Pith. sign in

REVIEW 4 major objections 6 minor 1 cited by

Shallow neural network yields regularization for ill-posed inverse problems

T0 review · 4 major / 6 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A two-layer ReLU network whose width is picked by the noise level provably regularizes ill-posed inverse problems.

desk verdict Novel idea, flawed lemma — width-as-regularization is worth refereeing, but H1 convergence claims and experiments need major repair. read the letter →

arxiv 2511.16171 v2 pith:ZP6AFZNE submitted 2025-11-20 math.NA cs.NA

Lan Wang , Qiao Zhu , Bangti Jin , Ye Zhang This is my paper
classification math.NAcs.NA MSC 65J2047A5268T0741A30
keywords ill-posedinverseproblemstwo-layerReLUnetworksBarronspaceMorozovdiscrepancyprincipleexpandingneuralnetworkmethodsiterativeregularizationTikhonovconvergencerates
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's central claim is that a two-layer ReLU network whose width is selected a posteriori by the Morozov discrepancy principle is a regularization method for general ill-posed equations A(f)=g with noisy data. The authors introduce expanding-network iterations: start with a few neurons, grow the network through a fixed schedule, and stop when the data misfit first meets the discrepancy threshold τδ; the selected width n(δ) then plays the role of the regularization parameter. They prove that under local Hölder continuity of order θ of A at the exact solution, the process terminates with n(δ)=O(δ^{-2/θ}) and the outputs converge to f† as δ→0. For a Tikhonov variant they derive convergence rates O(Φ(τδ)) under a variational source condition, matching classical rates. The practical upshot is a theoretical explanation of why small networks suffice for high-noise data and why larger widths can overfit.

What carries the argument

Two-layer ReLU networks f_n(x)=n^{-1}Σ a_j(b_j·x+c_j)_+ with parameters in a compact radius set M_r, viewed as empirical draws from the Barron space of ridge-unit superpositions with finite weighted L1 norm. The argument combines: an approximation bound giving H1 error O(∥f∥_{B1}/√n); the expanding family X_{n,r(n)} with radius growing in n; and the Morozov discrepancy principle (stop when ∥A(f)−gδ∥≤τδ) as the a posteriori rule that promotes the selected width n(δ) to the regularization parameter. Hölder continuity of A at f† converts the stopping bound into n(δ)=O(δ^{-2/θ}).

What would settle it

Take a sequence of two-layer ReLU units f_m(x)=a_m (b_m·x+c_m)_+ with |a_m|≤1, ∥b_m∥_1=1, c_m=0, and choose b_m to rotate among directions so that the gradients ∇f_m oscillate without converging. If a subsequence with ∥f_m−f_l∥_{H1(Ω)}→0 cannot be extracted, Lemma 2 is false, and the H1-convergence claims of Theorems 1–2 fail; the L2-convergence parts would still stand.

Watch

Extended reading notes

Core claim

The paper's central discovery is the identification of network width with the regularization parameter. For a continuous injective forward operator A that is locally Hölder of order θ at the exact solution f†, the expanding-network algorithm stops after finitely many steps with n(δ)=O(δ^{-2/θ}), and the resulting two-layer ReLU networks converge to f† in H1(Ω) as δ→0. A Tikhonov variant is shown to be well-posed and, under a variational source condition, to achieve E†(fδ_{n,α})=O(Φ(τδ)) when n(δ)≍δ^{-2/s} and α(δ)≍δ/Φ(τδ).

Load-bearing premise

The argument depends on Lemma 2's claim that every bounded sequence in Barron space has a subsequence whose derivatives (not merely values) converge; the proof given only guarantees convergence of the values, so if that stronger compactness fails the H1-convergence conclusions in Theorems 1 and 2 are unsupported.

Editorial extensions

If this is right

  • For any fixed noise level δ, a two-layer network with at most O(δ^{-2/θ}) neurons stably approximates the true solution; this is a constructive existence statement, not just an asymptotic one.
  • High-noise regimes call for small networks: the selected width grows as the noise shrinks, so a practitioner who increases width while noise is large is trading away stability.
  • The network-size-as-regularization view makes architecture selection a principled model-selection problem: the discrepancy principle supplies the stopping index, and the width is the smoothing parameter.
  • The Tikhonov variant inherits classical convergence-rate theory: under a variational source condition with index Φ, the error measured by E† is O(Φ(τδ)) when n and α are balanced as n≍δ^{-2/s}, α≍δ/Φ(τδ).
  • When the noise is high, the algorithm stops early (semi-convergence), so the selected width is a stability-oriented choice rather than the error-minimizing one.

