{"id":"34620b06-bff5-46b3-81f3-33a17792bff7","arxiv_id":"2509.09604","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A CNN maps single-energy LDOS maps back to the disorder potential in tight-binding models with test MSE 0.016 (1D) and 0.005 (2D).","lead":"This paper trains convolutional neural networks to recover the underlying potential landscape of a quantum lattice from a single-energy map of the local density of states, achieving low reconstruction error on synthetic 1D and 2D tight-binding models. The work suggests that STM-style LDOS images could be used to infer energy landscapes in disordered materials.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Normalization ambiguity between Eq. 3 and Appendix A2 leaves the reported reconstruction accuracy ambiguous; if per-sample normalization was used, the central claim overstates Hamiltonian recovery.","rationale":"The reader's weakest assumption about the idealized forward model is valid but concerns external applicability; the paper itself acknowledges it. A more immediate, internal concern is the normalization inconsistency. The main text's Eq. 3 suggests per-sample standardization, whereas Appendix A2 explicitly says global. This ambiguity affects whether the reported MSE reflects reconstruction of the physical potential or only its shape, and whether the OOD robustness plot actually tests amplitude generalization. It is a concrete, checkable issue that, if per-sample normalization was used, would invalidate the strongest central-claim wording. If global normalization was used (as the appendix states), the numerical results appear sound, the baselines are reasonable though weak, and the reader's CONDITIONAL verdict remains appropriate. I therefore keep the reader's verdict unchanged but add this specific technical condition. The paper deserves credit for detailed experimental sections, honest limitation statements, and plausible physical motivation; the concern is not about author integrity but about an ambiguity that must be settled before accepting the results at face value.","tokens_in":13459,"tokens_out":17474,"duration_ms":211142,"concrete_test":"Re-implement the training pipeline exactly as described in Appendix A2 (global normalization) and also with per-sample normalization. Compare test MSE on the normalized target and on the physical V after inverting the respective normalization. If per-sample normalization produces similar normalized MSE but much larger physical MSE (e.g., >0.1), the central claim fails. Additionally, inspect the code (or rerun) to determine which normalization was used for Fig. 5b; if per-sample, rerun the OOD experiment with global normalization and check whether L(V0=0.6) remains below 0.1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main text (Eqs. 3–4) defines V~ and ρ~ using per-sample mean/std, while Appendix A2 states a global training-set normalization. This is not a stylistic difference. If per-sample normalization is used, the CNN target has zero mean and unit variance for every sample, so the network does not learn the absolute disorder amplitude V0. The reported test MSE (0.016 in 1D, 0.005 in 2D) then measures shape recovery only; the claim of 'reconstructing the Hamiltonian' is overstated because the potential's scale is part of the Hamiltonian. The OOD study (Fig. 5b) varies V0, but per-sample normalization strips V0 from the target, so the robustness to changing V0 would be an artifact. The contradiction must be resolved: if the implementation follows Appendix A2, the main text is misleading; if it follows Eq. 3, the results do not demonstrate full Hamiltonian reconstruction. This directly affects the central numerical claim and the interpretation of the OOD and noise-robustness results.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the inverse problem of recovering a site-diagonal disorder potential V(r) of a single-particle nearest-neighbor tight-binding Hamiltonian (Eq. 1) from a single-energy spatial map of the LDOS (Eq. 2). The authors generate synthetic training pairs from correlated Gaussian random-field disorder in 1D and 2D, normalize both fields, and train convolutional image-to-image networks to map the normalized LDOS to the normalized potential. They report test MSE L≈0.016 (1D, L=128) and L≈4.6×10^-3 (2D, L=64), far below a linear fit and translation-augmented k-NN baselines (Table I). They also study robustness to added noise with noise-injected training and show modest out-of-distribution generalization around the training values V0=0.5, ξ=3. The paper frames the result as a step toward STM-based Hamiltonian reconstruction and explicitly limits the setting to noninteracting, single-orbital tight-binding with fixed hopping.","tokens_in":13739,"tokens_out":14060,"duration_ms":159187,"significance":"If the reported accuracy holds, the paper is a useful empirical demonstration that a CNN can approximate the inverse of the LDOS map within a disordered noninteracting tight-binding model, using modest data and compute (<1 GPU hour). Strengths include a clean train/validation/test split, hyperparameter tuning on validation, comparison against two baselines, honest worst-case examples, and an explicit statement of model limitations in Section IV. The appendix's global-normalization statement preserves the disorder amplitude in the target, so the reported MSE is not merely shape recovery. The paper does not overclaim uniqueness: it acknowledges that the inverse problem is ill-posed in general and frames the result as distribution-dependent approximate recovery. Within its stated scope, the contribution is solid, although it is an empirical benchmark rather than a new analytical method.","major_comments":[{"comment":"The normalization equations in the main text are ambiguous and, if read as per-sample standardization, would