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

Motivation

• DeepONets address limitations of finite-dimensional parametrizations, which incur input and output discretization errors, require knowledge of input measures, and may be unsuitable when governing physics is unknown but input-output data are available.These limitations motivate learning nonlinear operators directly between infinite-dimensional spaces.Source: §1 · pp. 2–3

Theoretical framework

• The paper extends DeepONet universal approximation from continuous operators on compact inputs to measurable operators without compactness, while seeking explicit computational-complexity estimates absent from earlier results.The earlier theorem guarantees arbitrary precision under continuity and compactness but does not specify sensor counts, network widths, depths, or parameter sizes.Source: §1 · p. 4

Error analysis

• DeepONet error is decomposed into encoding, approximation, and reconstruction errors, yielding rigorous upper and lower bounds whose optimal encoding and reconstruction rates depend on covariance-operator eigensystems.The architecture consists of an encoder, a finite-dimensional approximator, and a trunk-induced affine reconstructor.Source: §1 · p. 4

Model problems

• For a nonlinear forced ODE, variable-coefficient elliptic PDE, Allen–Cahn equation, and scalar conservation law, DeepONets achieve accuracy ϵ with network sizes growing algebraically in 1/ϵ, thereby breaking the curse of dimensionality.These examples span different inputs and Sobolev regularities and provide the paper’s first rigorous efficiency proofs for operator approximation.Source: §1 · p. 5

Generalization

• The paper derives a DeepONet generalization-error bound that scales asymptotically as 1/N up to logarithmic factors, where N is the number of training samples.This matches the standard finite-dimensional statistical-learning scaling despite the infinite-dimensional setting.Source: §1 · p. 5

Scope and limitations

• The analysis concerns best-approximation and sampling errors, while training imperfections, noisy or uncertain data, and training–evaluation distribution mismatch remain outside its scope.The results are formulated using ReLU networks, although the paper states that this activation choice is not essential to the main conclusions.Source: §1 · p. 5

2.1. Neural Networks.

• Neural networks are compositions of affine layer maps and scalar nonlinear activations; this work uses ReLU activation, σ(z) = max(z, 0), throughout.The network comprises input and output layers with K −1 hidden layers, and its trainable parameters are weights and biases.Source: §2.1 · p. 6

2. Deep Operator Networks

• DeepONets extend operator networks by replacing shallow branch and trunk networks with deep neural networks, while allowing a more general affine reconstruction than Lu et al. (2019).The formulation also extends directly from scalar functions to vector-valued systems.Source: §2.2 · pp. 6–7

2.2. DeepONets.

• A DeepONet encodes sensor evaluations E(u) = (u(x1), . . . , u(xm)), approximates them with A : Rm →Rp, and reconstructs outputs through a trunk-induced map R : Rp →C(U).The encoder, approximator, and reconstructor are composed as N = R ◦ A ◦ E; pointwise encoders are chosen for comparison with Lu et al. (2019) and detailed encoding-error analysis.Source: §2.2 · pp. 6, 7, 9
• The approximation problem assumes separable Banach spaces embedded in L2, a Borel probability measure with finite second moments, continuous inputs almost surely, and a measurable square-integrable target operator.Errors are measured in the L2(µ)-norm, and measurability is recovered despite pointwise encoding not being defined on spaces such as L2(D).Source: §2.2 · pp. 8–9
• Pointwise sensor encoding is a technical limitation because it is not well-defined on L2(D), motivating possible replacements by more general functionals while retaining the chosen architecture here.The framework focuses on perfect, noise-free input/output data, with finite-data generalization errors deferred to Section 5.Source: §2.2 · pp. 8–9

3.1. A universal approximation theorem.

• DeepONets approximate any Borel measurable G:C(D)→L2(D) with G∈L2(µ) to arbitrary L2(µ⊗dy) tolerance, including unbounded supports but not the stronger uniform compact-set guarantee.The extension removes compactness and continuity assumptions, matching the L2-distance used in training; quantitative complexity remains unspecified by this theorem.Source: §3.1 · pp. 9–10

3.2.1. Error decomposition and the curse of dimensionality.

• The total DeepONet error is bounded through encoding, approximation, and reconstruction errors, while arbitrary Lipschitz operators require complexity growing faster than algebraically in ε^-1.Encoding and reconstruction quantify finite-dimensional information loss, whereas approximation reduces to neural-network approximation of the projected mapping; the sensor count m(ε) drives the worse-than-algebraic scaling.Source: §3.2.1 · pp. 11–12

3.2.2. Reconstruction error.

• Optimal affine reconstruction and projection yield covariance-spectrum lower bounds on reconstruction and total errors, while neural-network trunk approximation and nonlinear push-forwards can limit attainable rates.For Sobolev-valued outputs, Fourier comparison gives algebraic reconstruction decay with trunk-net size, and the reconstruction and projection maps can have Lipschitz constants at most 2.Source: §3.2.2 · pp. 13–14

3.2.3. Encoding error.

• Random uniformly distributed sensors achieve almost optimal encoding-error decay up to a logarithmic factor when the covariance eigenbasis is bounded, while exponentially decaying input spectra yield bEE ≲ exp(−cℓm^(1/d)).The random-sensor rate depends only on covariance spectral decay and converges to zero almost surely; the exponential bound applies to equidistant-grid sensors for the specified trigonometric random-field measure.Source: §3.2.3 · pp. 15–16

3.2.4. Approximation error.

• Beyond Lipschitz regularity, holomorphic operators and PDE operators with exploitable internal structure provide classes for which DeepONets can avoid the curse of dimensionality.The holomorphic route is treated abstractly through prior neural-network approximation results and leads to quantitative estimates in Theorem 3.35, later applied to two concrete operators.Source: §3.2.4 · p. 16
• Approximation-error estimates with internal structure beyond smoothness are deferred to concrete parabolic and hyperbolic PDE operators discussed later.The abstract treatment focuses instead on projection, reconstruction, and encoding errors.Source: §3.2.4 · pp. 16–17

