Subdomain Neural Operator Model

• Subdomain neural operator models decompose global PDE operators into local components with explicit coupling mechanisms, improving geometry transferability and handling multiscale challenges.
• Key formulations use overlapping/non-overlapping domain decompositions, attention mechanisms, and reduced subspaces to achieve significant parameter efficiency and error reductions up to 96.8%.
• Empirical studies demonstrate that these models capture discontinuities and long-range interactions effectively, enabling faster convergence and robust solver reuse.

Searching arXiv for papers on subdomain neural operators and related domain-decomposition neural operator models. A subdomain neural operator model is a class of operator-learning constructions in which a global map between function spaces is represented through local or restricted components together with a mechanism for global communication or reconstruction. In the PDE setting, recent work realizes this idea through overlapping or non-overlapping domain decomposition, reusable local solution operators, subdomain-level attention, interface-aware latent embeddings, and geometry-adapted finite-dimensional subspaces. Taken together, these works suggest that subdomain neural operator modeling is most natural when the target operator is nonlocal but multiscale, when geometry transferability is harder than coefficient transferability, or when discontinuities and irregular domains make monolithic global parameterizations brittle or inefficient (Köhler et al., 16 Jun 2026, Huang et al., 1 Apr 2025, Ouyang et al., 23 Jun 2025, Feeney et al., 9 Jun 2025, Roy et al., 9 Apr 2026, Yang et al., 22 Apr 2025).

1. Conceptual basis and representative formulations

A recurring motivation is that many PDE solution operators are global and nonlocal, but their action is not uniform across scales. For a Poisson or diffusion operator, the inverse $A^{-1}$ is global, yet local, short-range coupling is strongest, while long-range, low-frequency coupling remains essential (Köhler et al., 16 Jun 2026). A second motivation is geometry generalization: operator learning on arbitrary global geometries is difficult because the input space includes the domain shape itself, and standard neural operators can struggle on unseen shapes, especially multiply connected domains with holes or corners (Huang et al., 1 Apr 2025). A third motivation is computational: in transformer-based operator models, attention cost is typically tied to discretization density, while FFT-based spectral convolutions assume a regular grid or flat geometry and can degrade on irregular domains (Feeney et al., 9 Jun 2025, Yang et al., 22 Apr 2025).

These motivations lead to several non-identical, but structurally related, model families.

Formulation	Decomposition unit	Global coupling mechanism
SNI (Huang et al., 1 Apr 2025)	overlapping subdomains	Schwarz-Richardson stitching
Hierarchical attention (Köhler et al., 16 Jun 2026)	overlapping subdomains plus a coarse space	two-level additive attention
Mondrian (Feeney et al., 9 Jun 2025)	non-overlapping subdomains	softmax-based inner products over functions
$\phi$-DeepONet (Roy et al., 9 Apr 2026)	one-hot representation of the domain decomposition	latent interface embedding in a modified trunk network
SUPRA (Yang et al., 22 Apr 2025)	finite-dimensional subspace	attention on subspace coordinates
NOEM (Ouyang et al., 23 Jun 2025)	FE and NOE subdomains	variational assembly

A common misconception is that the term denotes a single architecture. The literature instead uses the same local-to-global intuition in several ways: direct domain decomposition, attention over subdomain-restricted functions, interface-aware latent representations, or subspace restriction on general domains. This suggests that the unifying feature is not a specific backbone, but the replacement of a monolithic global operator by structured local or reduced components.

2. Schwarz-style local-to-global solvers and reusable subdomain operators

In "Operator Learning with Domain Decomposition for Geometry Generalization in PDE Solving" (Huang et al., 1 Apr 2025), the target domain is decomposed into overlapping subdomains $\{\Omega_k\}_{k=1}^K$, and a learned local operator replaces the exact local PDE solver inside an additive Schwarz iteration. The inference-time engine is called Schwarz Neural Inference (SNI). Its update is written as

$$\tilde{w}^{n+1}_k = \tilde{T}_k\circ\mathcal{G}^\dag\circ T_k(\Omega_k, B_k^n),$$

$$w^{n+1} = \sum_{k=1}^K R_k^\intercal \tilde{w}^{n+1}_k + (I - R_k^\intercal R_k)u^n, \qquad u^{n+1} = (1-\tau K)u^n+\tau w^{n+1}.$$

