PaperlayerView exact revision on arXiv

Source-linked AI summary

Neural Operator: Learning Maps Between Function Spaces

Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, Anima Anandkumar

arXiv:2108.08481v6cs.LGmath.NA

TL;DR

The paper addresses the difficulty of learning mappings between function spaces when standard neural networks depend on fixed discretizations. It introduces neural operators built from linear operator layers and nonlinear activations, with theoretical discretization-invariance and universal-approximation guarantees. Across PDE surrogate tasks, the proposed architectures perform strongly while enabling substantially faster inference than conventional solvers.

  • Problem

    Standard neural networks trained on discretized functions may not generalize well to different discretizations, limiting operator learning across grids and resolutions.

  • Method

    Neural operators compose linear integral or spectral operators with nonlinear activations and provide graph, low-rank, multipole graph, and Fourier parameterizations.

  • Results

    The architectures learn and generalize across different meshes, while Fourier neural operator inference is three orders of magnitude faster than the pseudo-spectral reference method.

  • Takeaways & Limitations

    Neural operators provide mesh-free surrogate models whose learned mappings can be shared across approximations at different discretization levels.

  • Takeaways & Limitations

    The universal-approximation theory analyzes a simplified architecture, while extending it to learned lifting, projection, and local operators remains an important non-trivial development.

Abstract

from arXiv · show

The classical development of neural networks has primarily focused on learning mappings between finite dimensional Euclidean spaces or finite sets. We propose a generalization of neural networks to learn operators, termed neural operators, that map between infinite dimensional function spaces. We formulate the neural operator as a composition of linear integral operators and nonlinear activation functions. We prove a universal approximation theorem for our proposed neural operator, showing that it can approximate any given nonlinear continuous operator. The proposed neural operators are also discretization-invariant, i.e., they share the same model parameters among different discretization of the underlying function spaces. Furthermore, we introduce four classes of efficient parameterization, viz., graph neural operators, multi-pole graph neural operators, low-rank neural operators, and Fourier neural operators. An important application for neural operators is learning surrogate maps for the solution operators of partial differential equations (PDEs). We consider standard PDEs such as the Burgers, Darcy subsurface flow, and the Navier-Stokes equations, and show that the proposed neural operators have superior performance compared to existing machine learning based methodologies, while being several orders of magnitude faster than conventional PDE solvers.

1. Introduction

Neural operators learn mappings between infinite-dimensional function spaces while remaining invariant across discretizations. The paper develops expressive and efficient operator architectures, proves theoretical guarantees, and demonstrates strong PDE surrogate performance and speed.

  • Motivation: Standard neural networks trained on discretized functions may fail to generalize beyond the training grid.This limitation arises because their finite-dimensional inputs and outputs depend on fixed discretizations.
  • Neural Operators: Neural operators directly map between function spaces and can use shared parameters across discretizations and resolutions without retraining.The framework is designed to accept varied discretizations while preserving the same operator model.
  • Discretization Invariance: Discretization invariance requires accepting arbitrary input points, evaluating outputs at arbitrary points, and converging to a continuum operator under refinement.The definition also fixes the parameter count so it does not grow unboundedly as resolution increases.
  • Neural Operators: Neural operators compose linear operator layers with nonlinear activations, enabling universal approximation of continuous operators.The linear layers can be parameterized through integral operators or spectral-domain multiplication.
  • Parameterizations: Four practical parameterizations are introduced: graph, low-rank, multipole graph, and Fourier neural operators.These designs target efficient approximation of infinite-dimensional operator layers across discretizations.
  • PDE Applications: The numerical study reports consistent gains over existing deep-learning methods and fast PDE surrogate inference.For two-dimensional Navier–Stokes, errors are < 1% at Reynolds number 20 and 8% at Reynolds number 200; FNO inference is three orders faster than the reference solver.

2. Learning Operators

The paper frames operator learning as approximating nonlinear maps between infinite-dimensional function spaces from finite input-output observations. It defines discretization invariance so models can accept arbitrary discretizations, produce outputs at arbitrary points, and converge under refinement.

