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Error estimates for DeepOnets: A deep learning framework in infinite dimensions

Samuel Lanthaler, Siddhartha Mishra, George Em Karniadakis

arXiv:2102.09618v3math.NA

TL;DR

Learning nonlinear operators between infinite-dimensional Banach spaces requires rigorous control of approximation and sampling errors beyond universal approximation results. The paper extends DeepONet theory to measurable operators, decomposes error across its architectural components, and analyzes the resulting bounds. Across four differential-equation examples, it proves algebraic complexity in the target accuracy, while noting scope limits for high spatial dimensions and other unmodeled error sources.

  • Problem

    Existing DeepONet theory offered universal approximation results but lacked rigorous efficiency and generalization guarantees for infinite-dimensional operator learning.

  • Method

    The paper extends universal approximation to measurable operators and decomposes DeepONet error into encoding, approximation, and reconstruction terms, with covariance-based and covering-number analyses.

  • Results

    For four differential-equation operators, the paper proves DeepONets can break the curse of dimensionality, achieving accuracy ε with network sizes growing algebraically in 1/ε.

  • Takeaways & Limitations

    The results provide rigorous evidence that DeepONets can efficiently approximate a broad class of nonlinear operators arising from differential equations.

  • Takeaways & Limitations

    The efficiency result is established for low spatial dimensions in one highlighted operator example, while larger spatial dimensions remain future work.

Abstract

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DeepONets have recently been proposed as a framework for learning nonlinear operators mapping between infinite dimensional Banach spaces. We analyze DeepONets and prove estimates on the resulting approximation and generalization errors. In particular, we extend the universal approximation property of DeepONets to include measurable mappings in non-compact spaces. By a decomposition of the error into encoding, approximation and reconstruction errors, we prove both lower and upper bounds on the total error, relating it to the spectral decay properties of the covariance operators, associated with the underlying measures. We derive almost optimal error bounds with very general affine reconstructors and with random sensor locations as well as bounds on the generalization error, using covering number arguments. We illustrate our general framework with four prototypical examples of nonlinear operators, namely those arising in a nonlinear forced ODE, an elliptic PDE with variable coefficients and nonlinear parabolic and hyperbolic PDEs. While the approximation of arbitrary Lipschitz operators by DeepONets to accuracy $ε$ is argued to suffer from a "curse of dimensionality" (requiring a neural networks of exponential size in $1/ε$), in contrast, for all the above concrete examples of interest, we rigorously prove that DeepONets can break this curse of dimensionality (achieving accuracy $ε$ with neural networks of size that can grow algebraically in $1/ε$). Thus, we demonstrate the efficient approximation of a potentially large class of operators with this machine learning framework.

1. Introduction

DeepONets address the challenge of learning nonlinear operators between infinite-dimensional spaces, where finite-dimensional representations introduce errors and general operators may incur exponential complexity. This paper develops rigorous error estimates and shows algebraic-size approximations for four differential-equation examples.

  • Motivation: Finite-dimensional parametrizations introduce non-vanishing input and output errors and may require explicit knowledge of the input measure.They also do not cover settings where governing physics are unknown but input-output data are available.
  • Motivation: Universal approximation results guarantee arbitrary accuracy for continuous operators on compact subsets, but do not by themselves establish efficient approximation.The paper identifies efficiency as a separate complexity question in infinite-dimensional settings.
  • Research gap: General Lipschitz operators may require neural networks whose complexity scales exponentially in 1/ε, creating a curse of dimensionality.The paper presents this as a worst-case heuristic supported by related lower-bound results.
  • Contributions: The paper decomposes DeepONet error into encoding, approximation, and reconstruction components and derives rigorous upper and lower bounds.The analysis also addresses measurable operators without compactness requirements and compares point-evaluation architectures with PCA-based approaches.
  • Results: For four prototypical differential equations, DeepONets achieve accuracy ε with network size growing algebraically in 1/ε, thereby breaking the curse of dimensionality.The examples span a nonlinear forced ODE, elliptic PDE, Allen-Cahn equation, and scalar conservation law.
  • Results: The paper also derives a generalization bound whose dependence on the number of training samples is consistent with standard finite-dimensional statistical-learning bounds up to logarithmic terms.Other errors, including optimization, noisy-data, and train-test mismatch errors, are outside the paper’s scope.