Reading between the lines

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

  • The paper's H1 convergence statements in Theorems 1 and 2 rest on Lemma 2, whose proof only establishes L2 compactness for bounded Barron sequences; if the stronger H1 compactness assertion fails, the stated convergence weakens to L2 and the Hölder-order rate may need re-derivation. The numerical experiments report only L2 errors, so they do not currently discriminate between these two conclusions
  • Reading the width law n(δ)≍δ^{-2/s} as a capacity-versus-resolution tradeoff suggests a generic rule: the resolving width grows like the inverse square of the detail scale δ^{1/s}; this could be tested in imaging pipelines by measuring reconstruction error versus width at controlled noise levels.
  • A concrete next step toward deep networks would require a parameter-bounded approximation theorem with explicit layerwise radius bounds; the paper identifies this as necessary for depth-adaptive rates, and a testable version would ask whether such bounds exist for standard ReLU DNNs of depth l and width n.
  • Because the discrepancy principle is known to over-regularize, a relaxed threshold (larger τ) might yield smaller true error at slightly larger width; the L-shaped curves in the experiments make this a testable and practically useful variation.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper proposes regularization methods for ill-posed operator equations A(f)=g using two-layer ReLU networks. Two iterative 'expanding neural network' algorithms are introduced; the network width n is increased adaptively and stopped by the Morozov discrepancy principle, so that the selected n itself acts as the regularization parameter and iteration index. Theorems 1 and 2 claim H1 (or L2) convergence of the reconstructions and a scaling law n(δ)=O(δ^{-2/θ}) under a local Hölder continuity assumption on A. A Tikhonov-type scheme with a Barron-norm penalty is then analyzed: Theorem 3 gives existence, stability, and convergence to an R-minimizing solution, and Theorem 4 claims convergence rates O(Φ(τδ)) under variational source conditions, with a specific power-law rate O(δ^μ) when Φ(t)=t^μ. Numerical experiments on a Fredholm integral equation, an auto-convolution Volterra equation, and a 2D EIT model illustrate the behavior of the three schemes and show that small networks are preferable at high noise levels.

Significance. If the results were correct, the paper would provide a constructive and theoretically grounded answer to a timely question: a simple shallow network can serve as a regularizer for general nonlinear ill-posed problems, with the network width selected a posteriori and quantitatively linked to the noise level. The Tikhonov part would extend variational source condition rates to the neural-network setting. The paper also ships reproducible code and gives concrete network-size choices. However, two load-bearing technical issues undermine the stated claims: Lemma 2's H1-compactness assertion is false, and the Young-type inequality used to derive convergence rates in Theorem 4 is invalid for the stated class of concave index functions. The manuscript can likely be repaired by weakening the convergence statements to L2 and correcting the rate analysis, but as written the advertised H1 convergence and power-type rates are not established.

