Neural Operator: Learning Maps Between Function Spaces

1. Introduction

• The framework addresses the failure of discretized neural networks to generalize beyond training grids by directly learning operators that can be discretized at different resolutions without retraining.Standard architectures may require new parameters to maintain comparable error under changed discretizations.Source: §1 · pp. 1–2

1.1 Our Approach

• Discretization invariance requires accepting arbitrary input points, evaluating outputs anywhere, converging under refinement, and using a fixed parameter count; neural operators satisfy these requirements whereas standard neural networks do not.This property enables application to data at any resolution and on any mesh, while graph neural networks and transformers may accept varying resolutions without converging to a continuum operator.Source: §1.1 · p. 2
• Neural operators compose linear operator layers with nonlinear activations to learn mappings between infinite-dimensional function spaces, with universal approximation for continuous operators on compact Banach-space sets.The framework replaces finite-dimensional neural-network layers with linear operators in function spaces.Source: §1.1 · pp. 2–3
• Four practical parameterizations—graph-based, low-rank, multipole graph-based, and Fourier operators—implement the framework through integral or spectral-domain operator layers.Graph operators use a Nyström extension for arbitrary grids, while Fourier operators enable efficient algorithms when fast transforms apply.Source: §1.1 · p. 3
• For two-dimensional Navier–Stokes flow maps, neural operators achieve < 1% error at Reynolds number 20 and 8% at Reynolds number 200, while FNO inference takes 0.005s versus 2.2s for the pseudo-spectral solver.The proposed methodology consistently outperforms existing deep-learning methods, and FNO retains accuracy in Bayesian inverse problems while remaining robust to testing noise.Source: §1.1 · p. 3

1.2 Background and Context

• Neural operators target PDE solution operators because PDEs are local but their solution maps are non-local, making learned integral or spectral operators a natural alternative to grid-limited neural networks.Existing models differ in their capabilities: PINNs represent one instance rather than a map between functions, interpolation-based networks lack universal operator approximation, and DeepONets can be incorporated as special cases or extended to DeepONet-Operator.Source: §1.2 · pp. 5–6

2.1 Generic Parametric PDEs

• For parametric PDEs, the solution operator G† maps coefficients or parameters a ∈ A to solutions u ∈ U through an operator equation L_a u = f.The framework includes weak elliptic operators such as −∇·(a∇) with homogeneous Dirichlet boundary conditions.Source: §2.1 · p. 6

2.2 Problem Setting

• Neural operator learning approximates a typically nonlinear map G†: A → U between Banach spaces of functions using finite-dimensional parameters and observed input-output pairs.The observations may be noisy, and approximation error is considered through average Bochner norms or uniformly over compact input sets.Source: §2.2 · p. 7
• The theory guarantees approximation below any desired tolerance with sufficient parameter dimension, but does not characterize error for fixed parameters or a fixed number of training samples.On fixed grids, neural operators coincide with ordinary neural networks; alternative basis-coefficient representations are noted but not pursued.Source: §2.2 · p. 7, §2.3 · p. 9

2.3 Discretization

• Training uses pointwise evaluations of input and output functions, while the discretized architecture can predict u(x) at arbitrary x ∈ D rather than only on the training grid.The architecture is trained on discretized data but maps into U, not into a discretized output space.Source: §2.3 · p. 8
• The proposed architecture is discretization-invariant: fixed parameters define operators whose outputs remain in the function space as input grids are refined.The paper proves this property for its architectures and reports approximately constant approximation error under refinement, enabling transfer across grid geometries and sizes.Source: §2.3 · pp. 8–9

Neural Operator Architecture

• The architecture lifts input functions, repeatedly combines local linear maps, nonlocal integral kernels, biases, and pointwise nonlinearities, then projects to the output function.These operations are defined directly in function space, so learned parameters can be reused across discretizations; hidden dimensions and intermediate domains are architectural hyperparameters.Source: §3 · pp. 9–10

Integral Kernel Operators

• The basic integral-kernel form uses a measure such as Lebesgue measure, while alternative measures can reduce computation or encode prior information.The basic form is the one analyzed theoretically and used most extensively in the numerical experiments.Source: §3 · p. 10
• Input-dependent kernels can outperform basic kernels on PDEs because they preserve the influence of the input function throughout all layers.The paper motivates this by noting that, without direct kernel dependence on a, its influence enters only through the initial condition and can diminish with depth.Source: §3 · p. 11
• A nonlinear kernel operator depending on the hidden function provides a continuous input/output-space generalization of the transformer architecture.Unlike the other kernel forms, this operator is nonlinear because its kernel depends on the input function v_t.Source: §3 · p. 11

