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REVIEW 4 major objections 3 minor 57 references

Imaging Anisotropic Conductivity from Internal Measurements with Mixed Least-Squares Deep Neural Networks

T0 review · 4 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper develops a mixed least-squares deep neural network that approximates the internal flux and the anisotropic conductivity tensor simultaneously from internal measurements of the solution, and proves error bounds of order…

desk verdict Solid anisotropic extension of the MLS-DNN method with convincing numerics, but the proof of the main error bound has a real gap and the theory only covers global minimizers, so the theorem is unproven as written. read the letter →

arxiv 2412.00527 v1 pith:2GGQK6HD submitted 2024-11-30 math.NA cs.NA

Siyu Cen , Bangti Jin , Xiyao Li , Zhi Zhou This is my paper · ORCID
classification math.NAcs.NA MSC 35R3065N2165N3068T07
keywords anisotropicconductivityinverseproblemdeepneuralnetworkmixedleast-squaresinternalmeasurementserrorestimateellipticPDEimaging
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that an anisotropic conductivity tensor in an elliptic boundary value problem can be recovered from internal measurements of the solution by training two deep neural networks --- one for the conductivity and one for the internal flux --- against a mixed least-squares loss that enforces the first-order system. If correct, this gives a practical, mesh-free route to anisotropic conductivity imaging in two and three dimensions without paired training data, and the analysis supplies error bounds for this DNN formulation. The authors prove an $O(\delta^{1/2})$ bound on the $L^2$ reconstruction error under a projected source condition, for both the population loss and its Monte Carlo empirical version, and support the claim with experiments in 2D and 3D, for Neumann and Dirichlet problems, full and partial internal data, and up to 10 percent noise.

What carries the argument

The load-bearing object is the mixed least-squares loss built from the first-order system $\sigma_i=A\nabla u_i$, $-\nabla\cdot\sigma_i=f_i$, with the relevant boundary condition, so that the conductivity and the flux are both unknowns fitted to the internal measurements. The loss uses a matrix-valued projection $P_K$, applied pointwise by eigenvalue decomposition and clipping eigenvalues to $[c_0,c_1]$, to keep $A_\theta$ inside the admissible set of uniformly elliptic symmetric matrices; the projection also enters the error analysis through the variational inequality that characterizes it. The convergence proof combines three tools: DNN approximation rates for $\tanh$ networks (giving the representation errors $\epsilon_A,\epsilon_\sigma$), the projected source condition $A^\dagger=P_K(\nabla u^\dagger\otimes\nabla\psi)$ that turns the $L^2$ error into computable residuals, and Rademacher-complexity bounds for the Monte Carlo empirical loss.

What would settle it

Choose a smooth full anisotropic tensor $A^\dagger$ that does not satisfy the projected source condition $A^\dagger=P_K(\nabla u^\dagger\otimes\nabla\psi)$ and run the MLS-DNN with exact data, taking $\gamma_A\to 0$ and large DNN capacity. If the empirical $L^2$ error $e(\hat A)$ stops decreasing at a positive plateau rather than following the predicted $O(\delta^{1/2})$ rate, the bound's premise --- a global minimizer plus the source condition --- is not what governs the reported accuracy.

