a strongly physics-shaped sense: local kernels model finite-speed propagation and dynamic conditioning models permittivity-dependent propagation rules.

Other variants hybridize convolution with spectral components. CFNO, proposed for full-chip mask optimization, splits layouts into non-overlapping tokens, applies a token-shared Fourier neural operator to each token, and then mixes token embeddings by token-wise convolution (Yang et al., 2022). DCNO, aimed at multiscale PDEs, alternates Fourier layers with dilated convolution blocks using a hierarchical dilation schedule such as $\mathcal{K}_l$4 (Xu et al., 2024). In both cases, convolution is used to restore or preserve local and high-frequency structure that low-mode spectral truncation alone tends to suppress.

The transfer-learning literature has treated CNO as a pretrained operator backbone that can be adapted with few target samples. “Convolutional-neural-operator-based transfer learning for solving PDEs” investigates fine-tuning, LoRA, and neuron linear transformation (NLT), with NLT updating pretrained weights by

$\mathcal{K}_l$5

That study reports that NLT achieves the highest surrogate accuracy across Kuramoto-Sivashinsky, Brusselator, and Navier-Stokes transfer settings (Fan et al., 19 Dec 2025).

CNO has also been embedded into inversion pipelines. In “Velocity model building from seismic images using a Convolutional Neural Operator,” the learned operator maps velocity information to RTM images and is frozen inside an optimization loop, where gradients with respect to the velocity input are obtained by automatic differentiation rather than by an adjoint-state solver (Ma et al., 24 Sep 2025). In that case the CNO serves as a differentiable surrogate forward operator rather than as an end-to-end inverse map.

5. Empirical performance and application areas

Empirical work on CNOs spans benchmark PDE operator learning, frequency-domain seismic wavefield prediction, time-domain photonic simulation, lithography mask optimization, multiscale elliptic and Helmholtz problems, and seismic inversion. The reported results consistently emphasize three properties: resolution robustness, better handling of high-frequency or localized structure than purely spectral baselines, and practical acceleration relative to classical solvers or numerical optimization pipelines.

Variant	Domain	Reported result
CNO benchmarks (Raonić et al., 2023)	Representative PDE Benchmarks	On Poisson, CNO outperforms FNO by nearly a factor of 20 in test error
PICNO (Ma et al., 22 Jul 2025)	Helmholtz wavefield prediction	Test relative $\mathcal{K}_l$6 errors: CNO $\mathcal{K}_l$7 vs PICNO $\mathcal{K}_l$8 at $\mathcal{K}_l$9 Hz
PIC2O-Sim (Ma et al., 2024)	Photonic FDTD simulation	51.16% lower average normalized $\mathcal{P}_l$0 error, 95.74% fewer parameters than FNO, and 308$\mathcal{P}_l$1–632$\mathcal{P}_l$2 speedup over Meep
CFNO (Yang et al., 2022)	Full-chip mask optimization	Throughput 138.9 $\mathcal{P}_l$3, versus 0.01 for levelsetGPU and 0.22 for A2-ILT
DCNO (Xu et al., 2024)	Multiscale PDEs	Relative $\mathcal{P}_l$4 error $\mathcal{P}_l$5: Darcy rough 0.531 and trigonometric 0.541
Seismic inversion with CNO (Ma et al., 24 Sep 2025)	Velocity model building from RTM images	About 15 seconds for a single velocity model on one NVIDIA A100 GPU, versus about 2.5 hours for the conventional FWI comparison

The benchmark study that introduced CNO reports strong performance across Poisson, wave, smooth and discontinuous transport, Navier-Stokes, Darcy flow, and compressible Euler, with especially notable gains in multiscale settings and out-of-distribution generalization. It also reports that CNO error is essentially invariant to resolution in the Navier-Stokes experiment, while FNO and U-Net are more resolution dependent, and that CNO tracks the output Fourier spectrum more faithfully, with less spurious spectral amplification than FNO and U-Net (Raonić et al., 2023).

The seismic Helmholtz literature offers a more targeted view of why physics-informed CNO variants matter. In a deliberately limited-data regime, PICNO is reported to reduce average relative $\mathcal{P}_l$6 error by $\mathcal{P}_l$7, $\mathcal{P}_l$8, and $\mathcal{P}_l$9 at $\Sigma_l$0, $\Sigma_l$1, and $\Sigma_l$2 Hz, respectively, relative to a purely data-driven CNO, with the advantage becoming stronger as frequency increases (Ma et al., 22 Jul 2025). The same study emphasizes improved out-of-distribution behavior on a velocity model with faults and sharp discontinuities absent from training (Ma et al., 22 Jul 2025).