major comments (4)
  1. [Section II, Lemma 2; Appendices A and B] The first assertion of Lemma 2, that bounded B1 sequences have H1-convergent subsequences, is false, and the proof only establishes H1→L2 compactness. A counterexample on Ω=(0,1) is f_k(x)=k^{-1}φ(kx) with φ∈C_c^∞: the B1 norm is uniformly bounded (by the spectral characterization ∫|ω||hat f_k(ω)|dω = O(1)) but ||f'_k||_{L2}=||φ'||_{L2}>0 while f_k→0 in L2, so no subsequence converges in H1. This lemma is used in the proofs of Theorem 1 and of Theorem 2 for R=R2 to conclude H1 convergence of the regularized solutions. The stated H1-convergence results are therefore unsupported. The arguments would still yield L2 convergence via B1↪H1↪L2 compactness; the theorems should be weakened accordingly or a genuinely stronger compactness assumption introduced.
  2. [Appendix D, Theorem 4] The rate proof uses the inequality αΦ(τt) ≤ t + τ^{-1}Φ^{-*}(τα) with Φ^*(α)=sup_{t≥0}(αt−Φ(t)). For a concave index function such as Φ(t)=t^μ with 0<μ<1, this conjugate is identically +∞, so Φ^{-*}(τα) is not defined and the bound (19) is vacuous. Consequently the claimed rates in Theorem 4(ii) and Corollary 1, including the power-law O(δ^μ), are not established. A correct Legendre-type inequality for concave Φ would produce different balancing exponents, not necessarily O(Φ(δ)). The authors should either state a restricted class of Φ for which the inequality holds or carry out the rate analysis with the correct convex conjugate of Φ^{-1}.
  3. [Section IV, Theorem 4] The opening convergence statement of Theorem 4 says that 'for any minimizer ... contains a subsequence ... such that f^δ_{n,α}→f† in L2 as n→∞ and δ→0'. This is false without conditions on α: if α is kept large, the minimizer is close to zero rather than to f†. The convergence does not follow from Theorem 3(c) unless one assumes α→0 and α^{-1}(δ+n^{-s/2})→0. The theorem statement needs to include such a parameter choice condition before the rate results.
  4. [Appendix A, proof of Theorem 1; Appendix B, proof of Theorem 2] The existence step asserts that the objective functional is continuous on X_{n,r(n)} because X_{n,r(n)} is sequentially compact in L2. This requires the forward operator A to be continuous from L2(Ω) to Y, and, for Algorithm 2, also appropriate continuity/lower semicontinuity of R in the same topology. The theorem statements only say A is 'continuous' without specifying the domain topology. In particular, R1(f)=C_{R1}||f||_{H1} is not continuous in L2. The proofs should either state these continuity hypotheses explicitly or argue directly on the compact parameter set M_r, where the networks and natural penalties are continuous in the parameters.
minor comments (6)
  1. [Section II, Lemma 2 statement] The second sentence says 'any bounded sequence in Barron space H^1(Ω)' — presumably it should read 'any bounded sequence in H^1(Ω)'.
  2. [Appendix A] There is a typo: 'Due to the Indeed, by the definition...' should be cleaned up.
  3. [Section V, discussion of Figure 1] The sentence 'n=n(δ) generally increases as n decreases' appears to have the variable reversed; presumably it should be 'as δ decreases'.
  4. [Section V] The claim that the observed stopping sizes are 'consistent with the theoretical estimate O(1/δ^2)' is not strongly supported by the figures: n(δ) varies only between 50 and a few hundred across five orders of magnitude of δ, and the trend is not monotone in the expected direction. A log-log plot of n(δ) versus δ with a fitted slope would be a more informative comparison.
  5. [Theorem 2] In condition (i), R1 is assumed locally Hölder continuous of order θ, but parts of the proof appear to treat R1 as Lipschitz. Please make the Hölder exponents and constants explicit throughout Appendix B.
  6. [References] Reference [47] lists the second author as 'L. A. S.', which appears to be an incomplete or corrupted author name.

Circularity Check

0 steps flagged · score 0.0 of 10

No meaningful circularity: the main theorems derive from external Barron-space approximation bounds, standard assumptions, and the discrepancy principle; self-citations are not load-bearing.

full rationale

The derivation chain is not circular. Theorems 1 and 2 obtain the stopping-index and network-size estimates by combining the discrepancy bound J≤τδ with the external Barron-space approximation bound (Proposition 3, cited to [45,46]) and standard injectivity/Hölder-continuity assumptions; no theorem assumes its own conclusion. Theorem 4 derives the rate E†=O(Φ(τδ)) from the variational source condition (Assumption 2) together with the minimization inequality (40) and standard balancing n≍δ^{-2/s}, α≍δ/Φ(τδ); the source condition is an assumption, not a fitted parameter. The numerical experiments use a priori fixed stopping parameters (τ=1.0001, α=(δ+n^{-1})^0.8) and do not fit free parameters to the reported errors. The self-citations ([31], [37], [44], [49], [50]) are contextual or illustrate classical frameworks and are not load-bearing. One non-circular correctness concern should be noted separately: Lemma 2 asserts that bounded sequences in Barron space have H1-convergent subsequences, but its proof only supplies L2 compactness (via B1↪H1 and H1↪L2), and the assertion is false; since Theorems 1 and 2 use this lemma for their H1-convergence statements, that part of the argument is unsupported. This is a mathematical error, not a circularity.