2. Deep Operator Networks

A DeepONet combines an encoder, a finite-dimensional neural approximator, and a trunk-induced affine reconstructor to map function inputs to function outputs. The paper formalizes this architecture for measurable operator-learning data and evaluates its L2(μ) approximation error.

  • Architecture: DeepONets replace shallow branch and trunk networks with deep neural networks while retaining the operator-network architecture.The networks use affine maps and scalar nonlinear activations; this work restricts them to ReLU activation.
  • Encoder: The encoder maps a function to its values at m sensor points, E(u) = (u(x1), . . . , u(xm)).Pointwise evaluation is well-defined on the continuous functions in the full-measure input set.
  • Architecture: The architecture therefore consists of three components: encoder, approximator, and reconstructor.Figure 1 presents this decomposition schematically.
  • Approximator: The approximator A maps the m-dimensional encoded data to p dimensions, and the branch net is the composition β(u) = A ◦ E(u).Thus the nonlinear approximation step acts between finite-dimensional spaces.
  • Reconstructor: The trunk net produces p+1 functions of the output coordinate, which define an affine reconstructor R from R^p into the output function space.The affine reconstruction is more general than the original Lu et al. DeepONet; setting τ0(y) constant recovers that version.
  • Error formulation: The approximation error is measured in the L2(μ)-norm between the target operator G and the DeepONet N.The formulation assumes embeddings of the input and output Banach spaces into L2(D) and L2(U).
  • Assumptions: The formal data setting uses a Borel probability measure with finite second moments, a measurable operator G, and square-integrable outputs.The main complexity analysis concerns best approximation, while finite-data generalization is treated separately.
  • Encoder: Point-evaluation encoders are technically restrictive, although more general functionals could model physical measurements or support other mathematical analyses.The paper retains point evaluations for comparison with prior numerical experiments and detailed encoding-error analysis.

3. Error bounds for DeepONets

The paper establishes universal approximation and quantitative error bounds for DeepONets, decomposing error into encoding, approximation, and reconstruction components. These bounds connect performance to covariance-operator spectra, while identifying both optimal-sensor results and limitations from spectral distortion and finite-dimensional approximation.

  • Universal approximation: Measurable operators on possibly unbounded supports admit DeepONet approximations with arbitrarily small L2(µ⊗dy) error.This extends prior continuous-operator, compact-domain guarantees, but uses a weaker distance aligned with practical training.
  • Error decomposition: The total DeepONet error is bounded through separate encoding, approximation, and reconstruction errors for Hölder or Lipschitz operators.The framework uses arbitrary encoders, finite-dimensional approximators, and affine reconstructors.
  • Approximation error: ε−m/2 network-size scaling is indicated for arbitrary Lipschitz operators, because achieving encoding error ε requires sensor dimension m(ε)→∞.Here m is the number of sensors, producing the stated curse-of-dimensionality scaling.
  • Reconstruction error: The covariance spectrum of the push-forward measure G#µ determines a lower bound on reconstruction and total approximation errors.The lower bound is expressed through eigenvalues λ1 ≥ λ2 ≥ … of the covariance operator associated with G#µ.
  • Spectral structure: Nonlinear push-forwards can destroy rapid covariance spectral decay, even when the input measure has exponentially decaying eigenvalues.Special operator structures can nevertheless yield fast spectral decay and help avoid the curse of dimensionality.
  • Reconstruction error: When G maps into Hs, trunk networks can achieve algebraic reconstruction-error decay under suitable scaling of their size.The result applies to nonlinear operators arising frequently in PDEs.
  • Encoding error: Random sensors achieve almost surely vanishing encoding error, with decay governed by covariance spectral decay and optimal up to logarithmic factors under bounded eigenfunctions.The result assumes the covariance operator has a bounded eigenbasis in L∞.
  • Reconstruction error: Optimal affine reconstruction is linked to finite-dimensional projection subspaces, whose projection errors are characterized by covariance-operator eigensystems.This characterization supports lower bounds and identifies optimal affine reconstruction choices.

