Backsolution: A Framework for Solving Inverse Problems via Automatic Differentiation

Koji Kobayashi; Tomi Ohtsuki

REVIEW 4 major objections 6 minor 29 references

Backsolution: A Framework for Solving Inverse Problems via Automatic Differentiation

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Backsolution recovers hidden potential and magnetization maps of a disordered conductor from two-terminal conductance data, using gradient descent with automatic-differentiation gradients.

desk verdict AD plus gradient descent is not new; the paper's value is the clean high-dimensional transport reconstructions, but single-run evidence and absent code/error metrics make it a proof-of-concept that needs revision. read the letter →

arxiv 2506.13210 v1 pith:VWI4PVGC submitted 2025-06-16 cond-mat.dis-nn cond-mat.mes-hall

Koji Kobayashi , Tomi Ohtsuki This is my paper

classification cond-mat.dis-nncond-mat.mes-hall

keywords inverseproblemsautomaticdifferentiationbacksolutionmagnetotransporttransfermatrixmethodpotentialprofilereconstructionreversemodelingdisorderedconductors

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

The paper introduces 'backsolution': treat the unknown parameters of a system as optimization variables and minimize the mean-squared error between a simulator's output and measured data, with all gradients obtained by automatic differentiation, a technique that computes exact derivatives of a computer program. The authors apply this to a transfer-matrix transport simulator and show that plain gradient descent reconstructs the full 144-parameter potential map and the 144-parameter magnetization-angle map from 651 two-terminal conductance values $G(\phi,E)$. The same recipe is applied to an ill-posed reverse-modeling task, generating disordered lattice configurations whose momentum-space images match a target even though the data do not determine a unique answer. The intended significance is that microscopic spatial information can be extracted from macroscopic transport measurements without microscopes, without training data, and without problem-specific tuning.

What carries the argument

The named method, 'backsolution,' is gradient descent on the mean-squared-error cost $f(p)$ with gradients obtained in a single forward pass by automatic differentiation. The engine is an automatic-differentiation library applied to a transfer-matrix simulator: the transfer matrix is the product of layer transfer matrices with QR orthonormalization every six layers, the transmission matrix follows from $t = (U^\dagger_{L,\mathrm{out}} T V_{R,\mathrm{in}})^{-1}$, and the observable is the Landauer conductance $G = \mathrm{Tr}(t^\dagger t)$. The load-bearing step is differentiating through this whole numerical pipeline, which turns a hard inverse problem into ordinary optimization and makes the method scale to more than 100 unknown parameters.

What would settle it

Run the 12x12 potential reconstruction from many random initial guesses with the mirror-symmetry-breaking notch removed; if gradient descent systematically lands on mirror-reflected profiles or stops at high cost instead of recovering the true map, the method's success depends on the notch and on a favorable cost landscape rather than on generic smoothness near the true solution.

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Extended reading notes

Core claim

The central claim is that gradient descent on the cost function $f(p) = \frac{1}{N}\sum_i [g(x_i;p) - y_i^{\mathrm{true}}]^2$, with the gradient computed by automatic differentiation through the simulator, is sufficient to reconstruct hidden spatial profiles from transport data. Even though the potential and conductance maps are rough owing to universal conductance fluctuations, the mean-squared-error cost is smooth and monotonic toward the global minimum when the target data cover a dense grid in magnetic flux and Fermi energy. With automatic differentiation supplying exact derivatives through a transfer-matrix simulator that uses QR orthonormalization for stability, the optimizer converges from a random initial guess to the true 144-parameter potential, and similarly to the magnetization-angle profile when the two spin channels are given different lead wavevectors. The authors also show that when the data are underdetermined, as in reconstructing a 920-parameter disordered lattice from its Fourier image, the same procedure produces plausible configurations consistent with the observations, so backsolution doubles as a reverse-modeling tool.

Load-bearing premise

The method assumes that the error surface is smooth and slopes toward the true answer near it, so plain gradient descent does not stall in a local minimum; the paper verifies this only on two-variable slices of a 144-variable problem and relies on a hand-placed notch to remove a mirror-reflection ambiguity.