Here $T_k$ and $\tilde{T}_k$ are preprocessing and postprocessing transformations, and the learned operator is trained on a family of basic local shapes rather than arbitrary global domains. In implementation, the mesh adjacency graph is partitioned with METIS, the subdomains are made overlapping by extension steps, training domains are random simple polygons bounded by the unit square, and GNOT is used as the base neural operator, although the framework is described as architecture-agnostic. The paper also provides a convergence theorem: if the classical additive Schwarz method converges and the learned local operators are uniformly close to the exact local solvers, then SNI converges to a fixed point, with

$$\|\tilde{u}^{n} - {u}^{n}\| \le \frac{1-\rho^n}{1-\rho}\,\tau t c, \qquad c'=\frac{\tau t c}{1-\rho}.$$

Empirically, relative error reductions of roughly 34.8% to 96.8% are reported across stationary problems and domains, with 54.1%–74.2% reductions on the heat problem.

The neural-operator element method (NOEM) takes a related but distinct route (Ouyang et al., 23 Jun 2025). Instead of iteratively stitching local solves to recover the global field, it inserts learned subdomain operators directly into a variational finite-element framework. The domain is partitioned as

$\Omega = \Omega^\text{FE} \cup \Omega^\text{NOE},$

and each NOE is a learned local Dirichlet-to-solution operator,

$$\mathcal{G}: u |_{\partial \Omega^\text{NOE}_j} \mapsto u|_{\Omega^\text{NOE}_j}.$$

The global approximation is

$\phi$0

and the unknown coefficients are obtained from

$\phi$1

This turns pretrained local neural operators into reusable elements inside a global Newton solve. The paper reports that, in a $\phi$2 multi-hole heat-transfer problem, the NOEM relative $\phi$3 error is 1.6%, and for 100 holes the method gives about 14 times speedup over FEM. In the 1D pedagogical problem, replacing 100 FEM elements on the middle segment by one NOE yields about 0.1% relative error.

The two frameworks reflect different uses of subdomains. SNI is an iterative Schwarz wrapper around local operator inference, whereas NOEM treats the local operator as a modular element inside a variational assembly. A plausible implication is that the former emphasizes geometry generalization by composition, while the latter emphasizes solver reuse and coarse global discretization.

3. Attention-based subdomain operators

A more explicitly operator-architectural formulation appears in "Hierarchical Attention via Domain Decomposition" (Köhler et al., 16 Jun 2026). The paper studies a 1D diffusion problem with homogeneous Dirichlet boundary conditions and interprets operator learning as approximating the inverse map $\phi$4. The baseline is a global softmax-free low-rank attention operator $\phi$5. The proposed subdomain neural operator replaces this dense factorization by a two-level additive structure: $\phi$6 Here $\phi$7 restricts to an overlapping subdomain, $\phi$8 is a partition-of-unity weight satisfying

$\phi$9

and $\{\Omega_k\}_{k=1}^K$0 is a coarse interpolation matrix. Local blocks $\{\Omega_k\}_{k=1}^K$1 handle short-range interactions inside overlapping subdomains, while the coarse block $\{\Omega_k\}_{k=1}^K$2 communicates low-frequency, long-range information. In the main experiment, the parameters are $\{\Omega_k\}_{k=1}^K$3, $\{\Omega_k\}_{k=1}^K$4, overlap $\{\Omega_k\}_{k=1}^K$5, local rank $\{\Omega_k\}_{k=1}^K$6, coarse rank effectively truncated to $\{\Omega_k\}_{k=1}^K$7, and global baseline rank $\{\Omega_k\}_{k=1}^K$8. The global baseline uses $\{\Omega_k\}_{k=1}^K$9 trainable parameters, while the Schwarz attention operator uses $$\tilde{w}^{n+1}_k = \tilde{T}_k\circ\mathcal{G}^\dag\circ T_k(\Omega_k, B_k^n),$$0, or about 8.6 times fewer parameters. At $$\tilde{w}^{n+1}_k = \tilde{T}_k\circ\mathcal{G}^\dag\circ T_k(\Omega_k, B_k^n),$$1, the final weighted MSE after 2000 steps is $$\tilde{w}^{n+1}_k = \tilde{T}_k\circ\mathcal{G}^\dag\circ T_k(\Omega_k, B_k^n),$$2 for the global model and $$\tilde{w}^{n+1}_k = \tilde{T}_k\circ\mathcal{G}^\dag\circ T_k(\Omega_k, B_k^n),$$3 for the Schwarz model; the mean relative $$\tilde{w}^{n+1}_k = \tilde{T}_k\circ\mathcal{G}^\dag\circ T_k(\Omega_k, B_k^n),$$4 error on 16 evaluation right-hand sides is $$\tilde{w}^{n+1}_k = \tilde{T}_k\circ\mathcal{G}^\dag\circ T_k(\Omega_k, B_k^n),$$5 versus $$\tilde{w}^{n+1}_k = \tilde{T}_k\circ\mathcal{G}^\dag\circ T_k(\Omega_k, B_k^n),$$6; and the relative Frobenius operator error is $$\tilde{w}^{n+1}_k = \tilde{T}_k\circ\mathcal{G}^\dag\circ T_k(\Omega_k, B_k^n),$$7 versus $$\tilde{w}^{n+1}_k = \tilde{T}_k\circ\mathcal{G}^\dag\circ T_k(\Omega_k, B_k^n),$$8. The paper describes this as a controlled study rather than a general PDE solver, but it makes the Schwarz analogy explicit at the level of both architecture and function.