4. Error bounds on DeepONets in concrete examples.

The paper decomposes DeepONet error into encoding, approximation, and reconstruction components, then derives explicit bounds for concrete nonlinear operators. Across the forced pendulum and elliptic PDE examples, suitable spectral decay and regularity yield algebraic or sub-algebraic complexity in the target tolerance, while assumptions and dimension restrict the scope.

  • General error framework: The total error separates into encoding, reconstruction, and finite-dimensional neural approximation errors, each controlled by covariance spectral decay or operator regularity.Encoding depends on the input covariance spectrum; reconstruction depends on the push-forward covariance spectrum and operator smoothness; approximation reduces to finite-dimensional neural-network approximation.
  • General error framework: Random sensors provide almost optimal encoding error under bounded eigenfunctions, while bespoke pseudo-spectral sensors recover optimal bounds up to constants.The random-sensor result includes a logarithmic factor, whereas more informed sensor choices remove that loss.
  • Forced gravity pendulum: For the forced gravity pendulum, error ε can be achieved with m ≳ ε^-1/k sensors and p ≳ ε^-1/r trunk outputs.The stated conditions assume the sensors resolve the characteristic length scale, mℓ ≫ 1.
  • Forced gravity pendulum: For the forced pendulum, DeepONet size grows sub-algebraically with ε^-1, and the paper concludes that the approximation breaks the curse of dimensionality.The conclusion holds for arbitrary natural numbers k and r, although the implied constants may deteriorate as they increase.
  • Forced gravity pendulum: For the forced pendulum with algebraically decaying input coefficients, total complexity still scales algebraically with ε^-1 and therefore avoids the curse of dimensionality.Changing the underlying measure changes the encoding-error decay, but not the algebraic total-complexity conclusion.
  • Variable-coefficient elliptic PDE: For the variable-coefficient elliptic PDE, DeepONets break the curse of dimensionality when the spatial dimension is low and sensors resolve the coefficient correlation length.The analysis explicitly treats d ∈ {1, 2, 3}; the sensor requirement includes m ≫ ℓ^-d|log(ε)|^d.

5. On the generalization error for DeepONets

The paper extends DeepONet analysis from best approximation to finite-sample training, defining empirical losses through sampling in both input-function and target spaces. Under boundedness and Lipschitz assumptions, it bounds generalization error and identifies how sample size, parameters, and weight bounds affect that guarantee.

  • Empirical loss construction: Training approximates the loss by sampling input functions and target-space evaluation points, producing an empirical loss for the DeepONet.The proposed sampling uses iid input functions from µ and target points uniformly from U, with deterministic quadrature as an alternative for target-space integration.
  • Generalization error: The generalization error measures the gap between the empirical optimizer and an optimizer of the population loss.The paper studies this gap for DeepONets, whose inputs lie in infinite-dimensional spaces.
  • Generalization bound: Theorem 5.3 bounds the generalization error under boundedness and Lipschitz continuity assumptions on the operator-related loss.The bound has a constant depending on µ, Ψ, and Φ, while remaining independent of B, dθ, and N in that constant.
  • Generalization bound: The bound scales, up to logarithmic factors, with the standard Monte Carlo rate in the number of samples from the infinite-dimensional input space.Thus increasing N reduces the generalization error, and the paper states that the curse of dimensionality is overcome for generalization error.
  • Scope of the guarantee: The estimate depends explicitly on the DeepONet parameter count, which may become large when reducing approximation error to a target tolerance.The bound can also increase with the weight bound B, although the paper notes this dependence is logarithmic.