Editorial extensions

If this is right

If the central claim is correct, microscopic potential and magnetization maps can be obtained from ordinary two-terminal conductance measurements, reducing reliance on direct imaging techniques.
The method scales to problems with over 100 unknown parameters and runs in minutes on a laptop, making it practical in regimes where Bayesian inversion becomes computationally expensive.
Because the same framework works when data are insufficient for uniqueness, it offers a general reverse-modeling approach, for example generating disordered atomic configurations consistent with a measured diffraction-type image.
Additional data or physical constraints can be folded into the cost function, so structure factors, excluded-volume constraints, or extra observables can sharpen the reconstruction.
No training dataset or problem-specific hyperparameter tuning is needed; any forward simulator that is differentiable by automatic differentiation becomes invertible by this recipe.

Reading between the lines

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

Going beyond the paper, the mirror-symmetry discussion implies that applying backsolution to real experiments will require a strategy for choosing among equivalent solutions, either by adding a known symmetry-breaking feature or by combining data from several lead configurations.
A natural extension is to apply the same pipeline to experimental rather than numerically generated target data; the main new challenge would be calibrating the lead parameters and noise model, which the paper fixes or treats as ideal.
Because the method uses plain gradient descent, standard accelerators such as momentum or Adam, or regularizers encoding prior knowledge of the profile, would likely improve convergence and broaden the range of cost landscapes where the method succeeds.
The framework should transfer to any inverse problem with a differentiable simulator and a smooth cost near the truth, such as reconstructing refractive-index or scattering profiles, with the same caveats about symmetry and local minima.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Short version: this paper repackages automatic differentiation plus gradient descent as 'backsolution.' The core loop is identical to the AD-based inverse design work the authors cite (Refs 15–19). What is actually new is the application: reconstructing 144-parameter potential and magnetization profiles from two-terminal magnetotransport data, plus a reverse-modeling example for disordered lattices. That is a useful demonstration, and the paper is honest about limitations — it flags local minima, non-uniqueness, and the need for smoothness near the global minimum.

Credit where due: the transfer-matrix simulator with QR decomposition is written in JAX, which is non-trivial, and the cost maps in Fig. 4 are a helpful sanity check. The authors also explicitly discuss the mirror-reflection degeneracy and the notch they add to break it. That is more transparent than much of the inverse-design literature.

The soft spots are real. The central claim — 'successfully reconstructs the true potential profile' — rests on one optimization run. There is no pixelwise error metric between reconstructed and true profiles, no repeated trials with different random initializations, and no report of how often the method lands in a local minimum. The smoothness condition in Sec. IV.B is verified only in two-parameter slices, not in the 144-dimensional space gradient descent actually traverses. The notch v_{0,0}=-1 is in both true sample and initial guess, a strong prior you would not have in an experiment. The stated noise robustness, δG<0.1, is asserted but not shown. And because target data are generated with the same transfer-matrix simulator, the demonstrations are self-consistent but say nothing about model misspecification or experimental systematics.

The reverse-modeling example is a nice illustration of AD on an ill-posed problem, but the initial guess is a slightly displaced triangular lattice with known density — a good prior. The claim that this 'has the potential to replace' reverse Monte Carlo is an overreach without a direct comparison.

The citation pattern is fine; the authors credit the AD-based inverse-design literature. My concern is not borrowing, it is unfulfilled evidence. If the authors supply code, quantitative reconstruction errors, multiple random runs, and a comparison to a standard inverse method, this becomes a solid proof-of-concept paper. As is, it is a credible but under-supported note.

Who is this for? Physicists working on transport inverse problems who might want a quick laptop-scale tool. A serious referee could extract value from the demonstration, but the paper needs those additions before I would trust the key claim.

Recommendation: I would engage with the paper — send it to peer review with the expectation of major revision, or ask the authors to add the missing statistics and code first.

Referee Report

4 major / 6 minor

Summary. The paper proposes "backsolution," a framework for solving inverse problems by minimizing a mean-squared-error cost between a numerical simulator's output and target observations, with gradients computed by automatic differentiation (JAX). The authors demonstrate the method on three problems: reconstructing a 144-site on-site potential profile of a 12x12 tight-binding conductor from a 651-point two-terminal magnetoconductance map; reconstructing a 144-site magnetization-angle profile from analogous magnetotransport data; and reverse-modeling 920 atomic positions of a displaced triangular lattice from a blurred Fourier-space image. The text argues that the method is general, scalable, and free of training data or problem-specific tuning, with the main condition being smoothness of the cost near the global minimum.