3.3. On the error due to projections of Hilbert spaces onto linear and affine subspaces.

• For Hilbert-space projections, optimal p-dimensional linear subspaces are characterized by the eigenspaces of the associated uncentered covariance operator, with minimal error determined by spectral decay.This result also guides the selection of suitable encoder sensors and bounds the encoding error.Source: §3.3 · pp. 17–18
• Optimal finite-dimensional affine subspaces consist of the measure mean translated along leading covariance eigenspaces, linking minimal projection error to covariance-eigenvalue decay.The affine characterization provides the basis for reconstruction-error bounds and optimal trunk-network design.Source: §3.3 · pp. 18–19

3.4. Bounds on the reconstruction error (3.3).

• The optimal affine reconstructor uses the output mean and eigenvectors associated with the largest covariance eigenvalues, while its optimal measurable projector is the corresponding orthogonal projection.The resulting reconstruction lower bound shows that output covariance spectral decay fundamentally limits accuracy for a fixed trunk dimension.Source: §3.4 · pp. 20–21
• The reconstruction upper bound separates covariance-spectrum truncation from trunk-network approximation, while a Fourier-based construction reduces DeepONet reconstruction analysis to estimating Fourier reconstruction error.This gives a practical alternative when the operator is known only through finite training measurements and the optimal eigensystem is inaccessible.Source: §3.4 · pp. 21–23

3.4.1. On the decay of spectrum for the push-forward measure.

• For Lipschitz continuous G, the push-forward covariance operator is trace class, but efficient DeepONet approximation depends on the precise rate of its eigenvalue decay.Exponential spectral decay would support efficient approximation, whereas trace-classness alone does not establish that rate.Source: §3.4.1 · pp. 23–24
• Even when the input covariance spectrum decays arbitrarily rapidly, a Lipschitz nonlinear operator can produce a push-forward covariance spectrum with any prescribed decay sequence.Thus, Lipschitz continuity alone cannot guarantee that rapid input spectral decay transfers to the operator outputs.Source: §3.4.1 · p. 24
• Two broadly applicable special cases yield fast spectral decay for covariance operators associated with push-forward measures.Source: §3.4.1 · p. 24

3.4.2. Reconstruction error for linear operators G : X →Y .

• For bounded linear G, a p-dimensional affine reconstruction space in L2(U) can bound reconstruction error using the covariance spectrum of the input measure and ∥G∥.Choosing reconstruction and projection as the orthogonal projection onto this space gives the stated reconstruction-error bound.Source: §3.4.2 · p. 24

3.4.3. Reconstruction error for operators with smooth image.

• For operators mapping into Hs(Tn), Fourier encoding with p degrees of freedom gives reconstruction error controlled by p^-s/n, with constants depending only on n and s.The Fourier projection retains modes with |k|∞≤N, where N≤p^1/n, and the truncation error satisfies ∥PNu−u∥L2(Tn)≤N^-s∥u∥Hs(Tn).Source: §3.4.3 · p. 25

3.5. Bounds on the encoding error (3.1).

• Encoding error decomposes into a covariance-spectrum component and an aliasing component, while random sensors provide optimal-up-to-logarithmic decay under suitable nonsingularity conditions.The framework uses covariance eigenfunctions, full-rank sensor evaluation matrices, and lower bounds on their smallest singular values; random sensors avoid requiring prior eigenfunction information.Source: §3.5 · pp. 27, 29

3.5.1. Examples.

• The examples formulate infinite-dimensional measures through Karhunen–Loève expansions, including Gaussian measures characterized by covariance operators and kernels.These expansions provide a practical way to sample probability measures on infinite-dimensional spaces.Source: §3.5.1 · pp. 29–30

3.5.2. Encoding error for a particular Gaussian measure.

• For a periodic Gaussian measure, pseudo-spectral encoding has super-exponentially decaying error, and uniformly random sensors achieve the same rate up to a logarithmic factor almost surely.The random encoder requires no information about the underlying measure when selecting sensors.Source: §3.5.2 · pp. 30–31

3.5.3. Encoding error for a parametrized measure.

• A parametrized random-field measure with sufficiently decaying trigonometric coefficients has support in C∞(D), while the framework also permits algebraically decaying coefficients.The construction uses centered coefficients in [−1,1] and converges uniformly in L2(D) under the stated summability condition.Source: §3.5.3 · p. 31
• For Fourier-parametrized random fields, the proposed encoder/decoder uses grid sensors, discrete Fourier recovery, and nonlinear shrinkage to enforce bounded parameters.The shrink operation preserves the Fourier-based construction while ensuring the recovered coefficients lie in [-1,1].Source: §3.5.3 · pp. 32–33
• When Fourier coefficients decay exponentially, the resulting encoding error decays exponentially in the number of sensors; the coefficient map is representable by a ReLU network of size O(m log(m)) and depth O(log(m)).The encoder/decoder therefore can achieve small encoding error with relatively few sensors while retaining an efficiently implementable neural representation.Source: §3.5.3 · pp. 33–34

3.6. Bounds on the approximation error (3.2).

• The approximation error measures the non-commutativity between the finite-dimensional approximator and the decoded operator, reducing its analysis to neural-network approximation of mappings R^m → R^p.The section therefore reviews finite-dimensional approximation results to bound DeepONet error.Source: §3.5.3 · p. 34, §3.6 · p. 34, §3.6.1 · p. 34

3.6.1. Regular high-dimensional mappings.

• For regular Lipschitz mappings, standard approximation bounds require network size O(p ε^-m), which grows exponentially with the sensor dimension m and produces a curse of dimensionality.Because m must increase as ε decreases to control encoding error, these generic bounds may not yield optimal DeepONet approximator sizes.Source: §3.6.1 · pp. 34–35