6. Discussion

The paper develops a rigorous DeepONet error framework and applies it to nonlinear operators arising from differential equations. Across four concrete examples, it proves curse-of-dimensionality-breaking approximation results and derives generalization bounds.

  • Framework: DeepONets are analyzed through rigorous approximation and generalization error bounds for nonlinear operators between infinite-dimensional spaces.The framework targets the infinite-dimensional setting underlying differential equations.
  • Universal approximation: Measurable operators G:C(D)→L2(U) can be approximated arbitrarily accurately without continuity or compactness assumptions.This extends the earlier universal approximation result to measurable mappings on non-compact spaces.
  • Error analysis: The total DeepONet error is decomposed into encoding, approximation, and reconstruction components, with bounds linked to covariance-operator eigenvalue decay.The reconstruction analysis also identifies eigenfunction-based trunk nets as optimal for the relevant projection problem.
  • Examples: All four differential-equation examples break the curse of dimensionality, with DeepONet size growing algebraically or sub-algebraically as error tolerance decreases.The examples include a forced gravity pendulum, an elliptic PDE, the Allen-Cahn equation, and a nonlinear hyperbolic PDE.
  • Generalization: The paper derives a generalization-error bound using covering-number estimates despite the overall infinite-dimensional setup.The bound concerns replacing population risk with the empirical loss used during training.
  • Error analysis: Randomly located sensors can achieve almost optimal encoding error, while general affine reconstructors and learned covariance eigenfunctions support near-optimal reconstruction.The method does not require a priori algorithmic knowledge of the underlying measure beyond the ability to sample from it.
  • Scope and outlook: The authors state that computational-complexity bounds are not fully sharp, especially for the Allen-Cahn equation, and identify recurrent and multiple-operator extensions as future work.They report almost sharpness for scalar conservation laws and believe some other examples are close to optimal.

Appendix B. On the definition of Error (2.11)

The appendix establishes that pointwise encoders used by DeepONets can be extended measurably from continuous functions to L2(D), making the error definition well-defined on the larger space.

  • Encoder extension: A Borel measurable extension E:L2(D)→Rm agrees with the pointwise encoder on continuous inputs.This resolves the mismatch between pointwise evaluation and the L2(D) domain used in the error formulation.
  • Encoder extension: The pointwise encoder E:C(D)→Rm evaluates an input at m fixed sensor locations.It maps u to (u(x1),…,u(xm)).
  • Construction: The construction represents point evaluation on continuous functions as a limsup of continuous functionals and assigns finite values to the measurable extension.The proof first treats one sensor and then extends the construction componentwise.
  • Supporting theorem: Lusin’s theorem supplies a compact subset on which a Borel measurable mapping between Polish spaces becomes continuous.The compact set can exclude an arbitrarily small amount of probability mass.

C.1. Proof of Theorem 3.1.

The proof of universal approximation clips an unbounded measurable operator, restricts it to a large compact set where it is continuous, projects its outputs finitely, and then applies continuous-operator approximation.

  • Network control: A ReLU clipping network bounds intermediate branch outputs while preserving inputs within the target radius, enabling a bounded DeepONet construction.The auxiliary network approximates the identity on ||x||ℓ2≤R1 and maps outputs into a radius R2.
  • Clipping and localization: A measurable operator G is first clipped at level M so that the clipping error vanishes as M increases.Dominated convergence justifies choosing M to make this contribution small.
  • Clipping and localization: Lusin’s theorem yields a compact K⊂C(D) on which the clipped operator GM is continuous while the excluded probability mass is small.The proof chooses µ(C(D)\K)<(ε/9M)^2.
  • Finite-dimensional reduction: Finite-dimensional projection of the compact image GM(K) produces a continuous map into C(U) with uniformly small output projection error.An orthonormal basis of continuous functions is used to select a projection Pκ.
  • Network approximation: The continuous-operator universal approximation theorem supplies an operator network approximating the projected map on K.The constructed network uses single-hidden-layer approximator and trunk networks before subsequent output-layer modifications.
  • Error bound: The proof combines the three error terms from N−G=R∘A∘E−R∘P∘G+[R∘P∘G−G] to establish the desired DeepONet error bound.The terms correspond to approximation, reconstruction, and encoding contributions in the paper’s decomposition.