  • PDE Setting: The PDE setting maps coefficient functions to solution functions through the solution operator G†.The paper describes this as the inverse of a parameterized linear PDE operator when defined.
  • Problem Setting: Operator learning approximates a typically nonlinear map G† between Banach spaces of functions using finite observations of input-output pairs.Inputs are sampled from a probability measure, and outputs may be corrupted with noise.
  • Approximation and Limitations: Universal approximation results make the approximation error arbitrarily small, while discretization invariance makes discretization error arbitrarily small with sufficiently fine resolution.The paper does not characterize error for fixed parameter dimension or fixed training-sample count.
  • Discretization Invariance: The proposed architectures map discretized inputs into the function space rather than a fixed output grid and remain consistent as input discretization is refined.They are also called function-space, mesh-invariant, or discretization-invariant architectures.
  • Results: The paper proves discretization invariance and numerically finds approximately constant approximation error as discretization is refined.This enables transferring solutions across grid geometries and discretization sizes with one fixed-parameter architecture.

3. Neural Operators

Neural operators replace finite-dimensional neural-network layers with function-space operations: local lifting and projection surround iterative layers combining linear maps, integral kernels, biases, and pointwise nonlinearities. This construction is independent of data discretization and supports several kernel choices, including parameter-dependent and nonlinear kernels.

  • Architecture: The architecture lifts the input function pointwise, iteratively updates hidden functions, and projects the final representation pointwise to the output function.The hidden update combines a local linear operator, a non-local integral kernel, and a bias before a pointwise nonlinearity.
  • Architecture: The neural operator is a composition of lifting, activation, local linear, integral-kernel, bias, and projection operators defined directly in function space.Its compact form is Gθ := Q ◦σT (WT−1 + KT−1 + bT−1) ◦· · · ◦σ1(W0 + K0 + b0) ◦P.
  • Integral Kernel Operators: The basic integral kernel operator uses a kernel function integrated against a measure, with alternative measures available to accelerate computation or encode prior information.The paper uses the Lebesgue measure in many cases and analyzes this kernel form theoretically and experimentally.
  • Integral Kernel Operators: A parameter-dependent kernel can outperform the basic kernel on some PDEs, including Darcy flow, when the solution strongly depends on the input parameters.Directly incorporating parameter dependence prevents the influence of the input from diminishing through successive layers.
  • Integral Kernel Operators: A kernel depending on the hidden function makes the integral operator nonlinear and can yield a continuous input/output-space generalization of transformers.This is obtained for a particular kernel and measure choice.
  • Approximation: The total error decomposes into approximation and discretization errors, both of which can be made arbitrarily small with suitable parameters and sufficiently fine discretization.The parameters remain fixed independently of the input discretization.

4. Parameterization and Computation

The paper develops efficient parameterizations of neural-operator layers to reduce the cost of discretized kernel integral operators. These include graph, low-rank, multipole, and Fourier constructions with differing computational strategies and complexity guarantees.

  • Motivation: Kernel integral operators require O(J^2) computation on J discretization points, motivating parameterizations that reduce dependence on discretization size.The quadratic cost arises from using all data points in kernel matrix-vector multiplications.
  • Graph Neural Operator: Graph Neural Operators combine a Nyström approximation, domain truncation, and graph neural networks to approximate the integral operator.The construction connects the integral-operator formulation to graph neural-network implementations.
  • Graph Neural Operator: Local truncation reduces the integral-operator cost to O(csJ^2), with cs < 1 determined by the interaction neighborhood.The method uses s(x) = B(x, r) ∩ D, restricting interactions to a fixed-radius neighborhood.
  • Graph Neural Operator: Standard CNN parameterizations are not consistent in function space because refining the discretization changes kernel support and can require infinitely many parameters.Doubling discretization points while halving spacing makes the effective support shrink; increasing filter size to compensate causes parameter growth.
  • Low-rank Neural Operator: Low-rank Neural Operators impose a tensor-product kernel structure, reducing computational complexity to O(J).The inner products can be evaluated independently of the output point, yielding linear complexity in the discretization size.
  • Multipole and Fourier Neural Operators: Multipole Graph Neural Operators use multi-scale discretizations and hierarchical low-rank structures for long-range interactions, while Fourier Neural Operators parameterize kernels in Fourier space.Fourier methods use FFTs and truncated modes, with complexity O(J log J) for the transform and inverse transform under the stated setting.

5. Neural Operators and Other Deep Learning Models

This section relates neural operators to DeepONets and transformers while emphasizing discretization invariance and representation choices. It also identifies computational and approximation trade-offs across these models.