3.6.2. Holomorphic infinite-dimensional mappings.

• For (b, ε, κ)-holomorphic maps with b ∈ ℓq, q ∈ (0,1), finite-dimensional ReLU approximations provide algebraically decaying error with network size controlled by the retained coefficient index set.These mappings arise naturally for nonlinear operators such as diffusion-equation data-to-solution maps with random coefficients.Source: §3.6.2 · pp. 35–36
• For holomorphic operator reductions, the DeepONet approximator has depth O(m log(m) log log(m)) and log-linear size in m and p while achieving very small approximation error.The result applies with the Fourier encoder/decoder and affine reconstruction, thereby overcoming the approximation curse of dimensionality in these cases.Source: §3.6.2 · p. 37

4. Error bounds on DeepONets in concrete examples.

• The concrete-example workflow estimates encoding error from covariance spectral decay, reconstruction error from operator smoothness, and approximation error from Sobolev regularity or holomorphy.Random sensors are almost optimal without further measure information, while bespoke sensors can attain optimal encoding bounds; four nonlinear differential-equation examples motivate the analysis.Source: §4 · p. 38

4.1.1. Problem formulation.

• The forced pendulum operator maps L2([0,T]) inputs to L2([0,T]) solution trajectories and is Lipschitz continuous.The input is an external force, the output is the solution vector, and the measure is defined through a Fourier random-field expansion with decaying coefficients.Source: §4.1.1 · p. 39

4.1.2. Bounds on the encoding error (3.1).

• For the pendulum’s Fourier-parametrized input measure, equidistant point sensors and the associated decoder yield an explicit encoding-error bound.The encoder uses m = 2N + 1 grid points, and the encoded Fourier coefficients can be represented by a neural network.Source: §4.1.2 · pp. 39–40

4.1.3. Bounds on the reconstruction error (3.3).

• A Legendre-polynomial trunk network achieves reconstruction error bounded by Cp^-r, with uniformly bounded reconstruction and projection Lipschitz constants.The construction has trunk-network size O(p^3) and depth O(p log(p)), so the reconstruction error becomes small with moderate trunk dimension.Source: §4.1.3 · p. 41

4.1.4. Bounds on the approximation error (3.2).

• Holomorphic extension of the pendulum solution map enables a neural approximator whose approximation-error bound is independent of sensor and reconstruction dimensions.The extension is established for admissible poly-ellipses by controlling the imaginary part of the complexified ODE solution.Source: §4.1.4 · pp. 43–44

4.1.5. Bounds on the DeepONet approximation error (2.11).

• For the forced gravity pendulum, DeepONets achieve approximation error ε with sensor and trunk dimensions scaling as ε^-1/k and ε^-1/r, and can break the curse of dimensionality.For fixed k and r, the resulting complexity grows algebraically with the error tolerance ε; the implied constants depend on k and r.Source: §4.1.5 · p. 44
• The curse-breaking conclusion persists for algebraically decaying input coefficients, although the encoding error then decays algebraically with the number of sensors.The overall complexity remains polynomial in the error tolerance because the other error components retain their favorable scaling.Source: §4.1.5 · p. 45

4.2.1. Problem formulation.

• The section next formulates an elliptic PDE with variable diffusion coefficients on a periodic domain, using a coefficient measure supported in L∞ and coercivity assumptions.The assumptions are intended to ensure coercivity and uniqueness of the PDE solution.Source: §4.2.1 · p. 45
• The elliptic operator mapping a variable coefficient to the PDE solution is Lipschitz, under coercivity, and its random-field inputs have decaying Fourier coefficients ensuring smoothness.The coefficient model covers Darcy flow and variable-conductivity applications; the assumed coefficient decay ensures uniform coercivity and support in C∞(Td).Source: §4.2.1 · p. 46

4.2.2. Bounds on the encoding error (3.1).

• For elliptic inputs, Fourier encoding has exponentially decaying error because the coefficient Fourier coefficients decay exponentially.The bound’s constant depends on coefficient regularity, decay, and s > d/2, but not on the number of encoded modes m.Source: §4.2.2 · p. 47

4.2.3. Bounds on the reconstruction error (3.3).

• Elliptic reconstruction exploits solution smoothness to provide trunk-network error bounds, with reconstruction and projection Lipschitz constants bounded by 2.The result applies under smooth coefficients, smooth forcing, coercivity, and suitable integrability of the coefficient distribution.Source: §4.2.3 · p. 47

4.2.4. Bounds on the approximation error (3.2).

• The elliptic coefficient-to-solution map is holomorphic in the random parameters, enabling neural-network approximation bounds whose constants are independent of sensor count m and trunk size p.Holomorphy requires ε < 1 − ∥b∥ℓ1, while the projection norm remains bounded independently of m and p.Source: §4.2.4 · p. 48

4.2.5. Bounds on the DeepONet approximation error (2.11).

• Combining elliptic encoding, reconstruction, and approximation estimates yields DeepONets that break the curse of dimensionality, with sub-algebraic size growth in ε^-1 for d ∈ {1, 2, 3}.The sensor choice requires resolving the coefficient correlation scale, with m ≫ ℓ^-d|log(ε)|^d; efficiency for large d remains open.Source: §4.2.5 · pp. 48–49

4.3. Nonlinear parabolic PDE: A reaction-diffusion equation.

• For the Allen–Cahn equation, bounded initial data preserve the L∞ bound, regularity yields a Lipschitz solution map, and Fourier sampling and reconstruction admit error bounds.The mapping sends initial data to the solution at time T, with Lip(G) ≤ C = e4T under the stated regularity conditions.Source: §4.3 · pp. 49–50, §4.3.1 · p. 50, §4.3.2 · p. 51

4.3.3. Bounds on the approximation error (3.2).

• Generic neural-network approximation of the m-dimensional Allen–Cahn surrogate requires size O(pε^-1/m), and with m ∼ log(ε^-1) sensors this incurs the curse of dimensionality.The limitation arises because only Lipschitz continuity, rather than holomorphy, is established for the operator.Source: §4.3.3 · p. 51