C.3. Proof of Theorem 3.10.

The proof characterizes the optimal finite-dimensional reconstruction subspace using the covariance operator’s eigenspaces. This links projection error minimization to the leading covariance eigenvalues.

  • Covariance structure: The covariance operator of a probability measure on a separable Hilbert space is compact and self-adjoint, so it admits an eigendecomposition.Its eigenvalues and eigenspaces provide the spectral structure used in the projection argument.
  • Projection optimization: Projection-error minimization over p-dimensional subspaces is equivalent to maximizing the covariance energy captured by the orthogonal projection.The proof rewrites the relevant objective using an orthonormal basis of the candidate subspace.
  • Optimal subspaces: An optimal p-dimensional reconstruction subspace is formed from leading covariance eigenspaces, with degeneracies handled by selecting subspaces within the eigenspace at the cutoff eigenvalue.The characterization tracks distinct eigenvalues and their associated eigenspaces.
  • Optimal subspaces: The optimal projection error is therefore governed by the covariance spectrum rather than by an arbitrary choice of basis.Equality conditions force full inclusion of eigendirections above the cutoff and exclusion below it.
  • Uniqueness: When the relevant leading eigenspaces determine the minimizing subspace uniquely, the optimal subspace is their direct sum.The corollary states this conclusion for dimensions selected across the eigenspace decomposition.

C.3.1. Proof of Theorem 3.10.

The proof identifies covariance-eigenfunction subspaces as optimal affine approximations and expresses the optimal projection error through the omitted eigenvalues.

  • The minimum projection error equals the sum of the covariance eigenvalues beyond the retained dimension, Pk>p λk.
  • The optimal p-dimensional subspace is spanned by eigenfunctions φ1, ..., φp of the covariance operator, ordered by decreasing eigenvalues.
  • Any affine subspace with a nonzero mean-offset term can be improved by translating it to eliminate that term.
  • The optimal affine subspace is centered at the mean E[v], reducing the problem to selecting an optimal p-dimensional linear subspace.

C.6. Proof of Lemma 3.16.

The proof controls encoder and reconstructor perturbations caused by approximate basis functions, obtaining an error bound proportional to their maximum approximation error.

  • The projection map is analyzed using orthonormal approximate basis functions and decompositions of inputs into projected and complementary components.
  • Each basis-function perturbation is bounded by 2ϵ under the assumed approximation condition.
  • The resulting projection perturbation scales as p3/2 maxj ∥τj − eτj∥L2, yielding an O(ϵ) contribution when the basis errors are controlled.
  • The reconstruction error satisfies bER ≤ bE eR + Cp3/2 maxj ∥τj − eτj∥L2 ≤ bE eR + Cϵ.
  • The proof also establishes a Lipschitz bound for the projection map and estimates the corresponding reconstruction stability.

C.8. Proof of Proposition 3.19.

The proof constructs Lipschitz maps with prescribed Gaussian-related distributions and uses covariance-eigenvector subspaces to obtain affine approximation and reconstruction bounds.

  • A non-degenerate Gaussian measure can be mapped by a Lipschitz function P:X→[0,1] onto the uniform measure on [0,1].
  • For suitable nonnegative coefficients γk, a Lipschitz map G:[0,1]→Y produces a pushforward measure with the prescribed covariance structure.
  • Composing the two maps yields a Lipschitz mapping G:X→Y with the desired pushforward construction.
  • The first p covariance eigenvectors define an affine input subspace V0 that satisfies the optimal projection characterization.
  • Choosing the reconstruction image as an affine subspace W and taking R∘P as its orthogonal projection gives the corresponding reconstruction estimate.