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

The paper introduces no new postulates or entities beyond the standard assumptions of regularization theory. The main external input is the Barron-space approximation theorem. The questionable internal assumption is the H1-compactness of bounded Barron sets (Lemma 2), which the provided proof does not establish. No parameters are fitted to data; all scalars (τ, c0, α, βn) are user-specified components of the algorithms.

assumptions (6)
  • standard math Proposition 3 (Barron approximation): for f∈B1 and horizon n, there is a two-layer ReLU network f_n with ||f−f_n||_{H1} ≤ C(Ω,d) n^{-1/2} ||f||_{B1}.
    External theorem cited from Li–Lu–Mathé–Pereverzev (2023) and E–Ma–Wu (2019). It is the engine converting network width n into approximation error and is not proved in this paper.
  • domain assumption Lemma 2, first part: any bounded sequence in Barron space B1 has a subsequence converging in H1(Ω).
    This is asserted with a proof that only yields L2-compactness. It is load-bearing for the H1-convergence conclusions in Theorem 1 and Theorem 2 (R=R2). Without a correct proof (or replacement by L2 convergence), those theorem statements are unsupported.
  • domain assumption Assumption 1 (A1.1–A1.5, A2): sequential lower semicontinuity and Lipschitz-type data-discrepancy conditions for D and A.
    Standard assumptions in Tikhonov regularization theory, invoked in Theorems 3 and 4.
  • domain assumption Assumption 2: a variational source condition at f† with index function Φ.
    Standard smoothness condition used to get convergence rates; it is an assumption on the exact solution, not derived.
  • domain assumption Algorithm 1 assumes a priori known bound B>cρ(f†) for the Barron norm of the exact solution.
    The a priori information about the exact solution is part of the method's setup; the proof requires r(n)→B and r(n)>cρ(f†) eventually.
  • domain assumption Local Hölder continuity of A at f† of order θ (Definition 2).
    Used to derive the quantitative stopping rates n(δ)=O(δ^{-2/θ}) in Theorems 1 and 2.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Shallow neural network yields regularization for ill-posed inverse problems." pith.science (2026). https://pith.science/paper/ZP6AFZNE

@misc{pith2026251116171,
  author       = {Pith},
  title        = {Pith review of: Shallow neural network yields regularization for ill-posed inverse problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZP6AFZNE}},
  note         = {Machine review of arXiv:2511.16171}
}
read the original abstract

In this paper, we develop a regularization theory for neural network approximations of general ill-posed operator equations with noisy data. Within the framework of iterative regularization, we introduce two expanding neural network methods (ENNs) under different a priori assumptions on the exact solution. Instead of prescribing a fixed architecture, ENNs adaptively select the number of neurons through an a posteriori stopping rule, so that the selected network size serves as a regularization parameter balancing approximation accuracy and stability with respect to data noise. We prove the regularization properties of the proposed ENNs and establish quantitative relationships between the selected network size and the noise level. Within the framework of variational regularization, we propose a neural network-based Tikhonov scheme and derive both convergence and convergence-rate results under mild assumptions. The resulting estimates account for the noise level, the network size, and the underlying smoothness expressed through general variational source conditions, thereby allowing greater flexibility than existing results. Numerical experiments demonstrate the effectiveness and robustness of the proposed algorithms. In particular, they show that, for highly noisy data, relatively small network architectures can already produce stable reconstructions, whereas excessively large architectures may degrade stability due to overfitting.

Figures

Figures reproduced from arXiv: 2511.16171 by the authors.

Figure 1
Figure 1. 1D linear integral equation: Relative L2 error of f δ n obtained by Algorithm 1 (top row), Algorithm 2 (middle row), and Tikhonov regularization (bottom row). Each column corresponds to a different random seed: 111, 666, 3333 (from left to right). The marker on each curve highlights the first network architecture that satisfied the stopping criterion for each noise level, with τ = 1.0001. The corresponding value of … view at source ↗
Figure 2
Figure 2. Auto-convolution equation: Relative L2 error of f δ n obtained by Algorithm 1 (top row), Algorithm 2 (middle row), and Tikhonov regularization (bottom row). Each column corresponds to a different random seed: 678, 765, 987 (from left to right). The marker on each curve highlights the first network architecture that satisfied the stopping criterion for each noise level, with τ = 1.0001. The corresponding value of n (… view at source ↗
Figure 3
Figure 3. EIT: Relative L2 error of f δ n obtained by Algorithm 1 (top row), Algorithm 2 (middle row), and Tikhonov regularization (bottom row). Each column corresponds to a different random seed: 20, 30, 40 (from left to right). The marker on each curve highlights the first network architecture that satisfied the stopping criterion for each noise level, with τ = 1.0001. The corresponding value of n (the stopping number) is i… view at source ↗

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Deep neural network yields regularization for ill-posed inverse problems

    math.NA 2026-05 unverdicted novelty 6.0 of 10

    Adaptive expanding DNN algorithms regularize ill-posed inverse problems with proofs of finite termination, convergence as noise vanishes, and asymptotic bounds on terminal network complexity.