4.3.4. A convergence finite difference scheme for the Allen-Cahn equation.

• An implicit-explicit finite-difference scheme for Allen–Cahn converges as Δt, Δx → 0 and preserves the bound ∥U^n∥ℓ∞ ≤ 1 under a stated timestep restriction.The scheme is introduced as a numerical method that neural networks may emulate to overcome the generic Lipschitz-approximation barrier.Source: §4.3.4 · p. 52
• For Allen–Cahn, neural networks emulate the finite-difference scheme with size O(n(m^2 + m|log(ε)|)) and depth O(1 + n|log ε|).The scalar nonlinearity itself can be approximated by a ReLU network whose size and depth grow logarithmically with accuracy.Source: §4.3.4 · p. 53

4.3.5. Bounds on the DeepONet approximation error (2.11).

• The Allen–Cahn DeepONet achieves error ε with m ≳ ε^-d and p ≳ ε^-d/4, yielding size ≲ ε^-6 for d = 2 and ε^-8 for d = 3.Thus the required network size grows algebraically in ε^-1, although higher-order numerical schemes could improve the explicit exponents.Source: §4.3.5 · p. 54, §4.3.4 · p. 54

4.4.1. Problem formulation.

• The scalar conservation-law operator maps L1 initial data to final-time solutions, is L1-contractive, and may produce discontinuities through shock formation.The approximation is formulated on BV-bounded inputs with entropy solutions and periodic boundary conditions in the one-dimensional prototype.Source: §4.4.1 · p. 55, §4.3.5 · p. 55

4.4.2. DeepONet approximation in the Banach space L1(D).

• In L1, local-average cell encoders permit discontinuous inputs, and neural networks emulate the Lax–Friedrichs scheme to construct DeepONets for scalar conservation laws.The encoder uses m equidistant cells, while the numerical emulation uses a neural-network approximated flux under a suitable CFL condition.Source: §4.4.2 · pp. 56–57
• For scalar conservation laws, error ε is achieved with p ∼ m and m ≳ ε^-1/α, giving total DeepONet size ∼ ε^-5 in the worst case α = 1/2 and ∼ ε^-2.5 when α ≈ 1.The resulting complexity is polynomial in ε^-1, so the construction breaks the curse of dimensionality.Source: §4.4.2 · p. 57

4.4.3. A lower bound on the DeepONet approximation error.

• For inviscid Burgers dynamics, Fourier spectral decay yields an L1 lower bound of order p^-1/2, so the upper rate α cannot exceed 1 for the constructed bounded-output DeepONets.Although the input measure is supported on a two-dimensional subspace and can have zero encoding error with two sensors, nonlinear dynamics produce a more complex push-forward measure.Source: §4.4.3 · pp. 58–60
• Applying the reconstruction-based argument to any reconstruction R yields the claimed lower bound on DeepONet approximation error.The proof begins by taking the support of µ and an arbitrary reconstruction, then concludes the lower bound for any such DeepONet.Source: §4.4.3 · p. 60
• The lower-bound proof uses spectral tail decay, with the eigenvalue sum satisfying k>p λ_k ≥ Cp^-1.The argument also invokes an interpolation inequality and Theorem 4.42.Source: §4.4.3 · p. 60
• For u ∈ BVM, both the target G(u) and DeepONet output N(u) are uniformly bounded in L∞(T) by M.The proof derives the target bound from ∥u∥L∞ ≤ M and assumes the same bound for N(u).Source: §4.4.3 · p. 60

5. On the generalization error for DeepONets

• Training approximates the DeepONet loss by sampling input functions and target-space evaluations, creating an empirical loss whose optimizer may differ from the population-loss optimizer.The empirical loss uses iid input samples from the underlying measure and sampled or quadrature-based target points.Source: §5 · pp. 60–61, §5.2 · p. 62

5.1. Deterministic choice of Y j

• For sufficiently regular outputs, deterministic quadrature can outperform random target sampling in low dimensions and can achieve exponentially decaying error when the loss integrand is complex analytic.The potential exponential decay is highlighted for suitable domains and quadrature rules, including toroidal domains with trapezoidal quadrature.Source: §5.1 · p. 61

5.2. Random choice of Uj, Y j

• Because the input function space is infinite dimensional, the analysis samples iid input functions and target points, making paired observations iid under the product distribution µ⊗Unif(U).The random construction uses N input samples and N target samples with Nu = N and Ny = 1.Source: §5.2 · p. 62
• Theorem 5.3 bounds the generalization error under boundedness and Lipschitz-continuity assumptions, with a constant independent of the weight bound B, parameter count dθ, and sample size N.The assumptions are stated to hold for the concrete operator examples considered in section 4.Source: §5.2 · p. 63
• Up to logarithmic factors, the generalization error scales with the Monte Carlo rate 1/√N, overcoming the curse of dimensionality for sampling from the infinite-dimensional input space.The bound is most useful in the underparametrized regime N ≫ dθ; dependence on parameter count and weight bounds can otherwise weaken or inflate the estimate.Source: §5.2 · pp. 63–64

6. Discussion

• DeepONets approximate measurable operators G:C(D)→L2(U) to arbitrary accuracy, without requiring continuity of G or compactness of subsets of the input space.The result extends the universal approximation theorem of Chen & Chen (1995) to the setting of an underlying measure µ.Source: §6 · p. 64
• The DeepONet error decomposes into encoding, finite-dimensional neural approximation, and affine reconstruction errors, enabling two-sided bounds tied to covariance-eigenvalue decay.Optimal reconstruction requires trunk nets to approximate covariance-operator eigenfunctions, while Sobolev regularity also yields reconstruction bounds.Source: §6 · pp. 64–65
• Random sensor locations achieve almost optimal encoding error, while standard finite-dimensional neural-network approximation results control the branch-network approximation error.A decoder construction establishes the sensor result, and holomorphic-map approximation theory is especially relevant when the induced operator map is analytic.Source: §6 · p. 65
• Across four differential-equation examples, DeepONets break the curse of dimensionality, with sub-algebraic size growth for the pendulum and elliptic PDE and polynomial growth for Allen–Cahn and scalar conservation laws.For scalar conservation laws, local-average encoders and an L1-Banach-space analysis handle discontinuous entropy solutions, and the upper bound is shown to be almost optimal.Source: §6 · pp. 65–66
• The generalization error decays, up to logarithmic factors, with the reciprocal square root of the number of input-function samples, overcoming infinite-dimensional dimensionality in this respect.The bound follows from covering-number estimates under general assumptions on the operator and branch and trunk networks.Source: §6 · p. 66
• Unlike PCA-autoencoder operator learning, DeepONets need no explicit covariance-operator eigenbasis because their trunk nets can implicitly learn a suitable output basis alongside the projected operator.The analysis is presented as more general than Bhattacharya et al. (2021), including curse-of-dimensionality mitigation.Source: §6 · p. 67
• The framework remains open to sharper complexity estimates, recurrent time-series extensions, additional PDEs such as Navier–Stokes, and multiple nonlinear operators.The Allen–Cahn complexity bound is specifically identified as needing sharper analysis, while scalar-conservation-law bounds are almost sharp.Source: §6 · pp. 67–68