Preprocessing

• Including coordinates, smoothed inputs, or derivative features in the input function can ease learning by supplying geometry and problem-specific information.The experiments use coordinate augmentation, while smoothing and derivative features are proposed for suitable operators and tasks.Source: §3 · p. 12

Discretization Invariance and Approximation

• Approximation and discretization errors can each be made arbitrarily small, so fixed-parameter neural operators can approximate operators to arbitrary accuracy on sufficiently fine discretizations.Universal approximation controls the approximation error, while discretization invariance controls the discretization error as the number of discretization points increases.Source: §3 · p. 12

4. Parameterization and Computation

• Neural-operator parameters can remain independent of discretization, enabling training on one discretization and prediction on another while maintaining comparable approximation error.The target is an intrinsic parameterization with error ϵ whose numerical approximation achieves approximately 2ϵ for sufficiently fine discretizations.Source: §4 · p. 12
• Kernel integration costs O(J^2) on a J-point discretization, making large-grid computation intractable, whereas pointwise terms require only O(J).The proposed parameterizations target this kernel bottleneck through assumptions such as sparsity, low rank, or hierarchical structure.Source: §4 · pp. 13–14

4.1 Graph Neural Operator (GNO)

• The Graph Neural Operator combines Nyström subsampling, domain truncation, and message passing to approximate kernel integration on arbitrary discretizations.Nyström reduces the quadratic cost to O(J′^2), while truncation yields O(c_sJ^2) with c_s≈r^d for radius r on D=[0,1]^d; composition can recover global influence with lower cost.Source: §4.1 · pp. 14–15

4.2 Low-rank Neural Operator (LNO)

• The Low-rank Neural Operator parameterizes the kernel as a rank-r tensor product, reducing evaluation complexity to O(rJ) and remaining consistent in function space.The construction generalizes from scalar to vector-valued functions and imposes rank-r structure on the kernel matrix.Source: §4.2 · pp. 16–17

4.3 Multipole Graph Neural Operator (MGNO)

• The Multipole Graph Neural Operator combines sparse short-range and low-rank long-range kernel approximations across multiscale discretizations, using a V-cycle to achieve linear computational complexity.Its hierarchy uses finer grids for local interactions, coarser grids for global interactions, and learned neural-network kernels for levelwise operators and transitions.Source: §4.3 · pp. 17–19
• MGNO represents discretizations with multilevel graphs, using orthogonal multipole graphs for uniform grids and generalized random graphs for arbitrary geometries, sampling, or adaptive refinement.Designing sub-discretizations appropriately yields O(J) complexity, and combining with a Nyström approximation yields O(J′) for J′ ≪ J.Source: §4.3 · p. 20

4.4 Fourier Neural Operator (FNO)

• FNO directly parameterizes convolution kernels by truncated Fourier coefficients, enabling discretization-invariant layers and FFT-based computation with O(J log J) complexity.The direct parameterization outperformed feed-forward alternatives, while linear parameterization had similar performance but greater computational and parameter costs.Source: §4.4 · pp. 21, 23, §4.5 · p. 24
• FNO’s fast implementation assumes uniform periodic grids or compatible boundary conditions, although Fourier continuations and zero-padding extend applicability to more general geometries.The underlying operations are defined more generally, but FFT efficiency requires the stated discretization conditions.Source: §4.4 · pp. 23–24

4.5 Summary

• The section’s summary contrasts GNO’s O(JJ′), LNO’s O(J), MGNO’s O(J), and FNO’s O(J log J) computational approaches.GNO subsamples J′ points, LNO uses a tensor-product kernel decomposition, MGNO uses multiscale kernel decomposition, and FNO uses Fourier-domain parameterization with FFT.Source: §4.5 · p. 24

5. Neural Operators and Other Deep Learning Models

• Neural operators encompass DeepONets and transformer attention as special cases, but inconsistent parameterizations can sacrifice discretization invariance.The framework recovers DeepONets through point-wise first-kernel parameterization and transformer blocks through particular nonlinear kernels and discretization.Source: §5 · p. 24, §5.1 · p. 25, §5.2 · p. 28