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Extended reading notes

Core claim

The central claim is that the mixed least-squares deep neural network recovers the anisotropic conductivity tensor $A^\dagger$ by minimizing the loss $J_\gamma(\theta,\kappa) = \sum_i(\|\sigma_{i,\kappa}-P_K(A_\theta)\nabla z_i^\delta\|^2_{L^2(\Omega)^d} + \gamma_\sigma\|\nabla\cdot\sigma_{i,\kappa}+f_i\|^2_{L^2(\Omega)} + \gamma_b\|n\cdot\sigma_{i,\kappa}-g_i\|^2_{L^2(\partial\Omega)} + \gamma_A\|P_K(A_\theta)\|^2_{L^2(\Omega)^{d,d}})$ for the Neumann problem, with a Dirichlet analogue, where $P_K$ is the orthogonal projection onto the admissible set of uniformly elliptic symmetric matrices. Theorem 3.1 gives $\|A^\dagger - P_K(A^*_\theta)\|_{L^2(\Omega)^{d,d}} \le c(\gamma_A^{-1}\eta^2+\eta+(1+\gamma_\sigma^{-1}+\gamma_b^{-1})\gamma_A)^{1/2}$ with $\eta$ combining DNN approximation error, noise level $\delta$, and the penalties, and Theorem 3.2 extends the bound to the empirical loss with Monte Carlo quadrature error terms. With the parameter choice $\gamma_A\sim\delta$, $\epsilon_A=\epsilon_\sigma\sim\delta$, the rate is $O(\delta^{1/2})$, comparable to output least-squares Galerkin methods. The experiments demonstrate reconstruction of full matrix-valued conductivities in 2D and 3D with data noise up to 10 percent, including from partial internal data.

Load-bearing premise

The error bounds apply only to an exact global minimizer of the nonconvex training loss, but the actual computations use a gradient optimizer that may stop at a local minimum; the paper states that optimization errors can be pronounced and that for exact data the reconstruction cannot be made arbitrarily accurate.

Editorial extensions

If this is right

  • An anisotropic conductivity tensor in 2D and 3D can be reconstructed from internal solution measurements without labeled training pairs, using an unsupervised DNN that fits flux and conductivity simultaneously.
  • With $\gamma_A\sim\delta$ and representation errors of order $\delta$, the reconstruction error in $L^2(\Omega)^{d,d}$ is of order $\delta^{1/2}$, the same rate as classical output least-squares Galerkin methods.
  • The method handles partial internal data: reconstructions remain accurate over the whole domain even when measurements are confined to a boundary band, where the paper's FEM comparison shows poor stability.
  • The mesh-free DNN formulation makes 3D implementation nearly identical to 2D, and the paper demonstrates 3D anisotropic reconstructions at 5 percent noise.
  • For both Neumann and Dirichlet problems, the reconstruction degrades only mildly at 10 percent noise, while the PINN variant requires boundary knowledge $A^\dagger|_{\partial\Omega}$ and second-order derivatives.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension would run the same scheme on conductivities that violate the projected source condition; the theory makes no prediction for them, so a numerical rate study would show whether the method's practical accuracy outruns the proved bound.
  • The reported insensitivity to the penalty parameter $\gamma_A$ suggests the DNN's implicit spectral bias is doing regularization work that the analysis assigns to the explicit $L^2$ penalty; separating these effects would require measuring the loss gap between the found local minimizer and the global one.
  • The same mixed least-squares structure could be adapted to power-density data $A\nabla u_i\cdot\nabla u_j$ or to time-harmonic problems, where internal data are phase-sensitive; the projected source condition would need re-derivation for those settings.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 3 minor

Summary. The paper develops a deep-learning approach (MLS-DNN) for recovering an anisotropic conductivity tensor A from internal measurements of solutions to an elliptic boundary value problem. The method is based on a mixed least-squares formulation in which the flux sigma and the conductivity A are represented by tanh neural networks, with the constraint A in K enforced by a matrix-valued projection. The authors prove error estimates for the L2 reconstruction error of the projected DNN conductivity for global minimizers of both the population loss (Theorem 3.1) and the empirical loss (Theorem 3.2), assuming a projected source condition, DNN approximation rates, and Rademacher-type statistical bounds. They also present numerical experiments for Neumann and Dirichlet problems in 2D and 3D, with full or partial data and up to 10% noise, comparing with FEM and PINN.