Reference graph

Works this paper leans on

58 extracted references · 1 canonical work pages · cited by 1 Pith paper

  1. [1]

    A. N. Tikhonov, A. V . Goncharsky, V . V . Stepanov, and A. G. Yagola,Numerical Methods for the Solution of Ill-Posed Problems. Dordrecht, Netherlands: Kluwer Academic Publishers, 1995

    1995

  2. [2]

    Approximation by superpositions of a sigmoidal function,

    G. Cybenko, “Approximation by superpositions of a sigmoidal function,”Math. Control Signals Syst., vol. 2, no. 4, pp. 303–314, 1989

    1989

  3. [3]

    Multilayer feedforward networks are universal approximators,

    K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,”Neural Netw., vol. 2, no. 5, pp. 359–366, 1989

    1989

  4. [4]

    Approximation capabilities of multilayer feedforward networks,

    K. Hornik, “Approximation capabilities of multilayer feedforward networks,”Neural Netw., vol. 4, no. 2, pp. 251–257, 1991

    1991

  5. [5]

    Universal approximation using feedforward networks with non-sigmoid hidden layer activation functions,

    M. Stinchcombe and H. White, “Universal approximation using feedforward networks with non-sigmoid hidden layer activation functions,” inInternational 1989 Joint Conference on Neural Networks. IEEE, 1989, pp. 613–617

    1989

  6. [6]

    Universal approximation bounds for superpositions of a sigmoidal function,

    A. R. Barron, “Universal approximation bounds for superpositions of a sigmoidal function,”IEEE Trans. Inf. Theory, vol. 39, no. 3, pp. 930–945, 1993

    1993

  7. [7]

    Benefits of depth in neural networks,

    M. Telgarsky, “Benefits of depth in neural networks,” inConference on learning theory. PMLR, 2016, pp. 1517–1539

    2016

  8. [8]

    The power of depth for feedforward neural networks,

    R. Eldan and O. Shamir, “The power of depth for feedforward neural networks,” inConference on learning theory. PMLR, 2016, pp. 907–940

    2016

Show all 58 references
  1. [9]

    Deep network approximation for smooth functions,

    J. Lu, Z. Shen, H. Yang, and S. Zhang, “Deep network approximation for smooth functions,”SIAM J. Math. Anal., vol. 53, no. 5, pp. 5465–5506, 2021

    2021

  2. [10]

    Optimal approximation rate of ReLU networks in terms of width and depth,

    Z. Shen, H. Yang, and S. Zhang, “Optimal approximation rate of ReLU networks in terms of width and depth,”J. Math. Pures Appl., vol. 157, pp. 101–135, 2022

    2022

  3. [11]

    Deep network approximation characterized by number of neurons,

    Z. Shen, “Deep network approximation characterized by number of neurons,”Commun. Comput. Phys., vol. 28, no. 5, 2020

    2020

  4. [12]

    Deep network approximation: Beyond RELU to diverse activation functions,

    S. Zhang, J. Lu, and H. Zhao, “Deep network approximation: Beyond RELU to diverse activation functions,”J. Mach. Learn. Res., vol. 25, no. 35, pp. 1–39, 2024

    2024

  5. [13]

    ReLU network with widthd+O(1)can achieve optimal approximation rate,

    C. Liu and M. Chen, “ReLU network with widthd+O(1)can achieve optimal approximation rate,” inInternational Conference on Machine Learning, 2024

    2024

  6. [14]

    The phase diagram of approximation rates for deep neural networks,

    D. Yarotsky and A. Zhevnerchuk, “The phase diagram of approximation rates for deep neural networks,” inAdv. Neural Inf. Process. Syst. (NeurIPS), vol. 33, 2020, pp. 13 005–13 015. [Online]. Available: https://proceedings.neurips.cc/paper/2020/file/979a3f14bae523dc5101c52120c53...