5.1 DeepONets

• DeepONet-Operator restores discretization invariance by replacing finite-dimensional branch inner products with function-space projections, while standard DeepONet error grows with input discretization.Parametrizing the projection functions with neural networks makes the model discretization invariant; numerical results in Section 7 compare this behavior with neural operators.Source: §5.1 · p. 26

5.2 Transformers as a Special Case of Neural Operators

• The attention mechanism in transformer models is a special case of a neural operator layer, with queries, keys, and values represented by learned functions.Standard attention remains memory- and computation-intensive because it evaluates a nested integral, whereas some efficient vision transformers using CNN tokenization are not discretization invariant.Source: §5.2 · pp. 28–29

6. Test Problems

• The test suite evaluates neural operators as mappings for parametric PDEs and evolution equations, including fixed-time and time-dependent solution paths.Inputs and outputs are modeled in Banach spaces of functions on bounded domains, with Monte Carlo relative errors defined using a probability measure.Source: §6 · p. 30

6.1 Poisson Equation

• The Poisson experiment tests analytical-solution approximation and whether Green’s-function inductive bias enables generalization from continuous training inputs to discontinuous inputs outside the support of µ.Training data use 1,000 examples generated on a uniform grid with 85 collocation points.Source: §6.1 · p. 31

6.2 Darcy Flow

• The Darcy Flow problem models a nonlinear solution operator for steady two-dimensional elliptic flow with random piecewise-constant coefficients representing subsurface permeability or material microstructure.The dataset contains 1,000 examples computed on a 421×421 grid and downsampled for other resolutions.Source: §6.2 · pp. 31–32

6.3 Burgers’ Equation

• The Burgers’ Equation problem learns a fixed-time solution operator for one-dimensional viscous flow with periodic boundaries and viscosity ν = 10−1.Solutions are generated with a pseudo-spectral split-step method on 8192 collocation points, using 1,000 training examples.Source: §6.3 · p. 32

6.4 Navier-Stokes Equation

• The Navier-Stokes experiments use both velocity and vorticity-streamfunction formulations, learning fixed-time maps and mappings from an initial trajectory segment to a later trajectory.Data are generated pseudo-spectrally on a 256×256 grid while varying viscosity, final time, and training-set size.Source: §6.4 · pp. 33–34

6.4.1 BAYESIAN INVERSE PROBLEM

• The Bayesian inverse problem recovers initial Navier-Stokes vorticity from noisy observations on a 7×7 interior grid by comparing pCN posterior means using the numerical solver or a neural operator.Observational noise has covariance Γ = (1/γ2)I with γ = 0.1; each posterior uses 5,000 burn-in and 25,000 generated samples.Source: §6.4.1 · p. 34

6.4.2 SPECTRA

• Navier-Stokes solution spectra exhibit a −5/3 decay across a viscosity-dependent wavenumber range, with that range expanding as viscosity decreases, consistent with turbulent-regime behavior.The statistically stationary spectral property is sustained for all positive times.Source: §6.4.2 · p. 34, Figure 6 · p. 35

6.5 Choice of Loss Criteria

• Training with relative L2 loss produces around half the testing error rate of training with MSE, while reducing overfitting through normalization and regularization.The study uses relative L2 error to evaluate performance across all problems.Source: §6.5 · p. 35

7.1 Poisson Equation

• The neural operator attains 10^-7 relative test error on Poisson using 1000 training examples, and its learned kernel closely approximates the analytic Green’s function.This extends approximation from the training norm toward uniform convergence over bounded sets.Source: §7.1 · pp. 36–37, Figure 7 · p. 37

7.2.1 DARCY FLOW / 7.2.2 BURGERS’ EQUATION

• FNO achieves the lowest Darcy error and nearly an order-of-magnitude lower Burgers error than benchmarks, with error invariant to resolution.On Burgers, replacing ReLU with GeLU reduces test error from 0.0018 to 0.0007.Source: §7.2.1 · p. 38, §7.2.2 · p. 39, Table 2 · p. 39

7.2.3 ZERO-SHOT SUPER-RESOLUTION. / 7.3.1 ZERO-SHOT SUPER-RESOLUTION.

• Neural operators enable zero-shot super-resolution: GNO transfers Darcy predictions from 16×16 to 256×256, while FNO-3D transfers Navier–Stokes predictions from 64×64×20 to 256×256×80.FNO is the only tested Navier–Stokes benchmark supporting both spatial and temporal super-resolution.Source: §7.2.3 · p. 39, Figure 9 · p. 40, §7.3.1 · p. 42