C.13. Proof of Theorem 3.9. √

The proof establishes random-sensor encoding bounds and analyzes Fourier encoders, including exponential decay for smooth inputs and aliasing effects for higher modes.

  • M(m)=⌈Cκm log(m)⌉ random sensors suffice for the stated probabilistic encoding estimate.
  • The random-sensor encoding error is controlled asymptotically through the growth of the sensor count and logarithmic factors.
  • For infinitely many sample sizes, the encoding error exceeds any fixed constant C, and this holds for all sufficiently large M.
  • With m=2K+1, the discrete Fourier encoder and decoder are admissible because E∘D=Id on the retained Fourier modes.
  • Higher-order Fourier modes alias onto lower modes on the sensor grid, producing a non-vanishing contribution to the encoding error.
  • For smooth inputs with exponentially decaying Fourier coefficients, the encoding error satisfies bEE≤C exp(−cℓN).

C.18. Proof of Corollary 3.34.

The proof transfers approximation and size estimates from an auxiliary network to the DeepONet approximator. Network composition and a final linear output layer add controlled complexity while preserving the stated depth and error bounds.

  • A linear output layer of size O(pN) represents the DeepONet output, increasing size by pN and depth by one.The resulting bounds follow from the corresponding estimates for the auxiliary network.
  • The proof composes networks for the encoded input map and finite-dimensional approximator to represent the full DeepONet map.The composition lemma supplies the required representation and complexity control.
  • The resulting network depth grows as C(1 + log(m) log log(m)), with C independent of m and p.The bound is stated for the constructed network after combining the preceding size estimates.
  • The approximation error is bounded by the auxiliary network error together with the trunk-function approximation term.The displayed estimate includes ϵ and the maximum L2(U) discrepancy between target and approximated trunk functions.
  • Under exponential coefficient decay, the relevant coefficient sequence belongs to ℓq(Zd) for every q ∈ (0, 1), enabling the stated complexity estimate.This follows directly from the exponential decay assumption.

Appendix D. DeepONet approximation of linear operators

This appendix analyzes DeepONet approximation of bounded linear operators through random sensor encoding, affine reconstruction, and finite-dimensional approximation. For a Gaussian-field elliptic example, covariance spectral decay yields rapidly decreasing approximation error and algebraic network requirements.

  • The appendix studies DeepONet approximation of bounded linear operators and motivates theory explaining observed exponential error decay as sensor count increases.A linear functional example is illustrated in Figure 2, while the general estimate is applied to an elliptic PDE.
  • Theorem D.2 uses iid uniform random sensors, optimal covariance-based trunk functions, and a trunk approximation whose reconstruction maps have bounded Lipschitz constants.With probability one over sensor locations, the resulting DeepONet satisfies an asymptotic approximation-error estimate.
  • For linear mappings, an exact affine finite-dimensional approximator exists for every affine encoder-decoder and reconstruction-projection pair.A shallow ReLU network can represent this affine map exactly, so the finite-dimensional approximation error vanishes.
  • Random sensor encoding error is controlled asymptotically with probability one, with a constant depending on the measure and domain-related eigenfunction bounds.The estimate is obtained for almost all numbers of iid random sensors.
  • The reconstruction analysis relates output covariance eigenvalues to input covariance eigenvalues and bounds the output covariance trace using the operator norm.This connects reconstruction error to spectral properties of the underlying measure.
  • For a Gaussian random field with exponentially decaying Fourier covariance coefficients, choosing p ∼ log(ϵ^-1)^d and m ∼ log(ϵ^-1)^{d(1+σ)} achieves overall error of order ϵ.The implied constants depend on the covariance length scale, dimension, and σ.
  • The parabolic norm C(2,4)([0,T]×Td) yields a less restrictive convergence result than the larger norm used in prior numerical analysis.The remainder estimate is adapted to available a priori estimates for parabolic equations.