    2020

  7. [15]

    Simultaneous neural network approximation for smooth functions,

    S. Hon and H. Yang, “Simultaneous neural network approximation for smooth functions,”Neural Networks, vol. 154, pp. 152–164, 2022

    2022

  8. [16]

    The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems,

    W. E and B. Yu, “The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems,”Commun. Math. Stat., vol. 6, no. 1, pp. 1–12, 2018

    2018

  9. [17]

    DGM: A deep learning algorithm for solving partial differential equations,

    J. Sirignano and K. Spiliopoulos, “DGM: A deep learning algorithm for solving partial differential equations,”J. Comput. Phys., vol. 375, pp. 1339–1364, 2018

    2018

  10. [18]

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,

    M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,”J. Comput. Phys., vol. 378, pp. 686–707, 2019

    2019

  11. [19]

    Weak adversarial networks for high-dimensional partial differential equations,

    Y . Zang, G. Bao, X. Ye, and H. Zhou, “Weak adversarial networks for high-dimensional partial differential equations,”J. Comput. Phys., vol. 411, p. 109409, 2020

    2020

  12. [20]

    Generative adversarial network: An overview of theory and applications,

    A. Aggarwal, M. Mittal, and G. Battineni, “Generative adversarial network: An overview of theory and applications,”Int. J. Inf. Manag. Data Insights, vol. 1, no. 1, p. 100004, 2021

    2021

  13. [21]

    KAN: Kolmogorov–arnold networks,

    Z. Liu, Y . Wang, S. Vaidya, F. Ruehle, J. Halverson, M. Soljacic, T. Y . Hou, and M. Tegmark, “KAN: Kolmogorov–arnold networks,” inProc. 13th Int. Conf. on Learning Representations (ICLR), 2025. [Online]. Available: https://openreview.net/forum?id=Ozo7qJ5vZi

    2025

  14. [22]

    Extensions of the deep galerkin method,

    A. Al-Aradi, A. Correia, G. Jardim, D. de Freitas Naiff, and Y . Saporito, “Extensions of the deep galerkin method,”Appl. Math. Comput., vol. 430, p. 127287, 2022

    2022

  15. [23]

    Deep convolutional ritz method: parametric PDE surrogates without labeled data,

    J. N. Fuhg, A. Karmarkar, T. Kadeethum, H. Yoon, and N. Bouklas, “Deep convolutional ritz method: parametric PDE surrogates without labeled data,” Appl. Math. Mech., vol. 44, no. 7, pp. 1151–1174, 2023

    2023

  16. [24]

    A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics,

    E. Haghighat, M. Raissi, A. Moure, H. Gomez, and R. Juanes, “A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics,”Comput. Methods Appl. Mech. Eng., vol. 379, p. 113741, 2021

    2021

  17. [25]

    A framework for data-driven solution and parameter estimation of pdes using conditional generative adversarial networks,

    T. Kadeethum, D. O’Malley, J. N. Fuhg, Y . Choi, J. Lee, H. S. Viswanathan, and N. Bouklas, “A framework for data-driven solution and parameter estimation of pdes using conditional generative adversarial networks,”Nature Computational Science, vol. 1, no. 12, pp. 819–829, 2021

    2021

  18. [26]

    Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations,

    M. Raissi, A. Yazdani, and G. E. Karniadakis, “Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations,”Science, vol. 367, no. 6481, pp. 1026–1030, 2020

    2020

  19. [27]

    Kolmogorov–Arnold-Informed neural network: A physics- informed deep learning framework for solving forward and inverse problems based on Kolmogorov–Arnold Networks,

    Y . Wang, J. Sun, J. Bai, C. Anitescu, M. S. Eshaghi, X. Zhuang, T. Rabczuk, and Y . Liu, “Kolmogorov–Arnold-Informed neural network: A physics- informed deep learning framework for solving forward and inverse problems based on Kolmogorov–Arnold Networks,”Comput. Methods Appl....

    2025

  20. [28]

    Gradient-based learning applied to document recognition,

    Y . LeCun, L. Bottou, Y . Bengio, and P. Haffner, “Gradient-based learning applied to document recognition,”Proc. IEEE, vol. 86, no. 11, pp. 2278–2324, 1998

    1998

  21. [29]

    U-net: Convolutional networks for biomedical image segmentation,

    O. Ronneberger, P. Fischer, and T. Brox, “U-net: Convolutional networks for biomedical image segmentation,” inMedical image computing and computer- assisted intervention–MICCAI 2015: 18th international conference, Munich, Germany, October 5-9, 2015, proceedings, part III 18, 2...