7.3 Navier-Stokes Equation

• FNO-3D performs best with sufficient Navier–Stokes data, while FNO-2D is best among the tested methods when data are insufficient; all methods exceed 15% error there.At higher spatial resolution, benchmark performance degrades while FNO maintains the same errors.Source: §7.3 · p. 40, Table 4 · p. 41

7.3.2 SPECTRAL ANALYSIS

• FNO recovers high-frequency Navier–Stokes components despite using kmax,j = 12 modes, achieving ≤1% error where truncation at 20 modes yields about 2%.Intermediate activations and the decoder recover frequencies omitted by individual Fourier layers.Source: §7.3.2 · pp. 42–43, Figure 12 · p. 43, Figure 13 · p. 43

7.3.3 NON-PERIODIC BOUNDARY CONDITION.

• FNO handles non-periodic boundaries because its linear transform preserves boundary information, achieving excellent accuracy on Darcy flow and time-dependent Navier–Stokes.Traditional Fourier methods do not generally have this capability.Source: §7.3.3 · p. 43

7.3.4 BAYESIAN INVERSE PROBLEM

• In Bayesian inversion, FNO matches the traditional solver’s posterior mean while reducing one evaluation from 2.2s to 0.005s and MCMC runtime from over 18 hours to 2.5 minutes.Including offline data generation and training, the FNO workflow remains faster at 12 hours.Source: §7.3.4 · p. 44, Figure 14 · p. 44

7.4.2 EXPRESSIVENESS / 7.4.3 COMPLEXITY / 7.4.4 REFINABILITY

• Among the four methods, FNO has the best overall expressiveness and speed, with complexity O(J log J), while LNO has O(J) complexity and graph methods depend on sampled nodes.GNO, LNO, and MGNO work with few parameters, whereas FNO requires more parameters for acceptable error.Source: §7.4.2 · p. 45, §7.4.3 · p. 45, §7.4.1 · p. 45

7.4.5 ROBUSTNESS

• The robustness study also evaluates noisy training and testing and an additional one-dimensional advection test with random square-wave inputs.Noise is independently sampled at every grid point, following the setting in Lu et al. (2021b).Source: §7.4.5 · p. 46
• About 10% error occurs on advection with 10% noise, while Darcy and Navier–Stokes remain below 10% error across scenarios; FNO is robust to test noise overall.These results are shown in the top half of Table 9 and Figure 15.Source: §7.4.5 · p. 46, Table 9 · p. 47
• FNO is less robust on advection because it is nonsmoothing, and on Burgers because steep fronts form despite smoothing.Source: §7.4.5 · p. 47
• Training with noise eliminates the clean–noisy performance gap, but may degrade clean-data performance; noise augmentation, early stopping, weight decay, and higher resolution improve robustness.Noise training can also reduce autoregressive error accumulation and support longer time horizons.Source: §7.4.5 · p. 47
• CNN or UNet branches can improve FNO performance on discontinuities, but they sacrifice discretization invariance, leaving discretization-invariant image-processing tools as future work.The composite model outperforms basic FNO on multiphase flow with high contrast and sharp shocks.Source: §7.4.5 · p. 47

8. Approximation Theory

• The proofs reduce neural-operator approximation to linear approximation, so they do not establish improved rates from nonlinear approximation despite the architecture’s nonlinear capacity.Exploiting nonlinear approximation for potentially better rates remains future work.Source: §8 · p. 49

8.1 Neural Operators

• The analyzed neural-operator class simplifies the practical architecture by removing lifting, projection, local linear, and final activation operators, although those learned components can improve practice.Extending the theorems to explain these empirical benefits is identified as a non-trivial development.Source: §8.1 · p. 50

8.2 Discretization Invariance

• Neural operators are discretization-invariant when Banach function spaces continuously embed into continuous functions on their domains.The proof approximates integral operators with Riemann sums and establishes uniform convergence of the discretization error over compact input sets.Source: §8.2 · p. 51