make the headline MSE (0.016/0.005) and the V0 out-of-distribution study misleading. Appendix A1 explicitly states that a global training-set normalization was used and that no sample-wise normalization was attempted, which resolves the substantive concern: the CNN target retains the V0 scale. However, the main text should be amended to refer to the global scheme, since as written a reader could reasonably interpret Eqs. (3)–(4) as per-sample and thereby misread the central numerical claim. This is a clarity issue in a load-bearing location, not a flaw in the implementation.","section":"§II, Eqs. (3)–(4), and Appendix A1"}],"minor_comments":[{"comment":"The 1D sample count is inconsistent: Section III.A says 'we generated 12000 data points', while Appendix A2 says 'The total number of data points was N=15000'; the baseline paragraph also says 12000 points. Please reconcile and specify the exact split used.","section":"§III.A vs Appendix A2"},{"comment":"The text says 'We show the results of NN-1D on four test LDOS profiles in Fig. 2', but Fig. 2 shows three profiles (best, median, worst). Please correct.","section":"§III.A"},{"comment":"The notation 'µ|E−E′|' should be 'µ(|E−E′|)' or the weighting function should be defined with explicit arguments, to avoid confusion with multiplication.","section":"Eq. (2), §II"},{"comment":"The superscript n in 'ρ~_E^n' appears before noise is introduced in the main text. Clarify whether it denotes a noised LDOS or is a typo.","section":"§II.C and Table II"},{"comment":"The caption says the LDOS input is shown '(right)' and the predicted potential is also '(right)'; one of these should be '(left)'.","section":"Fig. 2 caption"},{"comment":"The text refers to 'Fig. 7b and Fig. 7d', but Fig. 7 appears to have two panels (uncorrelated and correlated noise). Please renumber or relabel the subfigures consistently.","section":"Appendix B"},{"comment":"No code, data, or random seeds are made available. Given the ML methodology, releasing the data-generation and training code would substantially improve reproducibility.","section":"Global reproducibility"},{"comment":"The out-of-distribution averages use 10 realizations per grid point but no error bars or standard errors are shown. Adding uncertainty estimates would make the claimed L≲0.1 region more persuasive.","section":"Fig. 5b"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is a clean empirical study of a narrow inverse problem: recover the disorder potential from a single-energy LDOS slice in non-interacting tight-binding models. The result is real: the CNN achieves test MSE 0.016 (1D) and 0.005 (2D) on normalized potentials, far better than the linear-fit and k-NN baselines, and the robustness studies (noise, OOD) are honest with worst-case examples. This is a useful proof-of-principle for the STM community, and the paper is written with appropriate caveats about the idealized setting.\n\nThe main thing you'd worry about from the main text—per-sample vs global normalization—is resolved in the appendix. Eq. (3) looks like per-sample normalization, but Appendix A2 explicitly states a global training-set normalization and says 'We did not attempt sample-wise normalization.' So the network does learn the absolute disorder scale, and the OOD study varying V0 is not an artifact. The main text should be tightened to avoid the ambiguity, but the claims hold up.\n\nSoft spots: a sample-count inconsistency (12000 vs 15000 in the appendix), no code/data shipped, and the obvious gap between the idealized model and real STM surfaces. These are minor for what the paper claims; it never overpromises transfer to the lab. The baselines are fair, and the hyperparameter tuning is described well enough to reproduce.\n\nWho benefits: anyone working on inverse problems for STM or disordered materials, and ML-for-physics readers who want a careful benchmark problem. It deserves a serious referee. I'd send it out with a request to clean up the normalization notation, reconcile the sample count, and optionally state code availability.\n\nRecommendation: send to peer review; I'd expect it to come back as a solid contribution after minor revisions.","headline":"Solid empirical demonstration of LDOS-to-potential inversion; the normalization ambiguity is resolved by the appendix, and the paper deserves peer review.","tokens_in":14197,"tokens_out":2159,"would_cite":true,"duration_ms":25625,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A neural network can reconstruct a quantum lattice's disorder potential from a single-energy image of the local density of states.","keywords":["Hamiltonian reconstruction","local density of states","scanning tunneling microscopy","convolutional neural network","inverse problem","Anderson model","disorder","supervised learning"],"falsifier":"Apply the trained network to an LDOS image, taken from a scanning tunneling microscopy experiment on a surface whose actual potential landscape is independently known (for example, a quantum corral built by atom manipulation). If the network's predicted potential disagrees with the known landscape beyond the reported accuracy, the assumption that the simple tight-binding forward model captures the essential physics of the experiment would be falsified.","tokens_in":13367,"feed_emoji":"⚛️","tokens_out":9174,"duration_ms":106811,"temperature":0.7,"pith_summary":"This paper asks whether a quantum system's Hamiltonian can be recovered from a single spatial map of its local density of states (LDOS) at one energy. The authors train convolutional neural networks on synthetic data from 1D and 2D tight-binding models with random disorder, and find that the networks reconstruct the disorder potential with high accuracy: test MSE of 0.016 in 1D and 0.005 in 2D, far below linear-fit and nearest-neighbor