Significance. If properly validated, the framework would be a useful, simple baseline for inverse problems in quantum transport and beyond. The paper has several genuine strengths: it provides an end-to-end differentiable transfer-matrix implementation including QR-based stabilization, demonstrates the idea on three diverse test cases, explicitly discusses local-minimum and symmetry-degeneracy issues, and is candid about limitations. The principal weakness is that the validation is entirely synthetic: target data are generated with the same simulator used in the inverse loop, success is judged by the very MSE cost being minimized, and no quantitative parameter-space error metric is reported for the 144- and 920-parameter reconstructions. The central claim therefore needs substantially stronger evidence before the paper can be accepted.

major comments (4)

[§III.A and §IV.A] The claim that the 144-dimensional potential is "successfully reconstructed" rests on visual inspection of Fig. 2(c) and on the final value f < 10^-6. Because f is exactly the cost being minimized, a small f only shows that the simulator reproduces the target conductance map; it does not by itself establish that the recovered parameter vector is close to v_true unless identifiability is demonstrated. With Ndata = 651 and n = 144, uniqueness is not obvious even after the notch removes mirror symmetry. Please report a quantitative parameter-space error (e.g., mean absolute error, maximum absolute error, or correlation with v_true), and ideally a success rate over multiple random true potentials and initial guesses.
[§IV.B and Fig. 4] The key applicability condition—smooth and locally monotonic cost near the global minimum—is tested only in two-parameter slices of a 144-dimensional landscape (Fig. 4). Gradient descent in the full space can encounter directions not visible in any two-parameter slice. Please supplement with multi-start trials: for several random target profiles and random initial guesses, report the fraction of runs reaching f < 10^-6 and the final parameter-space error, or evaluate the cost along the actual optimization trajectory. This directly addresses whether the single successful run in Fig. 2 is typical or a favorable initialization.
[§IV.A and Fig. 2] The fixed notch v_{0,0} = -1 appears in both the true potential and the initial guess. This is a nontrivial prior: it fixes one of the 144 parameters exactly and breaks the mirror-reflection degeneracy of the two-terminal conductance. The paper should state unambiguously that Fig. 2 demonstrates reconstruction under this prior, and should test how the method performs when the notch is absent or its position is unknown, using the Monte Carlo restart strategy mentioned in §IV.A.
[§III.B and §V] The magnetization and lattice examples similarly lack quantitative validation. For Fig. 3(c), no error between the reconstructed angle field and θ_true is given, and the lead parameters k_up = 0.5 and k_down = 0.25 are treated as known; if these are actually hidden, the reconstruction problem changes. For Fig. 5(c), the match between predicted and true lattices is assessed only visually; reporting a structural metric such as a pair-correlation function or structure-factor residual would make the claim concrete.

minor comments (6)

[§III.A] The sentence claiming robustness to noise ("We have also confirmed that the method is robust in the presence of small noises δG < 0.1[e2/h]") is not supported by any figure, table, or quantitative result; please add the relevant data or soften the claim.
[§II.A] There is a typo: "for and if statements" should be "for and if statements" (the word "for" is repeated incorrectly).
[Fig. 4] The rows of Fig. 4 are not clearly labeled in the caption; the reader cannot tell which row corresponds to the example of Fig. 2, which to the sparse-mesh case, and which to the dense-mesh case without inferring from the text.
[§III.B] Please clarify whether the lead wavenumbers k_up and k_down are assumed known in the inverse loop or are also inferred; this affects the interpretation of the magnetization reconstruction.
[§V] The reverse-modeling example would benefit from a statement of how the predicted atomic configuration is compared with the true one (e.g., by a metric or by visual inspection only).
[References] Reference [8] contains a formatting artifact ("Ro¨ emer"); it should read "Römer".