8.3 Approximation Theorems

• The approximation constructs finite-dimensional input features, approximates their nonlinear transformation with a standard neural network, and expands the output through representers using integral-operator layers.Linear functionals are approximated by integration against smooth functions, while the output representation is realized through a single integral-operator layer.Source: §8 · p. 48, §8.3 · p. 52
• Under Assumptions 9 and 10, neural operators universally approximate continuous operators on compact subsets of broad Banach function spaces, including Lebesgue, Sobolev, and continuous-function classes.The results extend beyond prior Hilbert-space or continuous-function analyses and apply to more general Banach spaces on Lipschitz domains.Source: §8 · pp. 47–48, §8.3 · p. 52
• The theory also establishes density in Bochner L2 spaces for Hilbert-valued outputs under measurable, square-integrable target operators, while leaving analogous Banach-space results open.The Hilbert structure enables uniform norm control, and the argument uses compact-set restriction together with decay of the input measure outside that set.Source: §8.3 · p. 53

Finite-dimensional operators

• Finite-dimensional CNNs are mesh-dependent and resolution- or geometry-limited, whereas neural operators support grid-invariant error, mesh transfer, and zero-shot super-resolution without new data.Continuous convolutions and U-Net super-resolution reduce some limitations but remain constrained by grids or require fine-resolution training data.Source: §9 · pp. 53–54

DeepONet

• DeepONet uses branch and trunk networks for a standard linear operator approximation, whereas neural operators provide a nonlinear and constructively more expressive alternative.DeepONet builds on universal approximation results and has separate error estimates; the paper compares it numerically in Section 7.2.Source: §9 · p. 54

PINNs, DRM, DGM, and hybrid solvers

• PINNs, DRM, DGM, and hybrid solvers address individual PDE instances by retraining an optimization-based network for each parameter, while hybrid methods also require a known PDE.These methods are mesh-independent in relevant cases but do not learn a reusable solution operator; purely data-driven function-space maps are unavailable in the hybrid setting.Source: §9 · p. 54

Neural operators versus prior approaches

• Neural operators learn maps between infinite-dimensional spaces with one parameter set across discretizations, transfer solutions between meshes, require one training pass, and operate non-intrusively without PDE knowledge.After training on equation pairs, a new parameter requires only a forward pass; the target map may instead be learned from experimental or simulation data.Source: §9 · p. 55

Efficient neural operator parameterizations

• The graph, low-rank, multipole, and Fourier variants connect neural operators to Nyström, kernel, fast multipole, multiresolution, and spectral methods while targeting efficient discretization-invariant computation.GNOs add query locations as graph nodes; LNOs use factor networks and integration; MGNOs generalize multipole ideas to arbitrary graphs; FNOs operate in Fourier space with quasi-linear complexity.Source: §9 · pp. 55–57

Sources of error

• The review identifies operator-approximation, data-discretization, finite-sample, and optimization errors, but analyzes only the first and leaves parameter-growth and optimization-error guarantees open.Operator approximation error can decrease with more parameters and refined architectural approximations, while parameter growth may be superexponential and optimization may miss the empirical-risk minimum.Source: §9 · p. 57

10. Conclusions

• Neural operators learn function-to-function mappings through feature extraction, recurrent nonlinear nonlocal updates, and a final mapping into the output function.The architecture is designed for mappings between infinite-dimensional function spaces.Source: §10 · p. 57
• Four parameterizations produce mesh-free models that train and generalize across discretizations; Fourier neural operators achieve the best numerical performance among the experiments.Graph-kernel, low-rank, multilevel-graph, and Fourier approaches support different computational structures; Fourier methods are linear-log linear on structured grids, while low-rank and multiscale variants reduce quadratic integral-operator cost.Source: §10 · p. 57

10.1.1 NEW APPLICATIONS

• Future applications include accelerating repeated scientific-computing mappings, unknown-model tasks, turbulent flows, geological media, and physics simulation, with evaluation at arbitrarily large resolution.The operator setting also supports smaller-resolution training datasets and blackbox PDE surrogates for Darcy Flow, Burgers’ equation, and Navier-Stokes equation.Source: §10.1 · p. 58, §10.1.1 · p. 58

10.1.2 NEW METHODOLOGIES

• The full O(J2) integration method still outperforms the FNO by about 40%, motivating adaptive graph or probability approximations and alternative bases such as PCA or Chebyshev.Other proposed directions combine neural operators with numerical solvers or physics constraints for faster and more accurate approximation.Source: §10.1.2 · p. 58

10.1.3 THEORY

• The paper develops universal approximation theory for neural operators, but approximation architecture, parameter scaling, efficient PDE classes, and nonlinear approximation remain open theoretical problems.Operator approximation between Banach spaces is substantially more complex than approximation of functions between Euclidean spaces.Source: §10.1.3 · pp. 58–59