Significance. If the theoretical results were fully rigorous and connected to the implemented algorithm, the paper would offer an important extension of the Kohn-Lowe mixed formulation and of the isotropic MLS-DNN framework to anisotropic problems, with the first numerical demonstrations in 3D. The numerical study is genuinely useful: it shows stable recovery for several nontrivial anisotropic examples, including partial interior data, and the authors state that code will be released. The main caveats are that the error bounds apply only to exact global minimizers, which the ADAM-trained networks need not be, and that one step in the proof of Theorem 3.1 rests on an equality that is not implied by the source condition. These issues limit the theoretical support for the headline claim of accurate recovery up to 10% noise, although the empirical evidence stands independently.

major comments (4)
  1. [Section 3.1, proof of Theorem 3.1] The displayed equality (A†, A†−PK(A*θ)) = ((A†−PK(A*θ))∇u†, ∇ψ) is not an identity under Assumption 3.2; the assumption gives A† = PK(∇u†⊗∇ψ), not A† = ∇u†⊗∇ψ. From the projection inequality (2.4) one only obtains the inequality (A†, A†−PK(A*θ)) ≤ ((A†−PK(A*θ))∇u†, ∇ψ). Since the subsequent argument bounds the absolute value, replacing the equality by this inequality appears to repair the proof, but the current text contains a false equality and should be corrected.
  2. [Section 3 (Theorems 3.1 and 3.2) and Section 4] The theorems bound the reconstruction error of global minimizers of the population and empirical losses, but the reported reconstructions are obtained by ADAM on the nonconvex empirical loss. The paper itself states in Section 4 that 'the optimizer may get trapped in a local minimum, leading to pronounced optimization errors' and that for exact data 'the reconstruction cannot be made arbitrarily accurate.' Consequently, the O(δ^{1/2}) error bound is not established for the computed solutions, and the numerical results do not verify the theorem quantitatively. The authors should either add an optimization-error estimate or an explicit condition under which the ADAM solution is close to a global minimizer, or clearly separate the theoretical claim (for global minimizers) from the numerical demonstration (which is then purely empirical).
  3. [Section 4, noise generation] The theoretical noise model in (2.3) and (2.11) measures δ = ∥u† − zδ∥ in H^1(Ω) or H^{3/2}(Ω), while the experiments add i.i.d. Gaussian noise pointwise to ∇u† (Section 4: 'i.i.d. Gaussian noise is added pointwise to ∇u†'). A pointwise-perturbed gradient is generally not in H^1(Ω), so the experimental data do not satisfy the regularity assumptions of Theorem 3.2. If the noisy gradient is meant to be a smoothed version of raw data, the smoothing procedure and its effect on δ should be described; otherwise the numerical experiments do not directly test the noise-dependence predicted by the theory.
  4. [Lemma 3.2] Lemma 3.2 is central to the empirical-loss bound, but its proof is omitted and is referred to [30, Theorem 3.4]. The loss here involves the anisotropic projection PK(Aθ) and additional boundary terms, so it is not immediately obvious that the argument transfers verbatim. A proof or a precise statement of the required changes should be included, rather than delegating to the authors' previous paper.
minor comments (3)
  1. [Section 2.2, after (2.5)] The tuple γ is defined as (γσ, γb, γq) ∈ R^3_+, but γq never appears; the third component should be γA.
  2. [Section 4, paragraph after Table 1] The sentence 'This strategy is often used for tuning hyper-parameters in deep learning' refers to choosing hyperparameters that attain the smallest reconstruction error on the test examples; this selection procedure should be stated in the abstract or conclusions if the claimed accuracy is based on it.
  3. [Throughout] There are several typos: 'b e tween' after (2.4), 'reonstruction' in Table 2 heading, 'off-shelf' in Section 4, and 'Combing' in the proof of Theorem 3.1. These should be corrected.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction in the central error analysis; one technical lemma is deferred to the authors' prior work, but no fitted parameter is disguised as a prediction.

