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Quantitative approximation theorem of neural operators for the Dirichlet-to-Neumann map

Pu-Yun Kow, Andrea Aspri, Elisa Francini, Jenn-Nan Wang

arXiv:2610.35404v1math.APcs.NAmath.NA

TL;DR

Repeatedly evaluating the conductivity-to-DN-map forward problem can be computationally expensive in conductivity reconstruction, while general neural operators face a curse of parametric complexity. The paper constructs a neural-operator surrogate and derives quantitative complexity bounds for the DN map, reporting that this setting avoids that curse under suitable conditions.

  • Problem

    In Ensemble Kalman Inversion, each prediction can require solving a PDE for an ensemble member to evaluate the conductivity-to-DN-map forward map, which can be computationally expensive.

    Source: §4 · p. 20

  • Method

    The paper approximates a finite-dimensional projection of the DN map with neural operators and uses iterative conductivity-equation solutions to control construction error.

    Source: §2.1 · p. 5

  • Results

    The authors report that their DN-map neural operators avoid the general curse of parametric complexity under suitable conditions.

    Source: §1 · p. 2

  • Takeaways & Limitations

    In the examples considered, replacing the exact forward operator with a learned surrogate in Ensemble Kalman Inversion maintains reconstruction accuracy while reducing computational cost.

    Source: §4 · p. 20

    The coefficient construction assumes γ∈W2,∞(B) and positivity, so its stated setting requires this stronger regularity.

    Source: §2.2 · p. 7

Abstract

from arXiv · show

In this paper, we study neural operators for the Dirichet-to-Neumann map (DN map), which is encountered in the famous Calder\'on's problem. In many numerical methods used for reconstructing the unknown conductivity from the knowledge of the DN map need to evaluate the corresponding DN map at a given conductivity function, which involves solving the boundary value problem for the conductivity equation and taking the Neumann data. Such procedure is time consuming and not effective. Hence, it is a favorable strategy to construct a surrogate to replace the forward map mapping conductivity functions to corresponding DN maps. The main theme of this paper is to derive quantitative bounds on the parametric complexity of these neural operators. In particular, under suitable assumptions, their depth grows at most logarithmically with $\varepsilon^{-1}$, while the relevant spectral and projection ranks grow algebraically. It should be noted that general neural operators suffer from "the curse of parametric dimensionality". This work demonstrates the construction of neural operators for the DN map that does not suffer from such curse.

1. Introduction

The paper studies neural operators for the forward conductivity-to-DN-map problem, targeting quantitative parameter bounds that avoid the general curse of parametric complexity under suitable conditions. Its approach uses finite-dimensional projection and iterative construction, with the DN map’s Sobolev-space output requiring special care.

1. Introduction

  • The work aims to avoid the curse of parametric complexity for DN-map neural operators under suitable conditions.General neural operators characterized by regularity can require exponentially growing parameter counts as accuracy increases.Source: §1 · p. 2

1.1. Related works.

  • Prior quantitative approximation results cover several PDE operator-learning settings, while prior Calderón-map DeepONet results are qualitative or address a different map.The paper focuses on quantitative parametric-complexity bounds for approximating the forward map γ → Λγ.Source: §1.1 · p. 3, §1.2 · p. 4

1.2. Our results and contributions.

  • The target maps conductivity γ to its DN map, whose outputs act from H1/2(∂Ω) to H−1/2(∂Ω) and are distributions, not pointwise boundary values.The construction is independent of the discretization or mesh used to represent γ and the Dirichlet data.Source: §1.2 · p. 3
  • The proof projects the DN map onto a finite-dimensional subspace and approximates that reduced model, then uses iterative conductivity-equation solutions to control construction error and bound network depth.Projection error can be estimated in terms of projection rank.Source: §1.2 · pp. 3–4
  • The constructed operators are discretization-invariant and are applied with Ensemble Kalman Inversion to conductivity reconstruction from DN-map data.The paper also distinguishes its neural-operator surrogate focus from analytical work on conductivity reconstruction using finitely many measurements.Source: §1.2 · p. 4

2. The Dirichlet-to-Neumann map

The section reduces the normalized DN map to a finite-dimensional projection and then approximates its coefficients with neural operators. It also develops iterative solutions under conditions that support control of the approximation error.