Notation

• The paper uses lowercase letters for vectors and functions, uppercase letters for matrices or discretized transformations, and calligraphic letters for operators.These conventions are summarized in the table of notations for operator learning and neural operators.Source: Table 10 · p. 70

Functional analysis

• For Banach spaces, the paper defines norms, continuous dual spaces X*, and continuous linear-map spaces L(X; Y), using ∥·∥ for an unambiguous operator norm.The continuous dual X* consists of continuous linear functionals equipped with the operator norm.Source: Appendix A · p. 70

Domains and derivatives

• Domains are bounded, connected, topologically regular open subsets of R^d, with Lipschitz boundary regularity specified for d ≥ 2; weak derivatives are denoted by ∂^αf.For d = 1, any bounded open interval is a domain and is called a Lipschitz domain.Source: Appendix A · p. 71

Function spaces

• For a domain D, the appendix defines C^m(¯D) via restrictions of functions in C^m(Rd), notes unique bounded continuous extensions to ¯D, and treats C^m_b(D) and C^m(¯D) as Banach spaces.It also defines vector-valued spaces as n-fold Cartesian products, with analogous conventions for other spaces.Source: Appendix A · p. 71

Definitions

• The approximation property requires finite-rank operators to approximate the identity uniformly on every compact subset of a Banach space.Finite-rank operators are defined by having finite-dimensional range.Source: Appendix B · p. 72

Basis and approximation property

• Every Banach space with a Schauder basis has the approximation property, proved by finite-dimensional truncation on compact sets.The proof covers a compact set using finitely many points and selects a sufficiently large truncation index.Source: Appendix B · pp. 72–73

Basis invariance

• A continuous linear bijection transfers a Schauder basis from one Banach space to another.The transported basis follows from linearity and continuity of the bijection and its inverse.Source: Appendix B · p. 73

Approximation-property invariance

• A continuous linear bijection also transfers the approximation property by conjugating finite-rank approximants as W = TUT^-1.Compact sets are pulled back through the inverse map, approximated in the original space, and pushed forward to the target space.Source: Appendix B · p. 74

Finite-dimensional approximation

• Any continuous map G:X→Y between Banach spaces with the AP can be approximated on compact subsets by finite-dimensional encoding, continuous nonlinear processing, and finite-rank output reconstruction.The input encoding uses finitely many functionals on X, while the output reconstruction uses finitely many representers in Y; the approximation extends finite-dimensionalization previously restricted to separable Hilbert spaces.Source: Then the set · p. 75

Function-space isomorphisms

• A C^m-diffeomorphism between domain closures induces a continuous linear bijection between the corresponding C^m function spaces.In particular, C^m([0,1]^d) and C^m([-M,M]^d) are continuously linearly bijective for every M>0.Source: Then the set · p. 77
• An analogous continuous linear bijection exists between W^{m,1}((0,1)^d) and W^{m,1}((-M,M)^d), established using composition with an affine domain transformation.The proof uses derivative estimates and the change-of-variables formula, then establishes injectivity and surjectivity.Source: Then the set · pp. 77–78

Approximation property

• For Lipschitz domains, the spaces W^{m,p}(D) and C^m(¯D) used in the assumptions have the AP; W^{m,p}(D) moreover has a basis, while C^m(¯D) is shown to have the AP without necessarily establishing a basis.The proof uses isomorphisms, extension operators, known bases, and finite-rank approximation constructions.Source: Then the set · pp. 78–80

Approximation results

• Lemmas 27–31 establish that linear functionals on W m,p, C, and Cm can be approximated uniformly on compact sets by smooth kernel integral operators.The appendix states that these results are crucial because they approximate the functionals used in the neural-operator universality proof by elements of IO.Source: Appendix C · pp. 80, 81, 85, 86, 87

Lemma 27

• Lemma 27 uses a Riesz-representation argument, an isometric embedding of Cm into a Banach space, and Hahn–Banach extension to represent its functionals by finite signed Radon measures.The proof constructs the embedding through derivatives indexed by multi-indices and obtains the representation after extending the functional.Source: Appendix C · pp. 80–81

Lemma 28

• Lemma 28 approximates functionals on bounded subsets of W m,p(D) by C∞ kernels, covering both p > 1 through density and p = 1 through finite coverings and mollification.The argument controls approximation uniformly over the specified closed and bounded set, with compactness required when p = 1.Source: Appendix C · pp. 81–83