full rationale

The central derivation is not circular. The reconstruction is obtained by minimizing the mixed least-squares losses (2.5)/(2.7) or (2.12)/(2.13) over DNN parameters, and Theorems 3.1 and 3.2 bound the distance between A† and the projected DNN reconstruction in terms of the noise level δ, DNN approximation errors ϵA, ϵσ, and Monte Carlo sampling errors. Assumption 3.2, A† = P_K(∇u†⊗∇ψ), is a structural source condition on the unknown conductivity rather than a restatement of the algorithm's output; it controls one inner-product term in the proof, and the final bound still depends on the actual loss residuals. No fitted parameter is renamed as a prediction: the hyperparameters in Section 4 are tuned to minimize the reported benchmark error, but this is a standard experimental evaluation against known exact solutions, not a derivation of the theoretical rate. The only self-referential element is Lemma 3.2, whose proof is omitted with the sentence 'The proof follows identically as that of [30, Theorem 3.4]'; reference [30] shares three authors with this paper. This is reliance on a self-citation for a Rademacher-complexity quadrature bound, but that bound is a separate statistical-learning estimate and is not equivalent to the target conductivity-recovery result, so it does not make the central claim circular. Two non-circular limitations are explicitly flagged here and weighed in the verdict: the theorems apply to exact global minimizers of nonconvex losses, while ADAM may return a local minimum (Section 4 concedes that 'the optimizer may get trapped in a local minimum'), and the experiments add noise to ∇u† while the theory measures δ = ∥u†-zδ∥_{H1}; these are gaps between theory and numerics, not circularity.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim rests on regularity and source-condition assumptions (Assumptions 3.1-3.3), the unproved/self-cited Lemma 3.2, and a large set of hand-tuned algorithmic parameters (γA, γσ, γb, early stopping, network sizes). No new physical entities are introduced.

free parameters (5)
  • Regularization weight γA = 1e-5 (all experiments, Table 1)
    Chosen by trial-and-error; Remark 3.1 suggests γA ∼ δ, but experiments keep γA = 1e-5 for δ = 0, 1%, 10%. The reconstruction error varies by only a small factor across γA (Table 2a), so the method is not very sensitive, but the reported numbers use a hand-picked value.
  • Divergence and boundary penalty weights γσ, γb = e.g., (1,1) for noise-free, (10,2.5) for noisy in Examples 4.1/4.2; different values in other examples (Table 1)
    Selected by trial-and-error; the paper states hyper-parameters are tuned to minimize e(Â).
  • Early stopping epoch = e.g., 8e4 for exact data, 1e4 for 10% noise in Example 4.1 (Table 1)
    The stopping index is chosen by trial-and-error (Section 4.1: 'this index has been determined using a trial-and-error approach'). This directly affects the reported accuracy.
  • DNN depth and width (LA, WA, Lσ, Wσ) = LA=10, WA=32 for A; Lσ=5, Wσ=26 for σ; LA=5 for Examples 4.1 and 4.7
    Chosen empirically; the paper notes increasing depth/width increases expressivity but also parameters. Table 2(b) shows limited variation, so these are not critical, but they are hand-tuned.
  • Total variation penalty γtv = 1e-4 and 3e-3 for Example 4.4 exact/noisy; 8e-4 and 5e-3 for Example 4.10 exact/noisy
    Added ad hoc to promote piecewise constancy in Examples 4.4 and 4.10; not part of the theoretical analysis.
assumptions (6)
  • domain assumption Assumption 3.1: A† ∈ W^{2,∞}(Ω)^{d×d} ∩ K, f ∈ H^1(Ω), g ∈ H^{3/2}(∂Ω) (Neumann) or H^{5/2}(∂Ω) (Dirichlet).
    Ensures enough regularity for DNN approximation and elliptic regularity; not verified in experiments.
  • domain assumption Assumption 3.2 (projected source condition): A†(x) = P_K(∇u†(x) ⊗ ∇ψ(x)) for some ψ ∈ H^1_0(Ω) ∩ W^{2,∞}(Ω).
    Load-bearing for Theorems 3.1 and 3.2. The paper does not show that any numerical example satisfies this condition, so the error bounds may not apply to the experiments.
  • domain assumption Assumption 3.3: f ∈ L∞(Ω), g ∈ L∞(∂Ω), zδ ∈ W^{1,∞}(Ω) (and on boundary in Dirichlet).
    Required for Rademacher complexity bounds in Lemma 3.2; pointwise Gaussian noise as used in experiments does not literally produce W^{1,∞} data.
  • standard math Lemma 2.1 (DNN approximation rates in Sobolev spaces), cited from [19].
    Taken from the literature; standard approximation theory for tanh networks.
  • standard math Lemma 3.2 (Monte Carlo quadrature error bounds) is quoted from [30, Theorem 3.4] without proof.
    The paper says 'The proof follows identically as that of [30, Theorem 3.4]' and omits details. [30] shares authors with this paper, so this is a self-citation.
  • standard math Well-posedness of (1.1) and elliptic regularity, e.g., [35, Lemma 2.1] and [18].
    Standard PDE theory used to guarantee existence of solutions and H^3 regularity under Assumption 3.1.