"Mondrian: Transformer Operators via Domain Decomposition" (Feeney et al., 9 Jun 2025) develops a different attention-based construction. The domain is decomposed into $$\tilde{w}^{n+1}_k = \tilde{T}_k\circ\mathcal{G}^\dag\circ T_k(\Omega_k, B_k^n),$$9 non-overlapping subdomains, and attention is performed over the resulting sequence of subdomain-restricted functions rather than over grid points. For each subdomain,

$$w^{n+1} = \sum_{k=1}^K R_k^\intercal \tilde{w}^{n+1}_k + (I - R_k^\intercal R_k)u^n, \qquad u^{n+1} = (1-\tau K)u^n+\tau w^{n+1}.$$0

and the attention score matrix is defined by function-space inner products

$$w^{n+1} = \sum_{k=1}^K R_k^\intercal \tilde{w}^{n+1}_k + (I - R_k^\intercal R_k)u^n, \qquad u^{n+1} = (1-\tau K)u^n+\tau w^{n+1}.$$1

The attended output is

$$w^{n+1} = \sum_{k=1}^K R_k^\intercal \tilde{w}^{n+1}_k + (I - R_k^\intercal R_k)u^n, \qquad u^{n+1} = (1-\tau K)u^n+\tau w^{n+1}.$$2

Because the score matrix is $$w^{n+1} = \sum_{k=1}^K R_k^\intercal \tilde{w}^{n+1}_k + (I - R_k^\intercal R_k)u^n, \qquad u^{n+1} = (1-\tau K)u^n+\tau w^{n+1}.$$3, attention cost is decoupled from mesh resolution. The model also extends to hierarchical windowed attention and neighborhood attention, with the latter using a distance-based mask. In the Allen-Cahn experiments, the domain is partitioned into 64 subdomains using a $$w^{n+1} = \sum_{k=1}^K R_k^\intercal \tilde{w}^{n+1}_k + (I - R_k^\intercal R_k)u^n, \qquad u^{n+1} = (1-\tau K)u^n+\tau w^{n+1}.$$4 subdomain size; the best in-distribution $$w^{n+1} = \sum_{k=1}^K R_k^\intercal \tilde{w}^{n+1}_k + (I - R_k^\intercal R_k)u^n, \qquad u^{n+1} = (1-\tau K)u^n+\tau w^{n+1}.$$5 MAE is $$w^{n+1} = \sum_{k=1}^K R_k^\intercal \tilde{w}^{n+1}_k + (I - R_k^\intercal R_k)u^n, \qquad u^{n+1} = (1-\tau K)u^n+\tau w^{n+1}.$$6 for ViT-NO-SepMO and $$w^{n+1} = \sum_{k=1}^K R_k^\intercal \tilde{w}^{n+1}_k + (I - R_k^\intercal R_k)u^n, \qquad u^{n+1} = (1-\tau K)u^n+\tau w^{n+1}.$$7 for Swin-NO-MO; at $$w^{n+1} = \sum_{k=1}^K R_k^\intercal \tilde{w}^{n+1}_k + (I - R_k^\intercal R_k)u^n, \qquad u^{n+1} = (1-\tau K)u^n+\tau w^{n+1}.$$8, these remain $$w^{n+1} = \sum_{k=1}^K R_k^\intercal \tilde{w}^{n+1}_k + (I - R_k^\intercal R_k)u^n, \qquad u^{n+1} = (1-\tau K)u^n+\tau w^{n+1}.$$9 and $T_k$0. On 2D decaying Navier–Stokes, the best reported Swin-NO result is MAE $T_k$1 and rollout MAE $T_k$2. Mondrian therefore uses domain decomposition not as a Schwarz preconditioner, but as the tokenization mechanism of a transformer operator.