2. The Dirichlet-to-Neumann map

  • The DN map is formulated through the conductivity boundary-value problem, with the conductivity-to-DN map as the paper’s target.The discussion assumes the conductivity perturbation is supported inside the ball.Source: §2 · pp. 4–5

2.1. Low rank approximation.

  • The normalized DN map is projected onto a finite-dimensional subspace, with the projection incurring an error that can be bounded in terms of its rank.The section then targets neural-operator approximation of this reduced model.Source: §2.1 · pp. 5–6, §2.2 · p. 7

2.2. Coefficients of low rank approximation.

  • For the coefficient approximation in Proposition 1, J = O(ε^-1/2) under the stated regularity and dimension assumptions.The proposition assumes d = 2 or 3 and s > 1 + d/2.Source: §2.2 · p. 9

2.3. Iterative solutions.

  • The section proposes solving the transformed equation by Picard iteration and states that the resulting series converges in H2(B) to its unique solution.The iterative-solution analysis also gives a truncation rank N = O(ε^-2d/t) for fixed t in the stated range.Source: §2.3 · pp. 11, 13

3. Neural operators and parametric complexity

The neural-operator construction is organized around the analytical structure of the DN-map approximation. It combines input lifting, hidden transformations, and a projection that assembles truncated coefficients.

3. Neural operators and parametric complexity

  • On the specified conductivity class, the operators Ψq and Ψq,N have L2(B) operator norms at most η0, uniformly over γ and N.This uniform bound lets the constants in Proposition 4 depend on η0 and the a priori parameters, but not on the individual conductivity.Source: §3 · p. 15
  • The architecture has three components: a lifting step, hidden layers, and a final projection assembling truncated coefficients.The hidden layers combine pointwise transformations with non-local integral operators.Source: §3 · p. 15
  • In the lifting step, v(0)_j is defined as in (12), γ1 = γ^1/2 and γ2 = γ^-1/2 are defined, and Δγ1 is evaluated using AutoDiff.The hidden layer implicitly carries j, γ1, γ2, and their derivatives.Source: §3 · pp. 15–16
  • The projection assembles the output from two parts, including the truncated coefficient terms.The construction treats the coefficient integrals as learnable parameters, while the ranks can be chosen from the desired accuracy.Source: §3 · p. 16
  • With the rectified quadratic unit, the neural-operator depth can be estimated by the iteration count used in Proposition 4.The paper distinguishes neural-operator depth from the number of iteration steps.Source: §3 · p. 16

1 V (2) be the output layer, then

The quantitative approximation theorem bounds neural-operator complexity for the DN map: depth grows logarithmically with ε^-1, while the ranks grow algebraically under the stated assumptions.

Polynomial representation

  • A two-hidden-layer network with 8 neurons in the first layer and 4 in the second exactly represents the three-variable polynomial illustrated in Figure 2.Source: §1 · p. 17, Figure 2 · p. 18

Theorem 7

  • The theorem assumes d = 2 or 3, r > 0, s > 2 + d/2, and a fixed η0 in (0, 1).The estimates use the same ranks and iteration depth across the specified conductivity class.Source: §1 · p. 19
  • For fixed t in (0, 2 − d/2), the ranks satisfy N1 = O(ε−2d/t), N2 = O(ε−κ1), N3 = O(ε−κ2), and J = O(ε−1/2).The implied constants are uniform over the stated conductivity class and may depend on fixed a priori parameters and t.Source: §1 · p. 19
  • The ReQU realization uses at most 2L + 1 hidden layers, and the theorem gives logarithmic depth growth in ε^-1 for the target accuracy.The channel width per branch is bounded independently of ε.Source: §1 · p. 20

4. Application to the inverse problem

The section applies a learned neural-operator surrogate within Ensemble Kalman Inversion to reconstruct conductivity from normalized DN-map data, avoiding repeated costly forward solves in the examples considered.

EnKI reconstruction

  • Replacing the exact forward map with the learned surrogate reduces computational cost in EnKI, and the reported examples maintain reconstruction accuracy.The authors do not establish a theoretical bound on how forward-approximation error propagates to the reconstructed conductivity.Source: §4 · p. 20

Bayesian formulation

  • The inverse problem uses a K × K coefficient-matrix truncation of the DN map and models observations as noisy, with mean-zero Gaussian noise of known covariance.The observation space is Y := R^K×K, and the setup considers smooth conductivity.Source: §4 · pp. 20–21

Conductivity parameterization

  • A localized Gaussian random field and smooth cutoff construct conductivity fields that are strictly positive and whose perturbations from 1 are compactly supported within B.The cutoff equals 1 on B′ and 0 outside B′′, with B′ ⊂ B′′ ⊂ B.Source: §4 · p. 21

EnKI algorithm

  • EnKI iteratively updates an initial ensemble through prediction and analysis steps, then uses the mean parameter estimate to form γk+1 = exp(¯σk+1).The stated ensemble updates remain in the span of the initial parameter samples.Source: §4 · pp. 21–22

5. Numerical simulations

The simulations evaluate a neural-operator surrogate for predicting DN-map projection coefficients and for use in EnKI conductivity reconstruction. The surrogate’s coefficient predictions are stable in the mid-value range, while ML-assisted EnKI achieves lower MAE than the direct PDE toolbox on testing samples.