  • DeepONets: A neural operator with a point-wise parameterized first kernel and discretized integral operators yields a DeepONet.
  • DeepONets: DeepONet-Operator restores discretization invariance by replacing finite-dimensional input inner products with function-space inner products.Parametrizing the resulting functions w_j by neural networks makes the model discretization invariant.
  • Approximation: Nonlinear parameterizations generally outperform linear ones numerically, although theoretical understanding of operator nonlinear approximation remains limited.The paper notes that depth benefits are not captured by its current approximation analysis and leaves further theory for future work.
  • Function Representation: Neural-operator and DeepONet parameterizations differ in how they finite-dimensionalize input and output function spaces, inducing different biases and representation limits.The paper describes neural operators, PCA-based methods, DeepONets, graph operators, Fourier operators, and low-rank operators as using distinct representations.
  • Transformers: Transformer attention is a special case of a neural-operator layer obtained through a particular nonlinear integral kernel and Monte Carlo discretization.The correspondence is established for a pre-normalized, single-headed attention transformer block.
  • Transformers: Standard attention remains memory- and computation-intensive compared with the neural-operator architectures developed here, while some CNN-token vision transformers are not discretization invariant.The paper attributes the attention cost to evaluating a nested integral for each spatial point.

6. Test Problems

The paper studies neural operators as surrogate solution maps for elliptic and evolution PDEs, using Poisson, Darcy flow, Burgers, and Navier–Stokes test problems. The experiments use function-space settings with discretized domains and relative L2 error evaluation.

  • Problem Setup: The test problems evaluate approximation of solution operators for parametric PDEs and evolution equations over Banach spaces of functions on bounded domains.For evolution equations, the learned map may send an initial condition to a solution at a fixed time or to a time-dependent path.
  • Poisson Equation: The Poisson experiment tests exact-solution approximation and whether Green’s-function inductive bias enables generalization beyond the input distribution.The paper specifically considers discontinuous inputs outside the support of the training measure.
  • Darcy Flow: The Darcy experiment models steady two-dimensional subsurface flow with positive coefficient functions and fixed forcing.The coefficient distribution produces piecewise-constant functions with random interfaces, representing prototypical permeability fields.
  • Burgers Equation: The Burgers experiment learns a fixed-time solution operator for the one-dimensional viscous equation with periodic boundary conditions and viscosity ν = 10^-1.
  • Navier–Stokes Equation: The Navier–Stokes experiment studies a two-dimensional viscous incompressible flow using its vorticity–streamfunction formulation on the periodic unit torus.The forcing is fixed, and viscosity ν is varied across the spectral-decay analysis.
  • Navier–Stokes Equation: The Navier–Stokes data exhibit a -5/3 spectral-decay rate across a viscosity-dependent range of wavenumbers, consistent with the turbulent regime.The spectrum is examined at time t = 50, and the property is described as statistically stationary for all positive times.
  • Evaluation: All reported performance errors use relative L2 error, although mismatching training and testing norms can lead to overfitting in the training norm.

7. Numerical Results

The numerical studies show that neural operators maintain accuracy across resolutions, support zero-shot super-resolution, and often outperform benchmark methods. Fourier neural operators achieve strong accuracy, efficiency, and robustness, while limitations appear for discontinuous or sharply fronted solutions.

  • Neural operators exhibit resolution-independent error once the discretization is sufficiently fine, reflecting learned operator approximations rather than grid-specific fits.This residual error remains nonzero because the operators have finite parameters and are trained on finite data.
  • Darcy Flow and Burgers’ Equation: Fourier neural operators generally achieve the lowest errors on Darcy flow and nearly an order of magnitude lower relative error than benchmarks on Burgers’ equation.On Burgers’ equation, replacing ReLU with GeLU reduces test error from 0.0018 to 0.0007.
  • Zero-Shot Super-Resolution: Zero-shot super-resolution allows neural operators to train at lower resolution and evaluate at higher resolution without higher-resolution training data.Examples transfer GNO from 16×16 to 256×256 Darcy data and FNO-3D from 64×64×20 to 256×256×80 Navier–Stokes data.
  • Navier–Stokes Equation: FNO recovers Navier–Stokes high-frequency components despite using only kmax,j = 12 parameterized modes, achieving error ≤1% where 20-mode truncation gives around 2% error.The decoder and final network recover frequencies omitted during the Fourier convolution.
  • Efficiency: FNO inference takes 0.005s versus 2.2s for the conventional Navier–Stokes solver, while recovering nearly the same posterior mean in Bayesian inversion.The resulting MCMC takes 2.5 minutes with FNO versus over 18 hours with the traditional solver.
  • Complexity and Expressiveness: FNO has complexity O(J log J) with fast Fourier transforms, whereas LNO has O(J) complexity and full-node GNO and MGNO remain quadratic.GNO and MGNO complexity depends on the number of sampled nodes when sampling is used.