The contrast between these two attention models is instructive. Hierarchical Schwarz attention preserves an overlapping-subdomain plus coarse-space algebra directly modeled on two-level Schwarz, whereas Mondrian uses non-overlapping subdomains and exact softmax attention over function blocks. This suggests two distinct meanings of “subdomain attention”: one as a structured approximation to an inverse operator, the other as a resolution-agnostic transformer over restricted functions.

4. Interface-aware and discontinuity-capturing formulations

$T_k$3-DeepONet addresses a setting in which subdomains are induced by material interfaces, jump conditions, and piecewise-defined coefficients rather than by a purely geometric partition (Roy et al., 9 Apr 2026). The domain $T_k$4 is partitioned into $T_k$5 subdomains $T_k$6, and the target operator maps composite input functions $T_k$7 to composite outputs $T_k$8. The central ansatz is

$T_k$9

Discontinuous inputs are handled through multiple branch networks $\tilde{T}_k$0, one for each subdomain. Discontinuous outputs are handled through a modified trunk network that receives a latent interface embedding $\tilde{T}_k$1. The paper discusses scalar embedding, categorical embedding based on a one-hot vector,

$\tilde{T}_k$2

and nonlinear categorical embedding,

$\tilde{T}_k$3

Training is governed by a physics- and interface-informed loss combining PDE, boundary, and interface residuals.

The empirical motivation is that standard physics-informed DeepONet performs poorly on interface problems with discontinuities. On the 1D single-interface problem, standard DeepONet gives $\tilde{T}_k$4, IONet gives $\tilde{T}_k$5, and $\tilde{T}_k$6-DeepONet ranges from $\tilde{T}_k$7 for scalar embedding to $\tilde{T}_k$8 for nonlinear categorical embedding. On the 1D four-interface problem, standard DeepONet gives $\tilde{T}_k$9, IONet gives $$\|\tilde{u}^{n} - {u}^{n}\| \le \frac{1-\rho^n}{1-\rho}\,\tau t c, \qquad c'=\frac{\tau t c}{1-\rho}.$$0, and nonlinear categorical embedding with $$\|\tilde{u}^{n} - {u}^{n}\| \le \frac{1-\rho^n}{1-\rho}\,\tau t c, \qquad c'=\frac{\tau t c}{1-\rho}.$$1 gives $$\|\tilde{u}^{n} - {u}^{n}\| \le \frac{1-\rho^n}{1-\rho}\,\tau t c, \qquad c'=\frac{\tau t c}{1-\rho}.$$2. On the 2D diagonal-interface problem, standard DeepONet gives $$\|\tilde{u}^{n} - {u}^{n}\| \le \frac{1-\rho^n}{1-\rho}\,\tau t c, \qquad c'=\frac{\tau t c}{1-\rho}.$$3, IONet $$\|\tilde{u}^{n} - {u}^{n}\| \le \frac{1-\rho^n}{1-\rho}\,\tau t c, \qquad c'=\frac{\tau t c}{1-\rho}.$$4, and nonlinear categorical embedding with $$\|\tilde{u}^{n} - {u}^{n}\| \le \frac{1-\rho^n}{1-\rho}\,\tau t c, \qquad c'=\frac{\tau t c}{1-\rho}.$$5 gives $$\|\tilde{u}^{n} - {u}^{n}\| \le \frac{1-\rho^n}{1-\rho}\,\tau t c, \qquad c'=\frac{\tau t c}{1-\rho}.$$6. The paper also reports that $$\|\tilde{u}^{n} - {u}^{n}\| \le \frac{1-\rho^n}{1-\rho}\,\tau t c, \qquad c'=\frac{\tau t c}{1-\rho}.$$7-DeepONet variants have cost $$\|\tilde{u}^{n} - {u}^{n}\| \le \frac{1-\rho^n}{1-\rho}\,\tau t c, \qquad c'=\frac{\tau t c}{1-\rho}.$$8, compared with 1.26, 2.71, and 2.55 for IONet in representative benchmarks.