Appendix A. Notation for Standard Fourier basis

• Appendix A defines a normalized real Fourier basis on T^d from cosine modes for k1≥0 and sine modes for k1<0, equivalent to the real and imaginary parts of the complex basis.It also indexes the basis by j∈N through an enumeration κ of Z^d ordered monotonically by |k|∞.Source: Appendix A · p. 73

Appendix B. On the definition of Error (2.11)

• This extension makes the error (2.11) well-defined on L2(D), even when the encoder is originally defined only on continuous functions.Source: Appendix B · p. 73
• A pointwise encoder on C(D) admits a Borel measurable extension to L2(D) that agrees with the original encoder on C(D) ∩ L2(D).The result applies to encoders evaluating u at finitely many points x1, ..., xm in D.Source: Appendix B · p. 73
• For a single sensor, the point evaluation is represented on continuous inputs as a lim sup of continuous functionals.The proof reduces the general finite-sensor case to m = 1.Source: Appendix B · p. 73
• Because the approximating functionals are continuous, their lim sup is Borel measurable on L2(D).Source: Appendix B · p. 73
• The measurable extension is obtained by assigning the finite lim-sup value when available and a fixed value for infinite outcomes.The construction distinguishes finite values from ±∞.Source: Appendix B · p. 73

Appendix C. Proofs of Results in Section 3

• The universal approximation proof in Appendix C uses a version of Lusin’s theorem specialized to probability measures on Polish spaces.The theorem is cited as Bogachev (2007), Thm. 7.1.13.Source: §C.1 · p. 74

C.1. Proof of Theorem 3.1.

• A ReLU clipping network keeps the approximating DeepONet bounded while remaining close to the identity on the relevant coefficient ball, completing the measurable universal-approximation argument.The clipping lemma provides approximation within ϵ/12 on inputs of norm at most M + ϵ/3 and output norm at most 2M.Source: §C.1 · pp. 74, 76
• Theorem 3.1 begins by clipping the measurable operator G at level M, with the clipping error made arbitrarily small by dominated convergence.The bounded operator GM is then approximated instead of G.Source: §C.1 · pp. 74–75
• Because C(D) and L2(U) are Polish spaces, Lusin’s theorem supplies a compact K where the clipped operator GM is continuous, outside a set of small measure.The image GM(K) is compact, enabling uniform finite-dimensional projection.Source: §C.1 · p. 75
• Projecting GM(K) onto finitely many continuous basis functions yields a continuous finite-dimensional operator that uniformly approximates GM on K.Compactness of GM(K) provides κ such that the projection error is below ϵ/6.Source: §C.1 · p. 75
• The continuous projected operator is approximated on K by a DeepONet using one hidden layer in each branch and trunk network.Output-layer transformations can impose orthonormal trunk functions without changing the represented DeepONet.Source: §C.1 · pp. 75–76

C.2. Proof of Theorem 3.3.

• Theorem 3.3’s upper estimate decomposes N − G into three terms representing approximation, reconstruction, and encoding contributions.The decomposition is N − G = [R ◦ A ◦ E − R ◦ P ◦ G] + [R ◦ P ◦ G − G] = T1 + T2 + T3.Source: §C.2 · p. 77
• The three components are bounded separately, using Jensen’s inequality for the fractional-power term and the push-forward definition for the reconstruction term.The proof therefore derives the upper estimate by combining termwise bounds.Source: §C.2 · pp. 77–78

• Minimizing the projection error is equivalent to maximizing the covariance energy captured by the candidate p-dimensional subspace.The objective can be expressed using the orthogonal projection onto the candidate subspace and the covariance eigenbasis.Source: §C.3 · pp. 78–79
• The covariance operator is compact and self-adjoint, so it admits a discrete eigendecomposition with ordered eigenvalues and orthogonal eigenspaces.The same spectral structure applies to the uncentered second-moment operator via a finite-rank mean correction.Source: §C.3 · p. 79
• Every minimizing p-dimensional subspace is characterized by including all eigenspaces above the cutoff eigenvalue and selecting the remaining dimensions within the cutoff eigenspace.The proof derives this structure from the monotone eigenvalue-weighting bound and its equality conditions.Source: §C.3 · pp. 80, 82, 83
• If the cutoff does not create an eigenspace-selection ambiguity, the optimal subspace is uniquely determined by the leading covariance eigenspaces.The corollary identifies the minimizing subspace through the direct sum of the first n eigenspaces and states its uniqueness.Source: §C.3 · p. 83

C.3.1. Proof of Theorem 3.10.

• The optimal p-dimensional subspace is characterized spectrally by covariance-operator eigenspaces, with projection error equal to the sum of eigenvalues beyond p.For distinct eigenvalues, the relevant eigenspaces determine the minimizing subspace; in the simple-eigenvalue case it is span(φ1, . . . , φp).Source: §C.3.1 · pp. 83–85
• Theorem 3.10’s affine minimizer is centered at E[v], because the centered residual is orthogonal to the associated vector subspace.The proof shows that a nonzero first term can be removed by recentering, while the second term averages to zero.Source: §C.3.1 · pp. 84–85
• The minimizing affine-subspace problem reduces to minimizing projection error over p-dimensional vector subspaces for the induced measure.This reduction permits application of the covariance spectral characterization.Source: §C.3.1 · p. 85

C.4. Proof of Lemma 3.13.