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reconstructions are synthetic benchmarks validated against known ground truth, not predictions forced by construction.

full rationale

The paper's central demonstrations generate the target magnetotransport data with the same transfer-matrix simulator used in the inversion, but this is a standard self-consistency benchmark for an inverse solver rather than a circular derivation. The success claim is verified against the known hidden profiles (Fig. 2(c) and Fig. 3(c)), not merely against the minimized MSE cost; the converged cost f<1e-6 is accompanied by a visual comparison to v_true and m_true. No parameter is fitted to a subset of data and then presented as a prediction of a closely related quantity, and no result is defined in terms of the quantity it purports to derive. The self-citations ([24],[25],[26]) are background references for the transfer matrix method, random network leads, and related formalism; the needed equations (Eqs. (5)-(13)) are stated in the paper, so these citations are not load-bearing. The acknowledged notch is an explicit symmetry-breaking modification of the test problem, not a hidden equivalence. The remaining concerns — a single optimization run, 2D cost slices in a 144-dimensional landscape, and absence of pixelwise error metrics — are evidence-strength and robustness issues, not circularity, and do not make the derivation equivalent to its inputs.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The ledger shows that the method introduces no new physical entities, but its success rests on several domain assumptions: the simulator is accurate, the cost function is smooth near the solution, the parameterization exactly represents the true profile, and practical details such as learning rate and initialization are chosen by hand. The only free parameters are the learning rate and the symmetry-breaking notch, both set manually. No quantitative comparison or external benchmark is provided.

free parameters (2)

learning rate r = not specified
Used in Eq. (3) for the gradient descent update; the authors state only that it is 'appropriate', without giving values, schedule, or tuning procedure.
notch potential v_{0,0} = -1 (in energy units)
Introduced 'for numerical stability' to break the mirror-reflection symmetry of the two-terminal conductance; placed at the same site in both the true sample and the initial guess, which makes the problem uniquely solvable and removes the degeneracy that otherwise exists.

assumptions (5)

domain assumption The transfer matrix method with QR orthonormalization accurately computes the two-terminal conductance of the tight-binding models.
The simulators in Eqs. (14) and (15) are evaluated with the transfer matrix method described in Section II.B; any error in the simulator would propagate to the reconstructed profiles.
ad hoc to paper The MSE cost function f(p) is smooth and locally monotonic toward the global minimum near p_true.
Explicitly assumed in Section IV.B: "For stable convergence, the cost function must be smooth in the vicinity of the global minimum." The authors verify this only on two-parameter slices (Fig. 4), not in the full 144-dimensional space.
domain assumption The hidden parameters are exactly representable by the chosen variables (on-site potentials v_{x,y} or angles theta_{x,y}).
The reconstruction is limited to the parameterization used in the Hamiltonian; any true profile outside this form could not be recovered.
domain assumption The initial guess is close enough and, for the lattice example, the density and Gaussian width are known.
Section V assumes the density and sigma=0.3a are known, and the method's success depends on a good initial guess, as the authors note in Section IV.A.
domain assumption JAX's jax.grad and jax.numpy.qr compute gradients correctly for the numerical simulator.
The whole method depends on differentiable simulation; the authors rely on JAX's implementation without providing independent verification of the AD correctness.

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Cite this review

Pith. "Pith review of Backsolution: A Framework for Solving Inverse Problems via Automatic Differentiation." pith.science (2026). https://pith.science/paper/VWI4PVGC

@misc{pith2026250613210,
  author       = {Pith},
  title        = {Pith review of: Backsolution: A Framework for Solving Inverse Problems via Automatic Differentiation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VWI4PVGC}},
  note         = {Machine review of arXiv:2506.13210}
}

read the original abstract

We present a simple yet powerful framework for solving inverse problems by leveraging automatic differentiation. Our method is broadly applicable whenever a smooth cost function can be defined near the true solution, and a numerical simulator is available. As a concrete example, we demonstrate that our method can accurately reconstruct the spatial profiles in a conductor from magnetotransport measurements. Even if the given data are insufficient to uniquely determine the profiles, the same framework enables effective reverse modeling. This method is general, flexible, and readily adaptable to a broad class of inverse problems across condensed matter physics and beyond.

Figures

Figures reproduced from arXiv: 2506.13210 by the authors.