This architecture occupies an intermediate position within the broader subdomain neural operator landscape. It does not use explicit domain decomposition in the network architecture in the same way as IONet, yet it injects subdomain identity into both the operator parameterization and the loss. A plausible implication is that “subdomain” need not always mean spatially separate local solvers; it can also mean a latent representation of interface-induced function-space segmentation.

5. Geometry-aware subspaces and decomposed operator theory

SUPRA generalizes attention from vectors to functions on general domains (Yang et al., 22 Apr 2025). The starting point is an attention analogue in $$\|\tilde{u}^{n} - {u}^{n}\| \le \frac{1-\rho^n}{1-\rho}\,\tau t c, \qquad c'=\frac{\tau t c}{1-\rho}.$$9,

$\Omega = \Omega^\text{FE} \cup \Omega^\text{NOE},$0

which is then approximated in a finite-dimensional subspace spanned by $\Omega = \Omega^\text{FE} \cup \Omega^\text{NOE},$1 basis functions. With subspace coordinates $\Omega = \Omega^\text{FE} \cup \Omega^\text{NOE},$2,

$\Omega = \Omega^\text{FE} \cup \Omega^\text{NOE},$3

and the resulting attention is

$\Omega = \Omega^\text{FE} \cup \Omega^\text{NOE},$4

On irregular domains, the basis is chosen as the Laplacian eigensubspace,

$\Omega = \Omega^\text{FE} \cup \Omega^\text{NOE},$5

The paper argues that these eigenfunctions are defined directly on the physical domain, encode topology and geometry naturally, and generalize Fourier modes to arbitrary geometries. SUPRA has complexity

$\Omega = \Omega^\text{FE} \cup \Omega^\text{NOE},$6

compared with $\Omega = \Omega^\text{FE} \cup \Omega^\text{NOE},$7 for Fourier attention and $\Omega = \Omega^\text{FE} \cup \Omega^\text{NOE},$8 for Galerkin attention. Across five PDE benchmarks—Darcy, Navier–Stokes, Plasticity, Airfoil, and Pipe—the method reports relative $\Omega = \Omega^\text{FE} \cup \Omega^\text{NOE},$9 error reductions up to 33% while maintaining state-of-the-art computational efficiency. Representative errors ($$\mathcal{G}: u |_{\partial \Omega^\text{NOE}_j} \mapsto u|_{\Omega^\text{NOE}_j}.$$0) include Darcy 0.43, Navier–Stokes 6.25, Plasticity 0.04, Airfoil 0.34, and Pipe 0.31.

Although SUPRA is presented as a general-domain neural operator rather than a domain-decomposition method, it is closely related to subdomain modeling in the broader sense that it replaces raw pointwise interactions by geometry-adapted restricted coordinates. The paper explicitly states that its Laplacian basis makes it naturally a subdomain/general-domain operator model.

A parallel theoretical direction appears in "A Deep Learning Framework for Multi-Operator Learning: Architectures and Approximation Theory" (Weihs et al., 29 Oct 2025). That work is not explicitly about subdomain neural operators in the geometric or domain-decomposition sense, but it introduces decomposed operator architectures—$$\mathcal{G}: u |_{\partial \Omega^\text{NOE}_j} \mapsto u|_{\Omega^\text{NOE}_j}.$$1 and $$\mathcal{G}: u |_{\partial \Omega^\text{NOE}_j} \mapsto u|_{\Omega^\text{NOE}_j}.$$2—that separately encode parameterized operator identity $$\mathcal{G}: u |_{\partial \Omega^\text{NOE}_j} \mapsto u|_{\Omega^\text{NOE}_j}.$$3, input function $$\mathcal{G}: u |_{\partial \Omega^\text{NOE}_j} \mapsto u|_{\Omega^\text{NOE}_j}.$$4, and output coordinate $$\mathcal{G}: u |_{\partial \Omega^\text{NOE}_j} \mapsto u|_{\Omega^\text{NOE}_j}.$$5. For example,

$$\mathcal{G}: u |_{\partial \Omega^\text{NOE}_j} \mapsto u|_{\Omega^\text{NOE}_j}.$$6

and

$$\mathcal{G}: u |_{\partial \Omega^\text{NOE}_j} \mapsto u|_{\Omega^\text{NOE}_j}.$$7