    2015

  22. [30]

    Numerical solution of inverse problems by weak adversarial networks,

    G. Bao, X. Ye, Y . Zang, and H. Zhou, “Numerical solution of inverse problems by weak adversarial networks,”Inverse Probl., vol. 36, no. 11, p. 115003, 2020

    2020

  23. [31]

    Electrical impedance tomography with deep calder ´on method,

    S. Cen, B. Jin, K. Shin, and Z. Zhou, “Electrical impedance tomography with deep calder ´on method,”J. Comput. Phys., vol. 493, p. 112427, 2023

    2023

  24. [32]

    Error-constraint deep learning scheme for electrical impedance tomography (EIT),

    Q. Wang, H. Zhang, X. Li, X. Duan, J. Wang, R. Zhang, H. Zhang, Y . Ma, H. Wang, and J. Jia, “Error-constraint deep learning scheme for electrical impedance tomography (EIT),”IEEE Transactions on Instrumentation and Measurement, vol. 71, pp. 1–11, 2021

    2021

  25. [33]

    Solving ill-posed inverse problems using iterative deep neural networks,

    J. Adler and O. ¨Oktem, “Solving ill-posed inverse problems using iterative deep neural networks,”Inverse Probl., vol. 33, no. 12, p. 124007, 2017

    2017

  26. [34]

    Unsupervised knowledge-transfer for learned image reconstruction,

    R. Barbano, ˇZ. Kereta, A. Hauptmann, S. R. Arridge, and B. Jin, “Unsupervised knowledge-transfer for learned image reconstruction,”Inverse Probl., vol. 38, no. 10, p. 104004, 2022

    2022

  27. [35]

    Deep convolutional neural network for inverse problems in imaging,

    K. H. Jin, M. T. McCann, E. Froustey, and M. Unser, “Deep convolutional neural network for inverse problems in imaging,”IEEE Trans. Image Process., vol. 26, no. 9, pp. 4509–4522, 2017

    2017

  28. [36]

    NETT: Solving inverse problems with deep neural networks,

    H. Li, J. Schwab, S. Antholzer, and M. Haltmeier, “NETT: Solving inverse problems with deep neural networks,”Inverse Probl., vol. 36, no. 6, p. 065005, 2020

    2020

  29. [37]

    Deep unfolding as iterative regularization for imaging inverse problems,

    Z.-X. Cui, Q. Zhu, J. Cheng, B. Zhang, and D. Liang, “Deep unfolding as iterative regularization for imaging inverse problems,”Inverse Probl., vol. 40, no. 2, p. 025011, 2024

    2024

  30. [38]

    Learning a variational network for reconstruction of accelerated MRI data,

    K. Hammernik, T. Klatzer, E. Kobler, M. P. Recht, D. K. Sodickson, T. Pock, and F. Knoll, “Learning a variational network for reconstruction of accelerated MRI data,”Magn. Reson. Med., vol. 79, no. 6, pp. 3055–3071, 2018

    2018

  31. [39]

    A deep cascade of convolutional neural networks for dynamic MR image reconstruction,

    J. Schlemper, J. Caballero, J. V . Hajnal, A. N. Price, and D. Rueckert, “A deep cascade of convolutional neural networks for dynamic MR image reconstruction,”IEEE Trans. Med. Imaging, vol. 37, no. 2, pp. 491–503, 2017

    2017

  32. [40]

    H. W. Engl, M. Hanke, and A. Neubauer,Regularization of inverse problems. Springer Science & Business Media, 1996, vol. 375

    1996

  33. [41]

    Kaltenbacher, A

    B. Kaltenbacher, A. Neubauer, and O. Scherzer,Iterative Regularization Methods for Nonlinear Ill-Posed Problems. Berlin, New York: Walter de Gruyter, 2008

    2008

  34. [42]

    Data errors and an error estimation for ill-posed problems,

    Y . Yagola, A. Leonov, and V . Titarenko, “Data errors and an error estimation for ill-posed problems,”Inverse Probl. Sci. Eng., vol. 10, no. 2, pp. 117–129, 2002. 21

    2002

  35. [43]

    The method of extending compacts and a posteriori error estimates for nonlinear ill-posed problems,