• Lemma 3.13 decomposes each function uniquely under the direct sum L2(U) = V ⊥⊕V and uses the dual basis of the sensor functions.The decomposition and dual-basis inner products isolate the component controlled by the chosen span.Source: §C.4 · p. 85
• The reconstruction norm is minimized when the sum over the p basis coefficients vanishes, which supplies the lemma’s stated condition.This follows directly from minimizing the norm of the final term in the decomposition.Source: §C.4 · p. 86

C.5. Proof of Proposition 3.15.

• Proposition 3.15 applies Theorem 3.12 to the affine image bV0 := Im(R) of the reconstruction map R = Rτ.The proof then obtains the proposition’s estimate by taking square roots and applying a trivial inequality.Source: §C.5 · p. 86

C.6. Proof of Lemma 3.16.

• The proof bounds the projection error through the triangle inequality and an expansion in the assumed orthonormal basis.Orthonormality permits the projection to be written in the relevant coordinate expansion before estimating each contribution.Source: §C.6 · p. 86
• Each perturbed basis function satisfies ||τ_k − eτ_k||_L2 ≤ 2ϵ under the lemma’s assumptions.This estimate is used to control the aggregate projection and reconstruction errors.Source: §C.6 · p. 87
• The projection approximation satisfies bE_R ≤ bE_Ṙ + Cϵ after bounding basis perturbations and their propagated error.The intermediate perturbation estimate scales as p^(3/2) max_j ||τ_j − eτ_j||_L2, yielding the final Cϵ control.Source: §C.6 · p. 87
• The proof derives the Lipschitz estimate for P by comparing arbitrary u and u′ and estimating the resulting sums using orthonormality.The reconstruction estimate is then obtained by an analogous argument.Source: §C.6 · p. 88

C.7. Proof of Lemma 3.17.

• Each real trigonometric basis element is associated with a frequency index satisfying |k|∞ ≤ N, where N is the smallest natural number with p ≤ (2N + 1)^n.This frequency representation provides the coordinate structure used in the neural-network construction.Source: §C.7 · p. 88
• For any ϵ > 0, the proof constructs a neural network Tϵ = (Tϵ,1, . . . , Tϵ,p) approximating the trigonometric coordinates, with log(N) ∼ 1/n log(p) = O(log(p)).The construction invokes Elbrächter et al. (2021, Theorem III.9) for the componentwise approximation and size bounds.Source: §C.7 · p. 89
• The network is rescaled as τ = Tϵ/p^(3/2) to satisfy the required estimate (3.30).The p^(3/2) normalization matches the scaling appearing in the projection error bound.Source: §C.6 · p. 87, §C.7 · p. 89

C.8. Proof of Proposition 3.19.

• A bounded coefficient sequence yields a Lipschitz map G : [0, 1] → Y with the prescribed covariance, and composing it with the Gaussian-to-uniform map gives the required map X → Y.The construction defines the final map as G ◦ P.Source: §C.8 · pp. 89–90
• For a non-degenerate Gaussian measure, an affine one-dimensional projection followed by the error function pushes the measure to the uniform distribution on [0, 1].The projection is standardized to N(0, 1), which the error function maps to the uniform measure.Source: §C.8 · p. 90

C.9. Proof of Proposition 3.20.

• For Proposition 3.20, the first p covariance eigenvectors define an affine input subspace whose linear image lies in an affine output subspace of dimension at most p.If necessary, the image subspace is extended to dimension p, after which R ◦ P is chosen as the orthogonal projection onto it.Source: §C.9 · pp. 90–91

C.10. Proof of Theorem 3.8.

• Theorem 3.8 reduces the optimization over decoder-generated subspaces to the span of the first m eigenfunctions of the uncentered covariance operator.The proof represents the linear decoder image as a subspace generated by at most m vectors and invokes Theorem 3.10 for the minimizing space.Source: §C.10 · p. 91

C.11. Proof of Lemma 3.22.

• Lemma 3.22 uses that D ◦ E is the identity on span(φ1, …, φm) and that the decoder image is contained in this span to simplify the reconstruction expression.The proof also represents a random variable with distribution µ through covariance eigenfunctions and corresponding random coefficients.Source: §C.11 · pp. 92–93

C.12. Proof of Lemma 3.23.

• Lemma 3.23 bounds the spectral norm of 1 − AM by first controlling its entries and then applying a Frobenius-norm estimate.Each matrix entry is handled using Hoeffding’s inequality, yielding a simultaneous entrywise concentration statement.Source: §C.12 · pp. 93–94

C.13. Proof of Theorem 3.9. √

• Theorem 3.9’s random-sensor proof selects M(m) = ⌈Cκm log(m)⌉ with κ = 4 and controls the random encoding error asymptotically.The construction uses iid sensors sampled as coordinate projections on an infinite product of uniform distributions.Source: §C.13 · p. 94

C.14. Proof of Lemma 3.25.

• Lemma 3.25 analyzes the discrete Fourier encoder and decoder, showing exact recovery of retained modes while higher modes alias onto the retained Fourier grid.The resulting encoding error separates orthogonal projection loss from aliasing contributions under the Gaussian covariance expansion.Source: §C.14 · pp. 96–97

C.13. Proof of Theorem 3.9. √

• Across the proofs, exact Fourier-mode recovery, covariance eigenfunction expansions, and random-sensor concentration provide the mechanisms linking encoder size to approximation error.The arguments respectively address aliasing, Gaussian spectral structure, and almost-sure random-sensor bounds.Source: §C.13 · p. 94, §C.14 · p. 96

C.15. Proof of Lemma 3.26.

• Lemma 3.26 applies the Fourier covariance eigenstructure and Theorem 3.9 to obtain a uniform sensor-based estimate with γ = 1/(20C^2).The constant C is independent of ℓ and M, while the intermediate implied constant depends only on e^γ and |D|.Source: §C.15 · p. 98

C.16. Proof of Proposition 3.27.