The photonics literature emphasizes a different empirical axis: long autoregressive rollout under local causal dynamics. PIC2O-Sim reports test errors of 0.052 on MMI, 0.085 on MRR, and 0.086 on Metaline, outperforming FNO, F-FNO, KNO, NeurOLight, SimpleCNN, and SineNet while using 2.4M–4.4M parameters instead of the 146.4M–340M reported for FNO (Ma et al., 2024). Its ablations show that the convolutional field encoder, the dynamic local backbone, two-stage partitioned time bundling, and cross-stage hidden-state propagation all contribute materially to rollout fidelity (Ma et al., 2024).

For multiscale PDEs, DCNO reports strong results on rough elliptic, inverse, Navier-Stokes, and Helmholtz problems. On the Darcy rough and multiscale trigonometric elliptic benchmarks, the relative $\Sigma_l$3 errors $\Sigma_l$4 are 0.531 and 0.541 for DCNO, compared with 1.749 and 1.744 for FNO, and the paper states improvements over FNO of about 71% and 69% (Xu et al., 2024). The same work reports that DCNO’s relative error remains approximately invariant with input resolution, aligning it with the broader neural-operator objective of mesh-independent inference (Xu et al., 2024).

6. Limitations, ambiguities, and research directions

A recurrent ambiguity is terminological. In the original formulation, CNO is a bandlimited, alias-free, representation-equivalent neural operator with carefully defined activation and multiresolution transfers (Raonić et al., 2023). In later application papers, the same term may refer to a conventional convolutional backbone used in an operator-learning setting, sometimes with physical-space convolution, sometimes with dilated or dynamic kernels, and sometimes with additional Fourier branches or PDE-regularized objectives. This suggests that the field has not converged on a single canonical definition.

Several limitations are explicit in the theory. The constructive approximation bounds for CNN operator learning are derived for discretized outputs and 1D proofs, not for the full continuous infinite-dimensional setting, and the dense block still incurs a factor $\Sigma_l$5, so the method remains exposed to high-dimensional parameter dependence (Franco et al., 2022). Even in the broader neural-operator setting, localized-kernel theory shows that a naive fixed-stencil convolution does not automatically define a stable function-space operator under refinement; without appropriate scaling or continuous kernel parameterization, it collapses to a pointwise map (Liu-Schiaffini et al., 2024).

Generalization under distribution shift is also a live issue. The few-shot transfer study reports that a source-trained CNO can generalize poorly when applied directly to shifted target distributions: for Navier-Stokes, the relative $\Sigma_l$6 test error rises from $\Sigma_l$7 on source data to as high as $\Sigma_l$8 on target data without transfer (Fan et al., 19 Dec 2025). The same paper presents transfer adaptation, especially NLT, as a remedy rather than an intrinsic property of the original CNO (Fan et al., 19 Dec 2025).

Application papers disclose more domain-specific constraints. The physics-informed Helmholtz study notes that PICNO predictions can be somewhat smoother than the reference wavefields, likely because of limited training diversity and the smoothing tendency of convolutional operators (Ma et al., 22 Jul 2025). The mask-optimization CFNO framework assumes binary layout inputs and depends on a lithography simulator to decide whether a model-generated mask is better than the legacy label, which makes its litho-guided self-training pipeline domain-specific rather than generic (Yang et al., 2022). The seismic inversion framework based on a frozen CNO forward surrogate depends strongly on the quality of the initial/background model, on the representativeness of the synthetic training set, and, because it uses RTM images, does not capture more complex wave phenomena such as multiples or mode conversions (Ma et al., 24 Sep 2025).

The current research trajectory therefore points in several directions already visible in the literature: stricter treatment of locality and operator consistency, better high-frequency retention, physics-informed training objectives, adaptation under distribution shift, and embedding learned forward operators inside optimization loops. Across these directions, the underlying objective remains stable: to preserve the operator-learning character of neural operators while exploiting the locality, multiscale hierarchy, and implementation efficiency of convolutional architectures.