8. Approximation Theory

The paper develops approximation theory showing that neural operators are discretization-invariant and can universally approximate continuous operators under several function-space settings.

  • Approximation Theory: The analysis extends operator universal approximation results to more general Banach spaces beyond settings restricted to Hilbert or continuous-function spaces.The results address Lebesgue, Sobolev, continuous, and continuously differentiable function classes.
  • Approximation Theory: Neural operators approximate continuous operators through finite-dimensional input functionals, nonlinear finite-dimensional maps, and basis expansions of output functions.This construction uses linear maps into finite vectors, a nonlinear neural network, and output coefficients.
  • Approximation Theory: The proof theory relies on linear approximation and does not capture potential benefits of nonlinear approximation used by trained networks.Improved approximation rates from neural operators’ nonlinear nature remain an open direction.
  • Approximation Theory: The theoretical construction simplifies the practical architecture by removing lifting, projection, local linear operators, and the final activation, although these components help empirically.Extending the theorems to explain those practical benefits is identified as non-trivial.
  • Approximation Theory: Neural operators with fixed parameter counts satisfy discretization invariance when viewed as maps between suitable Banach function spaces.The theorem assumes continuous embeddings of the input and output spaces into continuous-function spaces.
  • Approximation Theory: The theory proves uniform-on-compacta approximation for continuous operators and density results under probability measures in separable Hilbert output spaces.The approximation theorems cover compact-set uniform convergence and L2-type settings under stated assumptions.

9. Literature Review

The literature review contrasts neural operators with mesh-dependent neural networks, PDE-specific methods, hybrid solvers, and related operator-learning approaches.

  • Literature Review: Finite-dimensional CNN approaches depend on training discretization, require architectural modifications across resolutions, and cannot query solutions at new domain points.Their error consistency across discretizations is not guaranteed.
  • Literature Review: PINNs, Deep Ritz, and Deep Galerkin methods parameterize individual PDE solutions rather than learning a reusable solution operator.They are mesh-independent but require a new optimization process for each new parameter.
  • Literature Review: Neural operators use one parameter set across discretizations, transfer solutions between meshes and discretization methods, and require only a forward pass for new inputs.They are purely data-driven and do not require knowledge of the underlying PDE.
  • Literature Review: The four parameterization families are graph, low-rank, multipole graph, and Fourier neural operators, offering different efficiency mechanisms.Graph methods connect continuum operators to GNNs, low-rank methods factorize kernels, multipole methods capture long-range interactions, and Fourier methods operate spectrally.
  • Literature Review: The paper studies operator-approximation error from finite parametrization and numerical approximation, which can decrease as parameters and approximation resolution increase.This differs from a claim that all practical error sources vanish.

10. Conclusions

The conclusions present neural operators as mesh-free function-to-function models with four nonlocal parameterizations and applications to PDE surrogate learning.

  • 10. Conclusions: The architecture extracts input features, iterates nonlinear nonlocal transformations on feature space, and maps the resulting features to an output function.The described network contains four steps, with the cited passage summarizing the principal stages.
  • 10. Conclusions: The four nonlocal operators use graph kernels, low-rank decomposition, multilevel graph structure, and Fourier-space convolution.Integral-operator variants support unstructured grids, while Fourier methods are efficient on structured grids.
  • 10. Conclusions: Neural operators provide black-box surrogate models for PDE solution maps, demonstrated on Darcy Flow, Burgers’ equation, and Navier-Stokes equation.Their black-box structure allows application to other problems without requiring the governing PDE.
  • 10. Conclusions: The full O(J^2) integration method outperforms FNO by about 40%, but at greater computational cost.The conclusion identifies more advanced integration and approximation schemes as potential improvements.
  • 10. Conclusions: The theory uses linear approximation techniques, while the role and scaling of nonlinear approximation in neural operators remain incompletely understood.The paper leaves architecture-dependent approximation efficiency and applicable PDE classes as open research questions.

Appendix A.