Lemma 29

• Lemma 29 replaces measure-based functionals on compact subsets of Cm by finite sums of derivative evaluations at distinct interior points.Boundary and coincident evaluation points are perturbed into distinct points in D while preserving the uniform approximation on the compact set.Source: Appendix C · pp. 85–86

Lemmas 30–31

• Lemmas 30 and 31 convert finite point-evaluation approximations on C and Cm into smooth kernel approximations using mollifiers, including separate kernels for derivative evaluations.For Cm, the construction applies the C-space argument independently to each derivative multi-index.Source: Appendix C · pp. 86–87

Appendix D

• Lemmas 32, 34, and 35 establish approximation of the construction’s input map, output map, and intermediate neural network by neural operators.The input map is represented through vector-valued constant functions, the output map maps constant functions to basis-expansion coefficients, and standard neural networks are represented exactly.Source: Appendix D · p. 88
• Lemma 32 approximates coefficient maps FJ : A → RJ on compact subsets using a neural network kernel under Assumption 9.The construction uses smooth-function approximation and finite-dimensional norm equivalence to obtain the required neural network approximation.Source: Appendix D · pp. 88–89
• Lemma 34 similarly approximates GJ′ : RJ′ → U on compact subsets by mapping neural-network outputs to basis-expansion coefficients.The proof handles function-space targets through smooth-function density and separate constructions for C^m2 and W^m2,p2 spaces.Source: Appendix D · pp. 89–90
• Lemma 35 exactly embeds any standard finite-dimensional neural network into the neural-operator class by using constant kernels and layerwise composition.The kernels implement affine transformations through integral operators, with analogous constructions continued across all network layers.Source: Appendix D · p. 91

Proof setup

• The proof reduces to D = D′ and A = U = C( ¯D), and uses continuity to consider only a single layer.This reduction is made without loss of generality through continuous embedding.Source: Appendix E · p. 92

Discretization construction

• Each discretization is associated with pairwise disjoint positive-measure partitions containing its distinct discretization points, inducing finite-dimensional maps ˆG_j.The maps are defined for parameters κ, b and inputs in R^{jd} × R^j.Source: Appendix E · p. 92

Approximation argument

• Compactness yields a finite ϵ-net, while uniform continuity and Riemann-sum convergence provide uniform control for sufficiently refined discretizations.The proof chooses approximation indices q and p_1, …, p_N and takes m ≥ max{q, p_1, …, p_N}.Source: Appendix E · pp. 92–93

Conclusion

• Applying the triangle inequality to these bounds establishes the desired result for every a ∈ K and completes the proof.The final estimate follows after combining the finite-cover and discretization approximations.Source: Appendix E · p. 93

• The target map is factored as G∘ψ∘F, with continuous linear maps F and G and a finite-dimensional continuous map ψ.This reduction follows from Lemma 22 after applying Lemma 26.Source: Appendix F · p. 94
• A sequence of neural operators approximates F while representing its outputs as spatially constant functions, reducing the intermediate range to R^J.Uniform continuity of ψ on a compact set controls the error introduced by replacing F with a sufficiently accurate approximation F_T.Source: Appendix F · p. 94
• Classical neural-network approximation of ψ on the compact image of F_T, together with exact neural-operator simulation of standard networks, yields the middle approximation.The constructed map preserves constant functions, enabling a final finite-dimensional neural network approximation of the output coordinates.Source: Appendix F · p. 94
• The proof combines the approximation errors through G and constructs an operator that remains uniformly bounded by 4M while staying ϵ-close on K.The bounded construction uses orthonormal output components and an additional neural network correction, then invokes Lemmas 34 and 35.Source: Appendix F · pp. 95–96

• The proof sets U = Hm2(D) and defines a truncation parameter R for the approximation argument.Source: Appendix G · p. 96
• Dominated convergence provides a sufficiently large R using the bound R(a)∥U ≤ ∥G†(a)∥U for every a ∈ A.Source: Appendix G · p. 96
• Lusin’s theorem yields a compact K ⊂ A on which the restricted mapping R|K is continuous.Source: Appendix G · p. 96
• Because K is closed, a generalized Tietze extension theorem produces a continuous mapping ˜G†, to which Theorem 11 is applied to obtain a neural operator G.Source: Appendix G · p. 96