    K. Dorofeev and Y . Yagola, “The method of extending compacts and a posteriori error estimates for nonlinear ill-posed problems,”J. Inverse Ill-Posed Probl., vol. 12, pp. 627–636, 2004

    2004

  36. [44]

    A coupled complex boundary expanding compacts method for inverse source problems,

    Y . Zhang, R. Gong, M. Gulliksson, and X. Cheng, “A coupled complex boundary expanding compacts method for inverse source problems,”J. Inverse Ill-Posed Probl., vol. 27, no. 1, pp. 67–86, 2019

    2019

  37. [45]

    Barron spaces and the compositional function spaces for neural network models,

    W. E, C. Ma, and L. Wu, “Barron spaces and the compositional function spaces for neural network models,”arXiv preprint arXiv:1906.08039, 2019

    1906 arXiv

  38. [46]

    Two-layer networks with the ReLU k activation function: Barron spaces and derivative approximation,

    Y . Li, S. Lu, P. Math ´e, and S. V . Pereverzev, “Two-layer networks with the ReLU k activation function: Barron spaces and derivative approximation,” Numer. Math., vol. 156, no. 1, p. 319–344, Nov. 2023. [Online]. Available: https://doi.org/10.1007/s00211-023-01384-6

    2023 doi

  39. [47]

    A. N. Tikhonov, L. A. S., and A. G. Yagola,Nonlinear Ill-Posed Problems. London, U.K.: Chapman and Hall: London, 1998

    1998

  40. [48]

    Saturation of regularization methods for linear ill-posed problems in Hilbert spaces,

    P. Math ´e, “Saturation of regularization methods for linear ill-posed problems in Hilbert spaces,”SIAM J. Numer. Anal., vol. 42, pp. 968–973, 2006

    2006

  41. [49]

    On the second order asymptotical regularization of linear ill-posed inverse problems,

    Y . Zhang and B. Hofmann, “On the second order asymptotical regularization of linear ill-posed inverse problems,”Applicable Analysis, vol. 99, pp. 1000–1025, 2020

    2020

  42. [50]

    A scaling fractional asymptotical regularization method for linear inverse problems,

    L. Yuan and Y . Zhang, “A scaling fractional asymptotical regularization method for linear inverse problems,”Advances in Computational Mathematics, vol. 51, no. 1, p. 1, 2025

    2025

  43. [51]

    Approximate source conditions for nonlinear ill-posed problems – chances and limitations,

    T. Hein and B. Hofmann, “Approximate source conditions for nonlinear ill-posed problems – chances and limitations,”Inverse Probl., vol. 25, p. 035003, 2009

    2009

  44. [52]

    A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators,

    B. Hofmann, B. Kaltenbacher, C. Poschl, and O. Scherzer, “A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators,”Inverse Probl., vol. 23, pp. 987–1010, 2007

    2007

  45. [53]

    Flemming,Variational Source Conditions Yield Convergence Rates

    J. Flemming,Variational Source Conditions Yield Convergence Rates. Cham, Switzerland: Springer International Publishing, 2018, pp. 15–20

    2018

  46. [54]

    Deautoconvolution in the two-dimensional case,

    Y . Deng, B. Hofmann, and F. Werner, “Deautoconvolution in the two-dimensional case,”ETNA, Electron. Trans. Numer. Anal., vol. 59, pp. 24–42, 2023

    2023

  47. [55]

    On the autoconvolution equation and total variation constraints,

    G. Fleischer, R. Gorenflo, and B. Hofmann, “On the autoconvolution equation and total variation constraints,”Z. Angew. Math. Mech., vol. 79, no. 3, pp. 149–159, 1999

    1999

  48. [56]

    Calder ´on’s inverse conductivity problem in the plane,

    K. Astala and L. P ¨aiv¨arinta, “Calder ´on’s inverse conductivity problem in the plane,”Ann. Math., vol. 163, no. 1, pp. 265–299, 2006

    2006

  49. [57]

    Singular solutions of elliptic equations and the determination of conductivity by boundary measurements,

    G. Alessandrini, “Singular solutions of elliptic equations and the determination of conductivity by boundary measurements,”J. Differ. Equations, vol. 84, no. 2, pp. 252–272, 1990

    1990

  50. [58]

    Barron space for graph convolution neural networks,

    S.-Y . Chung and Q. Sun, “Barron space for graph convolution neural networks,”arXiv preprint arXiv:2311.02838, 2023

    2023 arXiv

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.