The appendix defines notation for functions, vectors, operators, domains, derivatives, and standard Banach and Sobolev function spaces used throughout the paper.

  • Appendix A.: Lowercase symbols such as v and u denote vectors and functions, uppercase symbols such as W and K denote matrices or discretized transformations, and calligraphic symbols denote operators.Examples include G and F for operators.
  • Appendix A.: A Banach space is equipped with a norm and a continuous dual space consisting of continuous linear functionals.The appendix denotes the norm by ∥·∥X and the continuous dual by X∗.
  • Appendix A.: The paper uses L(X; Y) for continuous linear maps from X to Y and ∥·∥ for an operator norm when the spaces are unambiguous.This notation applies throughout the operator-learning formulation.
  • Appendix A.: Domains are bounded, connected, topologically regular open sets; Lipschitz domains additionally have locally Lipschitz-representable boundaries.In one dimension, any bounded open interval is treated as a Lipschitz domain.
  • Appendix A.: For vector-valued continuous functions, the C^m norm is the maximum of the componentwise C^m norms.The appendix defines C(D; R^n) as an n-fold Cartesian product.
  • Appendix A.: The notation W^m,p(D) denotes standard Sobolev spaces, with W^0,p(D)=L^p(D) and W^m,2(D)=H^m(D).The appendix refers to standard definitions for these spaces.

Appendix B.

This appendix defines the approximation property and develops foundational results connecting it to bases, finite-rank operators, and continuous linear bijections. It establishes that the AP is preserved under suitable isomorphisms and supports finite-dimensional approximation of continuous maps.

  • Finite-dimensional approximation: For spaces with the AP, every continuous map between them admits a finite-dimensional factorization over compact input sets.The construction maps inputs into finite-dimensional coordinates, applies a continuous finite-dimensional map, and reconstructs outputs.
  • Definitions: The approximation property requires finite-rank operators to approximate the identity uniformly on every compact set.A finite-rank operator has finite-dimensional image.
  • Basis and AP: A Banach space with a basis has the approximation property.The proof uses finite-dimensional truncations of the basis expansion on compact sets.
  • Basis and AP: A continuous linear bijection transfers the existence of a basis from one Banach space to another.The transported basis follows from linearity and continuity of the bijection and its inverse.
  • Basis and AP: A continuous linear bijection also transfers the approximation property between Banach spaces.The proof conjugates a finite-rank approximation U by defining W = TUT^-1.

Then the set

This section proves auxiliary compactness and isomorphism results used to obtain the AP for function spaces on general domains. It concludes that the relevant Sobolev and continuous-function spaces satisfy the AP under the stated assumptions.

  • Compactness: A compact image remains controlled when continuous maps converge uniformly on a compact set, yielding compactness of the associated infinite union.This compactness result is used in the proof of the finite-dimensional approximation lemma.
  • Function-space isomorphisms: A Cm-diffeomorphism between domain closures induces a continuous linear bijection between the corresponding Cm function spaces.The map is composition with the inverse diffeomorphism.
  • Function-space isomorphisms: The spaces Cm([0, 1]d) and Cm([-M, M]d), as well as the corresponding Wm,1 spaces, are continuously linearly bijective.These results follow from smooth coordinate transformations between the domains.
  • Approximation property: The spaces in Assumptions 9 and 10 have the approximation property, including Sobolev and Cm spaces on Lipschitz domains.The proof uses isomorphisms, bases, extension operators, and direct finite-rank constructions.
  • Approximation property: For Cm( D̄), finite-rank approximations are constructed through extension to a hypercube, basis truncation, and restriction back to the domain.The resulting operator satisfies the required uniform approximation on compact sets.

Appendix C.

This appendix develops approximation results for linear functionals on Sobolev and continuous-function spaces. It shows that such functionals can be represented or uniformly approximated on compact sets by smooth kernel integral operators.