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Cite this review

Pith. "Pith review of Imaging Anisotropic Conductivity from Internal Measurements with Mixed Least-Squares Deep Neural Networks." pith.science (2026). https://pith.science/paper/2GGQK6HD

@misc{pith2026241200527,
  author       = {Pith},
  title        = {Pith review of: Imaging Anisotropic Conductivity from Internal Measurements with Mixed Least-Squares Deep Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2GGQK6HD}},
  note         = {Machine review of arXiv:2412.00527}
}
read the original abstract

In this work we develop a novel algorithm, termed as mixed least-squares deep neural network (MLS-DNN), to recover an anisotropic conductivity tensor from the internal measurements of the solutions. It is based on applying the least-squares formulation to the mixed form of the elliptic problem, and approximating the internal flux and conductivity tensor simultaneously using deep neural networks. We provide error bounds on the approximations obtained via both population and empirical losses. The analysis relies on the canonical source condition, approximation theory of deep neural networks and statistical learning theory. We also present multiple numerical experiments to illustrate the performance of the method, and conduct a comparative study with the standard Galerkin finite element method and physics informed neural network. The results indicate that the method can accurately recover the anisotropic conductivity in both two- and three-dimensional cases, up to 10\% noise in the data.

Figures

Figures reproduced from arXiv: 2412.00527 by the authors.