**Figure 2.** FIG. 2. (a) True potential structure [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. (a) True magnetization structure [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: FIG. 5. (a) True, (b) initial, and (c) predicted lattice struc [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

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Reference graph

Works this paper leans on

29 extracted references · 25 canonical work pages

[1]

von der Linden, Maximum-entropy data analysis, Appl

W. von der Linden, Maximum-entropy data analysis, Appl. Phys. A 60, 155 (1995)

work page 1995
[2]

Mayer, R

M. Mayer, R. Fischer, S. Lindig, U. von Toussaint, R. Stark, and V. Dose, Bayesian reconstruction of sur- face roughness and depth profiles, Nucl. Instr. and Meth. in Phys. Res. B 228, 349 (2005)

work page 2005
[3]

von Toussaint, Bayesian inference in physics, Rev

U. von Toussaint, Bayesian inference in physics, Rev. Mod. Phys. 83, 943 (2011)

work page 2011
[4]

Vocciante, A

M. Vocciante, A. P. Reverberi, and V. G. Dov ` ı, A General Method for the Solution of Inverse Problems in Transport Phenomena, Chem. Eng. Trans. 43, 1615 (2015)

work page 2015
[5]

Tamura and K

R. Tamura and K. Hukushima, Method for estimating spin-spin interactions from magnetization curves, Phys. Rev. B 95, 064407 (2017)

work page 2017
[6]

Raissi, P

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. 378, 686 (2019)

2019
[7]

Kades, J

L. Kades, J. M. Pawlowski, A. Rothkopf, M. Scherzer, J. M. Urban, S. J. Wetzel, N. Wink, and F. P. Ziegler, Spectral Reconstruction with Deep Neural Networks, Phys. Rev. D 102, 096001 (2020)

work page 2020
[8]

J. J. Webb, R. Beanland, and R. A. Ro¨ emer, Large- Angle Convergent-Beam Electron Diffraction Patterns via Conditional Generative Adversarial Networks (2025), arXiv:2503.02852

work page Pith review arXiv 2025

Show all 29 references

[9]

Mukim, F

S. Mukim, F. P. Amorim, A. R. Rocha, R. B. Muniz, C. Lewenkopf, and M. S. Ferreira, Disorder information from conductance: A quantum inverse problem, Phys. Rev. B 102, 075409 (2020)

2020
[10]

Daimon, K

S. Daimon, K. Tsunekawa, S. Kawakami, T. Kikkawa, R. Ramos, K. Oyanagi, T. Ohtsuki, and E. Saitoh, De- ciphering quantum fingerprints in electric conductance, Nat. Commun. 13, 3160 (2022)

2022
[11]

Yokoi, Y

N. Yokoi, Y. Tanaka, Y. Nonaka, S. Daimon, J. Haruyama, and E. Saitoh, Diffusion model for ana- lyzing quantum fingerprints in conductance fluctuation (2025), arXiv:2506.08617 [cond-mat.mes-hall]

2025 arXiv
[12]

G. J. Percebois, A. Lacerda-Santos, B. Brun, B. Hack- ens, X. Waintal, and D. Weinmann, Reconstructing the potential configuration in a high-mobility semiconductor heterostructure with scanning gate microscopy, SciPost Phys. 15, 242 (2023)

2023
[13]

Franceschetti and A

A. Franceschetti and A. Zunger, The inverse band- structure problem of finding an atomic configuration with given electronic properties, Nature 402, 60 (1999)

1999
[14]

Fujita, Y

H. Fujita, Y. O. Nakagawa, S. Sugiura, and M. Oshikawa, Construction of hamiltonians by supervised learning of energy and entanglement spectra, Phys. Rev. B 97, 075114 (2018)

2018
[15]

S. Yu, Y. Gao, B.-B. Chen, and W. Li, Learning the Effective Spin Hamiltonian of a Quantum Magnet, Chin. Phys. Lett. 38, 097502 (2021)

2021
[16]

Inui and Y

K. Inui and Y. Motome, Inverse Hamiltonian design by automatic differentiation, Commun. Phys. 6, 37 (2023)

2023
[17]

Williams and E

I. Williams and E. Polizzi, Automatic Differentiation for Inverse Problems with Applications in Quantum Trans- port (2023), arXiv:2307.09311

2023 arXiv
[18]