5.1. Simulation of the DN map.

  • The dataset contains 4,000 samples, split into 3,200 training and 800 testing samples, with closely aligned training and testing coefficient statistics.The reported means are 353.43 and 351.90, and standard deviations are 45.36 and 46.75, respectively.Source: §5.1 · p. 22
  • A learning rate of 0.01 gives the lowest reported MAPE, 11.11%, while a three-hidden-layer model gives the lowest reported MAPE across tested depths, 10.54%.The four-layer model performs worse, which the authors suggest may reflect overfitting on the 4,000-sample dataset.Source: §5.1 · pp. 23–24, Figure 3 · p. 23
  • For harmonic inputs k = 1, 2, 3, voltage profiles show radial symmetry, while current variations are larger near the boundary and increase for higher-frequency inputs.The authors connect radial symmetry and projection-map smoothness to avoiding overfitting and improving training efficiency.Source: §5.1 · p. 24, Figure 4 · p. 25, Figure 5 · p. 26
  • For projection coefficients, predictions consistently underestimate ground truth, but errors are more manageable at lower target values.For the cited lower-target examples, MAE ranges from 10.48 to 25.33 and MAPE from 2.89% to 6.72%; overall, estimates are stable and reasonably accurate in the mid-value range.Source: §5.1 · p. 24, Table 1 · p. 25

5.2. Simulation of the inverse problem.

  • For one testing sample, ML evaluation gives 80.82% MAPE versus 88.50% with the PDE toolbox.The authors note that this modest single-sample reduction is consistent with the gap in Table 2; they attribute the large MAPE to using only K = 3 coefficients and expect more observations to reduce it.Source: §5.2 · p. 26, Figure 6 · p. 27
  • ML-assisted EnKI consistently achieves lower MAE than the direct PDE toolbox on testing samples.The authors attribute the improvement likely to smoother ML-generated outputs suppressing discretization noise and providing more stable observational data.Source: §5.2 · p. 27, Table 2 · p. 27

Appendix A. Harmonic functions

Appendix A describes boundary Laplace–Beltrami eigenfunctions and interior Dirichlet eigenfunctions on balls, then uses their indices and harmonicity to organize estimates.

Appendix A. Harmonic functions

  • On the unit sphere in R^3, the Laplace–Beltrami eigenfunctions have eigenvalue −ℓ(ℓ+1); on the circle, the corresponding eigenvalue is −ℓ².The boundary eigenfunctions form an orthonormal basis of L2(∂B) for the circle case.Source: Appendix A · p. 28
  • For the sphere, double indices (ℓ, m) are flattened as j = ℓ² + (ℓ + m), while in R2 the degree scales as ℓ ∼ j^(1/(d−1)).The R2 indexing distinguishes cosine and sine terms.Source: Appendix A · p. 28
  • The harmonic functions are written in spherical or polar coordinates, and their gradient estimates use harmonicity, Green’s identity, and bounds depending only on degree.The identity also applies when ℓ = 0 and w_j is constant.Source: Appendix A · pp. 29–30
  • In R2, normalized Dirichlet eigenfunctions on the ball are expressed using Bessel functions, with eigenvalues associated to the positive zeros of J_k.The zeros satisfy J_k(j_kl) = 0.Source: Appendix A · p. 30

Appendix B. Discretization-invariance property

Appendix B establishes discretization invariance for the neural-operator architecture under compactness assumptions, using convergence of the discretized integral operators.

Appendix B. Discretization-invariance property

  • The architecture in Definition 5 is discretization-invariant for conductivity functions in a compact set with respect to the W^(2,∞)(B) norm.A uniformly W^(3,∞)(B)-bounded subset is given as an example.Source: Appendix B · p. 31
  • The result allows γ to be represented by discretizations such as finite-element or spectral projections.The discretized versions are considered for γ in the specified compact set.Source: Appendix B · p. 31
  • The proof reduces convergence to the integral operators and uses q(L) → q in L∞(B) as L → ∞.Here q(L) is defined from the discretized conductivity and its Laplacian.Source: Appendix B · p. 31
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