Figure 1
Figure 1. The reconstructions for Example 4.1 with exact data in (b) and noisy data (δ = 10%) in (d). From the top to bottom, the results are for A11, A12 and A22, respectively [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. The evolution of the loss (top) and the reconstruction error [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. The reconstructions for Example 4.2 with exact data in (b) and noisy data (δ = 10%) in (d). From the top to bottom, the results are for A11, A12 and A22, respectively. mild deterioration in reconstruction quality when 10% data noise is present. This shows its robustness against data noise. Next, we present a more ill-posed problem of using partial internal data. Example 4.3. The domain Ω = (0, 1)2 , the measurement … view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: The reconstructions for Example 4.3 with exact data in (b) and noisy data (δ = 10%) in (d). From the top to bottom, the results are for A11, A12 and A22, respectively. guess is set as follows: A0 11 = (2 + sin(4πx2)/2)(1 +x1(x1 −1)), A0 12 = (1 + sin(2πx2) 2/2)(1 + x1(…
Figure 5
Figure 5. Figure 5: The reconstructions for Example 4.3 with exact data in (b) and noisy data (δ = 10%) in (d) using the FEM. From the top to bottom, the results are for A11, A12 and A22, respectively. MLS-DNNs for noisy data. This improvement is attributed to the a priori knowledge A† |∂…
Figure 6
Figure 6. Figure 6: The reconstructions for Example 4.3 with exact data in (b) and noisy data (δ = 10%) in (d) using PINN. From the top to bottom, the results are for A11, A12 and A22, respectively. constancy in the reconstruction [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]
Figure 7
Figure 7. Figure 7: The reconstructions for Example 4.4 with exact data in (b) and noisy data (δ = 5%) in (d). From the top to bottom, the results are for A11, A12 and A22, respectively. Example 4.6. The domain Ω = (0, 1)3 , the measurement ∇z δ i on the region ω = Ω \ (0.2, 0.8)3 , A† = …
Figure 8
Figure 8. Figure 8: The reconstructions for Example 4.5 with exact data in (b) and noisy data (δ = 5%) in (d). From the top to bottom, the results are for A11, A12, A13, A22, A23 and A33, respectively. in reconstruction quality, except for a slight deformation in the conductivity landscap…
Figure 9
Figure 9. Figure 9: The reconstructions for Example 4.6 with exact data in (b) and noisy data (δ = 5%) in (d). From the top to bottom, the results are for A11, A12, A13, A22, A23 and A33, respectively. levels regardless of the noise level δ, further corroborating the robustness of the app…
Figure 10
Figure 10. Figure 10: The reconstructions for Example 4.7 with exact data in (b) and noisy data (δ = 10%) in (d). From the top to bottom, the results are for A11, A12 and A22, respectively. (a) δ = 0% (b) δ = 1% (c) δ = 10% [PITH_FULL_IMAGE:figures/full_fig_p028_10.png]
Figure 11
Figure 11. Figure 11: The evolution of the loss (top) and the reconstruction error [PITH_FULL_IMAGE:figures/full_fig_p028_11.png]
Figure 12
Figure 12. Figure 12: The reconstructions for Example 4.8 with exact data in (b) and noisy data (δ = 10%) in (d). From the top to bottom, the results are for A11, A12 and A22, respectively. deterioration in the reconstruction quality even when 10% noise is present in the data. This 29 [PI…
Figure 13
Figure 13. Figure 13: shows that the reconstruction results are accurate across the entire domain Ω for both exact and noisy data, including the central region where no observational data ∇z δ i is provided. This example again shows the stability of the MLS-DNN approach for partial interna…
Figure 14
Figure 14. Figure 14: The reconstructions for Example 4.9 with exact data in (b) and noisy data (δ = 10%) in (d) using the FEM. From the top to bottom, the results are for A11, A12 and A22, respectively. Also we present the results by PINN for Example 4.9, which involves minimizing an empi…
Figure 15
Figure 15. Figure 15: The reconstructions for Example 4.9 with exact data in (b) and noisy data (δ = 10%) in (d) using PINN. From the top to bottom, the results are for A11, A12 and A22, respectively. Example 4.10. The domain Ω = (0, 1)2 , A† = [PITH_FULL_IMAGE:figures/full_fig_p032_15.png]
Figure 16
Figure 16. Figure 16: The reconstructions for Example 4.10 with exact data in (b) and noisy data (δ = 5%) in (d). From the top to bottom, the results are for A11, A12 and A22, respectively. x3 + 1 3 (x 3 1 + x 3 2 − x 3 3 ), u † 4 = −x1 + x2 + x3 + 1 3 (−x 3 1 + x 3 2 + x 3 3 ), u † 5 = −u…
Figure 17
Figure 17. Figure 17: The reconstructions for Example 4.11 with exact data in (b) and noisy data (δ = 5%) in (d). From the top to bottom, the results are for A11, A12, A13, A22, A23 and A33, respectively. These experiments clearly demonstrate that the proposed DNN approach can accurately r…
Figure 18
Figure 18. Figure 18: The reconstructions for Example 4.12 with exact data in (b) and noisy data (δ = 5%) in (d). From the top to bottom, the results are for A11, A12, A13, A22, A23 and A33, respectively. References [1] G. Alessandrini. An identification problem for an elliptic equation in…

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