baselines. The networks also tolerate moderate noise and generalize to nearby disorder strengths and correlation lengths. If the result extends to experiments, scanning tunneling microscopy maps of the LDOS could become a direct probe of a sample's unknown energy landscape.","feed_headline":"A CNN reconstructs disorder potentials from a single-energy LDOS image","feed_subtitle":"STM images at one energy could become direct measurements of a material's hidden potential landscape.","key_machinery":"The key instrument is an image-to-image convolutional neural network (CNN): a convolutional network that maps one image to another image, rather than to a class label. It takes the normalized LDOS field at one energy as input and outputs the normalized disorder potential field. The network uses convolutional layers with ReLU activations and, in 2D, residual connections, with circular padding to respect periodic boundary conditions. Training data is generated by exact diagonalization of the tight-binding Hamiltonian, followed by a windowed sum of eigenstate densities, and the loss is the mean-squared error between predicted and true normalized potentials. The argument works because the LDOS i","core_discovery":"The paper's central claim is that the inverse problem—recovering the potential V(r) from a single-energy LDOS image ρ_E(r)—is practically solvable by supervised learning for a class of single-particle lattice Hamiltonians. Using an image-to-image CNN trained on normalized pairs (ρ_E, V), the authors achieve normalized mean-squared errors of 0.016 (1D, L=128) and 0.005 (2D, L=64), outperforming baseline approaches by more than an order of magnitude. The reconstruction quality degrades gracefully under noise when the network is trained with synthetic noise regularization, and remains low for disorder amplitudes V0 between 0.3 and 0.6 and correlation lengths ξ between 2 and 4.5. The authors con","pith_inferences":["If validated on experimental STM data, the same image-to-image paradigm could be extended to reconstruct other spatially varying Hamiltonian parameters, such as hopping modulations or local interaction strengths, from LDOS maps at multiple energies.","The observed failure modes—charging rings near potential minima and plateaus near maxima—suggest that incorporating physical priors (like the sign of the potential or spectral bounds) as additional loss terms could sharpen reconstruction in extreme regions.","A multi-energy input (stacking LDOS slices at several bias voltages) could resolve ambiguities that a single slice leaves hidden, an extension the authors mention but do not test; this is a natural next experiment within their framework.","The method's reliance on a specific forward model means its success on real materials will depend on how faithfully a tight-binding model captures the sample; surfaces with strong electron-electron correlations may require the network to be retrained on data from a more complete model."],"forward_implications":["A single STM conductance image could be used to estimate the local potential energy landscape of a surface, enabling downstream calculations of transport and other observables.","The low data requirement (about 1000 samples in 2D) suggests the method may be feasible when synthetic data generation is expensive, such as for interacting models.","Noise-regularized training (injecting synthetic noise) maintains reconstruction accuracy even when test noise reaches the LDOS signal amplitude, indicating robustness to realistic experimental conditions.","The network retains accuracy for disorder strengths and correlation lengths outside its training range (e.g., V0 ∈ [0.3,0.6], ξ ∈ [2,4.5]), implying that exact knowledge of the disorder prior may not be necessary.","Because the model uses only a single energy slice, experiments need not perform full spectroscopic mapping to extract the Hamiltonian's potential term."],"fun_headline_variants":["AI turns one STM image into a quantum Hamiltonian","Single-energy LDOS is enough: CNN recovers hidden Hamiltonian","Neural network cracks inverse problem from one LDOS map","One STM image reveals material's hidden landscape via deep learning"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the single-particle, nearest-neighbor tight-binding model with a Gaussian random-field site potential used to generate all training data is an adequate description of the real system; if electron-electron interactions or multi-orbital physics are significant, the reconstructed 'potential' may not correspond to the physical energy landscape.","fun_headline_variants_meta":{"raw":{"variants":["AI turns one STM image into a quantum Hamiltonian","Single-energy LDOS is enough: CNN recovers hidden Hamiltonian","Neural network cracks inverse problem from one LDOS map","One STM image reveals material's hidden landscape via deep learning"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00018,"raw_usage":{"total_tokens":1110,"prompt_tokens":685,"completion_tokens":425,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":429,"completion_tokens_details":{"reasoning_tokens":369}},"tokens_in":429,"tokens_out":425,"duration_ms":5244,"temperature":1.0,"reasoning_tokens":369,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T18:44:20.622110+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Apply the trained network to an LDOS image, taken from a scanning tunneling microscopy experiment on a surface whose actual potential landscape is independently known (for example, a quantum corral built by atom manipulation). If the network's predicted potential disagrees with the known landscape beyond the reported accuracy, the assumption that the simple tight-binding forward model captures the essential physics of the experiment would be falsified.","supporting_citations":[],"review_version":1}