• Proposition 3.27 identifies the reconstruction term as a pseudo-spectral Fourier projection and invokes standard H^s(T^d) projection estimates for s > n/2.The argument uses bounded parameters Y_j ∈ [−1,1] and defines the reconstructed coefficients from the encoded function.Source: §C.16 · p. 98

C.17. Proof of Theorem 3.28.

• Theorem 3.28’s estimate is uniform in Y because the Fourier-coefficient decay bound is independent of N and Y ∈ [−1,1]^J.The proof combines the exponential coefficient decay with the relation between the Fourier cutoff N and sensor count m.Source: §C.17 · p. 99
• Theorem 3.28 obtains b_EE ≤ C exp(−cN) for parameterized functions with exponentially decaying Fourier coefficients, with N ≲ m^(1/d).The estimate follows from m = (2N + 1)^d and constants independent of N and Y.Source: §C.17 · p. 99

C.18. Proof of Corollary 3.34.

• C.18 proves that the target network is obtained by adding a linear output layer of size O(pN) to the network from Proposition 3.33.The target is a linear combination of the original network outputs with coefficients in R^p.Source: §C.18 · p. 99
• C.18 derives size(N) ≤ size(bN) + pN and depth(N) ≤ depth(bN) + 1.These bounds follow directly from the added output layer.Source: §C.18 · p. 99
• C.18 transfers the claimed approximation-error estimate from Proposition 3.33 to the constructed network N.The proof applies the preceding bound for the projected composition on sensor values y ∈ [−1, 1]^J.Source: §C.18 · p. 99

C.19. Proof of Theorem 3.35.

• C.19 uses a composition lemma stating that two neural networks N1 and N2 can be represented jointly as a network for N2 ◦ N1.This composition result is the main neural-network calculus tool used in the theorem proof.Source: §C.19 · p. 100
• The resulting network satisfies depth(N) ≤ C(1 + log(m) log log(m))), with C > 0 independent of m and p.The size bound is obtained from Corollary 3.34 together with the composition construction.Source: §C.19 · p. 100
• C.19 composes the network for u ↦ bYκ(u) with the network N to represent u ↦ A(u) = (N ◦ bYκ)(u), with constants independent of m and p.The independence from p follows because bYκ does not involve the projection P.Source: §C.19 · p. 100
• C.19 bounds the approximation error bEA using Corollary 3.34 and verifies the required summability because exponential decay implies αk ∈ ℓq(Zd) for every q ∈ (0, 1).The proof first states an approximation-error estimate and then applies the exponential-decay assumption (3.52).Source: §C.19 · p. 100

Appendix D. DeepONet approximation of linear operators

• The numerical linear-functional example exhibits exponential test-error decay as the number of random sensors increases, with double-descent resonances when sensor and training-sample counts coincide.The experiment uses Gaussian random-field inputs, uniformly sampled sensors, and mean-square training loss.Source: Appendix D · pp. 100–101
• Theorem D.2 bounds DeepONet approximation of bounded linear operators with random sensors, optimal covariance-based reconstruction, and a shallow ReLU branch approximator, with probability one over sensor locations.The linear approximation error vanishes because the finite-dimensional operator can be represented exactly by an affine map, while encoding and reconstruction errors determine the estimate.Source: Appendix D · pp. 101–102
• For the periodic elliptic operator with Gaussian input spectrum decaying exponentially in Fourier modes, accuracy ϵ is achieved with p ∼log(ϵ^-1)^d and m ∼log(ϵ^-1)^{d(1+σ)}.The trunk and branch network size bounds depend on the covariance length scale ℓ, dimension d, and auxiliary parameter σ > 0.Source: Appendix D · pp. 103–104

E.2. Proof of Lemma 4.3.

• The proof of Lemma 4.3 uses induction and Gronwall’s inequality to bound every time derivative of the nonlinear ODE solution by a constant depending on derivative order, nonlinearity, and final time.The resulting bound is uniform in the forcing input.Source: §E.2 · p. 105

E.3. Proof of Lemma 4.4.

• Legendre-polynomial projection and the ODE regularity estimate control the reconstruction error in Lemma 4.4, with a constant independent of the projection dimension p.The reconstruction corresponds to the optimal projection associated with the Legendre trunk functions.Source: §E.3 · p. 106

E.4.–E.6. Proofs of Lemmas 4.6–4.8.

• For the pendulum ODE, sufficiently small imaginary forcing keeps the imaginary solution component below one, which prevents finite-time blow-up and yields global existence on [0,T].The subsequent complex-parameter argument establishes boundedness and holomorphic dependence of the solution on each parameter.Source: §E.4 · pp. 106–107, §E.5 · p. 107

E.8.–E.9. Proofs of Lemma 4.16 and Theorem 4.22.

• The elliptic regularity proof differentiates the variable-coefficient equation, applies energy estimates and induction, and obtains Sobolev bounds whose constants depend explicitly on coefficient norms and ellipticity parameters.The supplied proof also invokes standard parabolic Hölder-space and Schauder estimates for the later parabolic theorem.Source: §E.8 · pp. 108–109, §E.9 · p. 110

E.9. Proof of Theorem 4.22.

• Theorem 4.22’s proof bootstraps parabolic estimates to show that C4,α initial data produce a solution in C4,α(U), with a bound depending on the initial-data norm.The argument combines an L∞ bound, strong Lp regularity, Sobolev embedding, and Schauder estimates.Source: §E.9 · p. 111

E.10. Proof of Corollary 4.23.

• The Corollary 4.23 proof studies the difference of two bounded solutions, whose equation has a uniformly bounded coefficient F(v,v′), and applies an energy estimate.The difference is w = v − v′, and the coefficient satisfies |F(v,v′)| ≤ 4.Source: §E.10 · p. 111

E.11. Proof of Theorem 4.27.

• Theorem 4.27 uses a local truncation-error estimate in the parabolic norm C(2,4), which is less restrictive than the larger norm used by Tang & Yang (2016).The remainder estimate is based on the available parabolic a priori bounds, while the rest of the proof follows Tang & Yang (2016, Theorem 4.1).Source: §E.11 · pp. 111–112

E.12. Proof of Lemma 4.28.