Inui and Y

K. Inui and Y. Motome, Inverse Hamiltonian design of highly entangled quantum systems, Phys. Rev. Res. 6, 033080 (2024)

2024
[19]

Hirasaki, K

Y. Hirasaki, K. Inui, and E. Saitoh, Inverse magnetocon- ductance design by automatic differentiation, Phys. Rev. B 110, 214201 (2024)

2024
[20]

Amari, A theory of adaptive pattern classifiers, IEEE Transactions on Electronic ComputersEC-16, 299 (1967)

S. Amari, A theory of adaptive pattern classifiers, IEEE Transactions on Electronic ComputersEC-16, 299 (1967)

1967
[21]

D. E. Rumelhart, G. E. Hinton, and R. J. Williams, Learning representations by back-propagating errors, Na- ture 323, 533 (1986)

1986
[22]

Bradbury, R

J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. Van- derPlas, S. Wanderman-Milne, and Q. Zhang, JAX: com- posable transformations of Python+NumPy programs (2018), http://github.com/jax-ml/jax

2018
[23]

J. B. Pendry, A. MacKinnon, and P. J. Roberts, Uni- versality classes and fluctuations in disordered systems, 7 Proc. R. Soc. Lond. A 437, 67 (1992)

1992
[24]

Slevin, P

K. Slevin, P. Markoˇ s, and T. Ohtsuki, Reconciling con- ductance fluctuations and the scaling theory of localiza- tion, Phys. Rev. Lett. 86, 3594 (2001)

2001
[25]

Kramer, T

B. Kramer, T. Ohtsuki, and S. Kettemann, Random Net- work Models and Quantum Phase Transitions in Two Dimensions, Phys. Rep. 417, 211 (2005)

2005
[26]

Kobayashi, T

K. Kobayashi, T. Ohtsuki, and K.-I. Imura, Disordered weak and strong topological insulators, Phys. Rev. Lett. 110, 236803 (2013)

2013
[27]

P. A. Lee and A. D. Stone, Universal conductance fluc- tuations in metals, Phys. Rev. Lett. 55, 1622 (1985)

1985
[28]

P. A. Lee, A. D. Stone, and H. Fukuyama, Universal con- ductance fluctuations in metals: Effects of finite temper- ature, interactions, and magnetic field, Phys. Rev. B 35, 1039 (1987)

1987
[29]

R. L. McGreevy, Reverse Monte Carlo modelling, J. Phys.: Condens. Matter 13, R877 (2001)

2001

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

von der Linden, Maximum-entropy data analysis, Appl

W. von der Linden, Maximum-entropy data analysis, Appl. Phys. A 60, 155 (1995)

work page 1995

[2] [2]

Mayer, R

M. Mayer, R. Fischer, S. Lindig, U. von Toussaint, R. Stark, and V. Dose, Bayesian reconstruction of sur- face roughness and depth profiles, Nucl. Instr. and Meth. in Phys. Res. B 228, 349 (2005)

work page 2005

[3] [3]

von Toussaint, Bayesian inference in physics, Rev

U. von Toussaint, Bayesian inference in physics, Rev. Mod. Phys. 83, 943 (2011)

work page 2011

[4] [4]

Vocciante, A

M. Vocciante, A. P. Reverberi, and V. G. Dov ` ı, A General Method for the Solution of Inverse Problems in Transport Phenomena, Chem. Eng. Trans. 43, 1615 (2015)

work page 2015

[5] [5]

Tamura and K

R. Tamura and K. Hukushima, Method for estimating spin-spin interactions from magnetization curves, Phys. Rev. B 95, 064407 (2017)

work page 2017

[6] [6]

Raissi, P

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. 378, 686 (2019)

2019

[7] [7]

Kades, J

L. Kades, J. M. Pawlowski, A. Rothkopf, M. Scherzer, J. M. Urban, S. J. Wetzel, N. Wink, and F. P. Ziegler, Spectral Reconstruction with Deep Neural Networks, Phys. Rev. D 102, 096001 (2020)

work page 2020

[8] [8]

J. J. Webb, R. Beanland, and R. A. Ro¨ emer, Large- Angle Convergent-Beam Electron Diffraction Patterns via Conditional Generative Adversarial Networks (2025), arXiv:2503.02852

work page Pith review arXiv 2025