E.15. Proof of Proposition 4.31.

The proof constructs a neural approximator for the Allen–Cahn solution operator by combining a numerical scheme with interpolation and projection. Its error is controlled by spatial discretization, sensor reconstruction, and network approximation.

  • The approximator A is built from a neural approximation of a convergent numerical scheme for the Allen–Cahn solution operator.The construction targets A ≈ P ◦ G ◦ D and uses constants independent of m, n, and ϵ.
  • Local linear interpolation maps sensor values in Rm back to continuous input functions while reproducing the values at the sensor grid points.Composing interpolation with projection gives the finite-dimensional reconstruction map used by the approximator.
  • The solution-operator discrepancy is bounded by Cm^-1/d plus the maximum sensor-point discrepancy between the original and reconstructed inputs.The constant depends on the input through its C4,α norm.
  • For an equidistant grid, choosing n = ⌈Tm2/d⌉ and ϵ = m^-2/d aligns the numerical discretization and neural approximation scales.The grid relation is stated as Δx ∼ m^-2/d.
  • The construction yields an approximator network with p = m whose size is bounded by the stated neural approximation estimates.The final network is obtained by composing the numerical-scheme network with the linear projection-interpolation map.
  • The ReLU construction used for interval localization has an architecture independent of the interval endpoints and width parameter.Its piecewise-linear form is linear near the interval boundaries.

E.17. Proof of Theorem 4.38.

The proof constructs a DeepONet for the scalar conservation-law setting by approximating localized trunk functions and bounded branch outputs. The resulting complexity and error bounds combine truncation, trunk approximation, and branch-network estimates.

  • The approximator A maps sensor values to p branch outputs, setting Nj ≡ 0 for j > m.This finite-dimensional construction is paired with trunk functions that approximate the corresponding localized basis functions.
  • Each trunk network τj approximates an indicator function 1Cj in L1 within ϵ, while indices beyond m use zero trunk functions.The collection forms the trunk network used in the DeepONet reconstruction.
  • The branch outputs are bounded uniformly on the support of the input measure, allowing the trunk approximation errors to be summed over the retained components.The proof separately estimates the first truncation term and each term in the finite sum.
  • For any ϵ > 0, a DeepONet exists with size bounded independently of p and m by the stated branch and trunk complexity estimates.The construction establishes the theorem’s approximation and complexity claim.
  • The covariance operator of the push-forward measure is represented through the solution’s translation structure, yielding a stationary convolution kernel.Stationarity identifies the Fourier basis as covariance eigenfunctions, with eigenvalues given by Fourier coefficients of the kernel.

Appendix F. Proof of Theorem 5.3

The appendix proves Theorem 5.3 through covering-number arguments, moment estimates, and bounds under the stated assumptions. The proof tracks constants and establishes the claimed inequalities and generalization-error bound.

  • Covering argument: Covering points θ1, …, θK approximate every θ in [−B, B]^dθ within ℓ∞-distance ϵ.A measurable index map selects the first covering point satisfying the required distance condition.
  • Moment estimates: For p ≥ 1, the pth moment of a maximum is bounded by K times the largest individual pth moment.This estimate is used alongside the covering construction to control the supremum over parameter values.
  • Final estimate: The proof combines bounds from Lemmas F.5 and F.6 with assumptions on Ψ and Φ, then applies Lemma F.8 to obtain the target estimate.The constants are tracked through substitutions and intermediate estimates before concluding inequality (F.2).
  • Constant dependence: The resulting constant C is expressed as C(µ, Ψ, Φ) and is independent of dθ, B, and N.The stated dependence incorporates constants determined by the Gaussian tail of µ and the upper bound on Ψ and Φ.
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