• Lemma 4.28 constructs ReLU networks approximating multiplication and the cubic nonlinearity η3 − η, with size growing logarithmically in 1/ϵ.The construction uses Yarotsky’s square approximation and composition; with smooth activations, the corresponding network size can instead remain uniformly bounded in ϵ.Source: §E.12 · pp. 112–113

E.13. Proof of Lemma 4.29.

• Lemma 4.29 represents one numerical time step by a ReLU network of size O(m2 + m|log ϵ|) and depth O(|log ϵ|).The construction combines the nonlinear approximation with the fixed matrix-vector map R∆x,∆t, whose layer has size O(m2).Source: §E.13 · pp. 113–114

E.14. Proof of Lemma 4.30.

• Lemma 4.30 controls the accumulated numerical-scheme error by using the boundedness of U^k and R∆x,∆t, then summing the error recurrence and applying Gronwall’s inequality.The proof defines Ek = ∥U^k − eU^k∥ℓ∞ and uses E0 = 0.Source: §E.14 · p. 114

E.15. Proof of Proposition 4.31.

• The proof constructs A by composing the neural approximation of the convergent numerical scheme with local interpolation and the projection operator P.With bL = P ◦ L, the approximator is A = bL ◦ bN = P ◦ L ◦ bN.Source: §E.15 · pp. 114–115
• Choosing equidistant-grid parameters n = ⌈Tm2/d⌉ and ϵ = m−2/d yields the neural network approximation used in the construction.The grid relation ∆x ∼ m−2/d determines these choices.Source: §E.15 · p. 115
• Local linear interpolation recovers grid values exactly, with L(v)(xj) = vj, and projection gives a linear map bL from Rm to Rp.This linear structure allows the composed approximator to be represented as a neural network while preserving the size bound of bN.Source: §E.15 · p. 115
• The proof bounds the reconstruction and discretization errors by combining Lipschitz estimates, the numerical-scheme error, and the shared discrete initial data from E ◦ D = Id.The resulting comparison includes ∥G(u) − G ◦ D ◦ E(u)∥L∞ ≤ C m−1/d plus the grid-point discrepancy term.Source: §E.15 · p. 116
• The proposition concludes that an approximator network A : Rm → Rp exists with p = m and the stated approximation estimate.The constants are controlled uniformly over supp(µ) through the relevant regularity norms.Source: §E.15 · pp. 115–116

E.16. Proof of Lemma 4.37.

• For Lemma 4.37, a ReLU combination defines the interval function [a,b] and is linear near both endpoints.The construction assumes ϵ < b − a and uses four shifted ReLU terms.Source: §E.16 · p. 116
• The interval function [a,b] is represented by the same neural network architecture for every choice of a, b, and ϵ.Thus, the architecture is parameter-independent even though its parameters vary with the interval specification.Source: §E.16 · p. 116

E.17. Proof of Theorem 4.38.

• Theorem 4.38 is proved by constructing trunk and branch networks from an auxiliary neural network and approximations of indicator functions.The construction sets unused components to zero and defines the trunk through approximations τ_j of 1_Cj.Source: §E.17 · p. 117
• The resulting approximation error combines a truncation contribution Cm^-α with indicator-approximation terms controlled by the choice of τ_j.The proof separately estimates the first term, the terms for j ≤ m, and any remaining terms for j > p.Source: §E.17 · p. 117
• The coefficient outputs remain uniformly bounded, with |N_j(E(u))| ≤ M for all u ∈ BVM, enabling control of the approximation terms.This bound is invoked from Theorem 4.35 in the proof’s error estimate.Source: §E.17 · p. 117
• There exists a constant C independent of p and m such that, for every ε > 0, a DeepONet satisfies the stated size and error bounds.The proof chooses ε = m^-αp^-1 and concludes the claimed L1 estimate using the trunk and branch complexity bounds.Source: §E.17 · p. 118

E.18. Proof of Lemma 4.41.

• Lemma 4.41 represents the covariance operator ΓG#µ through a kernel obtained from the translated solution and a change of variables.The translated initial data produce a translated solution, after which the relevant expression is written as a convolution.Source: §E.18 · pp. 118–119
• The covariance kernel is stationary, k(x, x′) = g(x−x′), so its eigenfunctions are the Fourier basis {e_k}_{k∈Z}.Stationarity supplies the Fourier eigenfunctions, while the corresponding eigenvalues are identified through the Fourier coefficients of g.Source: §E.18 · p. 119

Auxiliary lemmas

• The auxiliary lemmas control approximation over parameter space through ℓ∞ coverings, covering numbers, and moment estimates for maxima of random variables.Lemma F.2 uses an ε-net, Lemma F.3 bounds expectations by K times the largest moment, and Lemmas F.4–F.5 combine these ingredients.Source: Appendix F · pp. 119–121

Auxiliary estimates

• A standard covering-number estimate supplies a bound with a constant independent of ε, B, and d.The appendix cites Welti (2020) for the covering-number fact and uses it in the subsequent estimates.Source: Appendix F · p. 121

Probabilistic estimates

• Gaussian-tail and Lp estimates for Ψ yield the remaining probabilistic control needed in the theorem’s proof.The appendix derives Lp bounds as a function of p and invokes Gaussian-measure tail behavior to control the relevant quantities.Source: Appendix F · pp. 122–123

Proof of Theorem 5.3

• Theorem 5.3 follows by combining Lemmas F.5 and F.6 with assumptions on Ψ and Φ to derive the stated generalization-error bound.The proof explicitly applies the two lemmas for p ≥ 2 and uses constants κ and C controlling Ψ and Φ.Source: Appendix F · pp. 122–123
• The final constant can be written C = C(µ, Ψ, Φ), independent of dθ, B, and N, under the stated assumptions.The proof identifies dependence on µ, Ψ, and Φ while excluding dependence on the parameter dimension, domain bound, and sample size.Source: Appendix F · p. 123