The paper proves universal approximation in continuous and measurable settings and derives explicit scaling laws for Lipschitz operators. This is relevant because it provides a rigorous approximation-theoretic template for decomposed operator learning even when the decomposition is not a physical partition of $$\mathcal{G}: u |_{\partial \Omega^\text{NOE}_j} \mapsto u|_{\Omega^\text{NOE}_j}.$$8.

Another adjacent construction is the domain-aware KANO/2FBNO framework for 2BSDE families on bounded Euclidean domains (Furuya et al., 3 Nov 2025). It is not formulated as a subdomain decomposition method, but it uses localized kernels, wavelet/Besov machinery, and domain-adapted operator approximation. For compact perturbation sets, the solution map $$\mathcal{G}: u |_{\partial \Omega^\text{NOE}_j} \mapsto u|_{\Omega^\text{NOE}_j}.$$9 is uniformly approximable in $\phi$00, and in a structured semilinear subclass the paper gives logarithmic-depth and polynomial-width complexity bounds. This suggests that localized, geometry-aware operator representations form a broader mathematical context for subdomain neural operator models.

6. Empirical patterns, limitations, and scope

Across these works, several empirical patterns recur. First, structural locality tends to improve data efficiency or parameter efficiency. The Schwarz attention operator reaches lower final weighted MSE than a global low-rank baseline while using significantly fewer parameters (Köhler et al., 16 Jun 2026). SNI achieves strong geometry generalization by transferring a local operator across subdomains rather than learning every target geometry directly (Huang et al., 1 Apr 2025). NOEM reuses a single pretrained local operator across many global solves and across multiple-hole geometries without additional tuning or training (Ouyang et al., 23 Jun 2025). Second, explicit treatment of long-range communication remains necessary. Hierarchical attention adds a coarse space (Köhler et al., 16 Jun 2026), Mondrian retains global softmax attention over subdomains (Feeney et al., 9 Jun 2025), and SUPRA preserves global attention but performs it in a reduced function-space basis (Yang et al., 22 Apr 2025). Third, discontinuities require dedicated mechanisms. $\phi$01-DeepONet augments the trunk with a latent interface embedding and splits the branch side across subdomains because a vanilla DeepONet trunk spans a smooth basis for the output space (Roy et al., 9 Apr 2026).

The limitations are equally consistent. The hierarchical Schwarz-attention study is explicitly limited to a one-dimensional Poisson problem, synthetic mixed Fourier signals, and a linear softmax-free model, with higher-dimensional and nonlinear extensions left as future work (Köhler et al., 16 Jun 2026). SNI depends on local operators trained on a family of basic shapes and on preprocessing that maps local problems into the training range (Huang et al., 1 Apr 2025). Mondrian notes that true zero-shot superresolution remains difficult and that new subdomain behaviors may appear as the physical domain grows (Feeney et al., 9 Jun 2025). $\phi$02-DeepONet reports that larger latent dimensions can worsen performance due to over-parameterization, and its OOD accuracy degrades as the GRF length scale decreases (Roy et al., 9 Apr 2026). SUPRA reports that training can fail without normalization and that basis choice matters, especially on irregular domains (Yang et al., 22 Apr 2025). NOEM shows that the global error is strongly correlated with the local NO error, so local operator fidelity remains the governing factor (Ouyang et al., 23 Jun 2025).

The scope of the term is therefore narrower than “any neural operator on a bounded domain,” but broader than “Schwarz iteration with a neural local solver.” A precise reading of the recent literature suggests three core criteria. First, the model replaces a monolithic global map by structured local, restricted, or reduced components. Second, it includes an explicit mechanism for coupling those components—coarse correction, softmax attention, iterative stitching, variational assembly, or latent interface embedding. Third, it exploits that structure to address one of the principal failure modes of global operator models: geometry transferability, attention cost, interface discontinuity, or solver reuse. Within that frame, subdomain neural operator models form a technically diverse but conceptually coherent family of local-to-global operator-learning methods.