  • Functional representation: Linear functionals on Cm spaces admit representation through finite signed Radon measures.The representation is obtained using an embedding into a vector-valued continuous-function space and Hahn–Banach extension.
  • Kernel approximation: For bounded domains, functionals on Wm,p(D) can be uniformly approximated over suitable compact or closed bounded sets by integral operators with C∞ kernels.The construction treats p > 1 through duality and density, and p = 1 using total boundedness, mollification, and Lusin-type arguments.
  • Kernel approximation: The p = 1 case uses finite nets of the target set together with mollified representatives to control approximation errors uniformly.The argument applies convolution with standard mollifiers to weak derivatives and associated dual functions.
  • Kernel approximation: Functionals on Cm( D̄) are approximated by finite sums of point evaluations and then by smooth kernel integrations.Evaluation points can be selected distinctly inside D while preserving the approximation on compact sets.
  • Kernel approximation: Mollifiers convert point evaluations into localized integral averages whose errors can be made arbitrarily small on compact sets.Nonnegativity and unit integral of the mollifiers support convergence to function values.

Appendix D.

This appendix shows that the finite-dimensional components of the abstract approximation construction can each be represented or approximated by neural operators. It then connects standard finite-dimensional neural networks to neural-operator layers.

  • Input map: The input finite-dimensional map can be approximated by a neural operator that produces a vector-valued constant function.Its coefficients are obtained from smooth-kernel approximations of linear functionals.
  • Output map: The output finite-dimensional map can be approximated by a neural operator that maps a constant coefficient function into a basis expansion.Smooth approximations of the output basis functionals provide the kernel representation.
  • Conclusion: The three components of the finite-dimensional factorization can therefore be approximated by neural operators under the stated assumptions.These components are the input representation, finite-dimensional nonlinear map, and output reconstruction.
  • Layer construction: A finite-dimensional neural-network layer can be encoded as a neural-operator layer using a constant kernel implementing its linear weight matrix.Setting κ0(x, y) = 1/|D| W0 makes the integral operator reproduce the matrix action W0w.
  • Layer construction: Repeating the kernel construction across layers embeds standard finite-dimensional neural networks into neural operators.The construction preserves the affine transformation and activation sequence of the original network.

Appendix E.

Appendix E proves the relevant approximation result by reducing it to a single-layer case, discretizing compact input sets, and controlling approximation errors through continuity and Riemann-sum convergence.

  • The proof assumes D = D′ and uses continuous embedding to work with A = U = C(¯D).
  • Each discretization is paired with disjoint positive-measure partitions whose cells contain unique discretization points.
  • For an operator G with parameters κ and b, the construction defines discretized maps ˆG_j on sampled inputs and outputs.
  • Compactness provides finitely many representatives a_1, . . . , a_N that approximate every a ∈ K within a prescribed κ-dependent tolerance.
  • Riemann-sum convergence and uniform continuity supply discretization and activation-error bounds once the refinement index exceeds suitable thresholds.
  • Repeated application of the triangle inequality combines the bounds and completes the proof.

Appendix F.

Appendix F constructs a neural operator approximating a continuous operator by factoring through finite-dimensional continuous maps, approximating those maps with neural networks, and controlling the resulting errors on compact sets.

  • Lemma 26 yields a factorization G_1 = G ◦ ψ ◦ F through continuous linear maps F and G and a finite-dimensional continuous map ψ.
  • The approximating maps F_t can be chosen so that F_t(a)(x) = w_t(a)1(x), identifying their outputs with finite-dimensional vectors.
  • Uniform continuity on compact sets controls the approximation error between F and F_t through a modulus of continuity.
  • A classical neural-network approximation theorem approximates ψ uniformly on the compact set F_t(K), and neural operators can exactly mimic the resulting standard networks.
  • Combining the approximations produces ˜G in the stated neural-operator class, while separately establishing uniform boundedness by 4M.
  • The construction approximates the extended map ˜ψ on compact ranges, with bounds such as |ψ(q) − ˜ψ(q_1)(x)|_1 ≤ ϵ 6∥G∥.

Appendix G.

Appendix G extends the approximation argument to a setting with U = H^{m_2}(D), using truncation, compact restriction, continuous extension, and Theorem 11.

  • The proof works in the Sobolev space U = H^{m_2}(D) and introduces a truncation parameter R > 0.
  • Dominated convergence is used to choose R large enough to control the truncated operator relative to G†.
  • Lusin’s theorem provides a compact set K ⊂ A on which the truncated operator R|K is continuous.
  • A generalized Tietze extension theorem extends R|K to a continuous mapping ˜G† beyond K.
  • Applying Theorem 11 to the extension yields a neural operator G in the class N O_N(σ_1, σ_2, σ_3; D, D′).
  • The final displayed bound depends on R and the measure of A \ K, indicating the approximation argument retains an error contribution outside the compact set.
Loading 2108.08481v6…