{"id":"317e190b-a60b-4260-bbd1-bb518ea5997f","arxiv_id":"2506.10219","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Compact neural networks compute ground and excited steady states of the 1D nonlinear Schrödinger equation, and superpositions of those states show chaotic dynamics with near-Kolmogorov scaling.","lead":"Neural networks are used here to compute hard-to-find excited states of the nonlinear Schrödinger equation, the equation behind optical solitons and ultracold atom clouds. The computed states are then used to check chaotic wave behavior, which could give physicists a new tool for studying wave turbulence.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Excited states are not demonstrated to satisfy the NLSE: the first-excited residual is admitted to remain O(1), and higher states have no residual or quantitative benchmark, so the central NNQS claim lacks its key verification.","rationale":"The reader's weakest assumption concerns the loss landscape and the absence of convergence guarantees. My concern is more specific and is grounded in the paper's own reporting: the text admits that the first-excited NLSE residual is not minimized, so the main deliverable—excited states satisfying Eq. (2)—is not evidenced. The compact tanh fit (MSE 6.1e-6) is a fit to the network output, not a check of the NLSE, and the ground-state agreement with imaginary time evolution does not extend to higher saddle points. I do not view this as fraud or as a definitely wrong method; the approach is plausible and could work with reduced learning rate, quantitative residual reporting, and release of code. But as written, the paper does not establish its central claim. The conditional verdict is therefore appropriate, and the condition should explicitly require residual benchmarks for every reported state and quantitative comparison against Ref. [12] before the excited-state and chaos claims are accepted.","tokens_in":11228,"tokens_out":5524,"duration_ms":72208,"concrete_test":"Compute for each reported ψ_i (i=1,...,9; g=1,10,100) the normalized residual R_i = || -½ψ_i'' + Vψ_i + gψ_i^3 - μ_i ψ_i ||_2 / ||ψ_i||_2 on the same 4097-point grid, with μ_i chosen as the projection ∫ψ_i[-½ψ_i''+Vψ_i+gψ_i^3] dx, and compare to the residual obtained by solving Eq. (2) with the continuation method of Ref. [12] on the same grid. If R_i is not below ~1e-3 for all states, the paper's claim that Eq. (7) computes steady states is not supported; the first-excited result is already suspect because its L_NLSE remains O(1).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that minimizing the composite loss in Eq. (7) yields stationary states of Eq. (2). The only term explicitly enforcing the steady-state equation is L_NLSE. In Section III.B, the authors state that for the first excited state at g=100 the NLSE residual loss 'oscillates persistently about unity' and that a reduced learning rate 'would be necessary' for accuracy. An O(1) residual means the converged network output is not a solution of Eq. (2); the paper still presents it as a first-excited-state solution. For higher states (i=2,...,9, Fig. 4), no residuals or quantitative comparisons are given; the text only says they agree with Ref. [12]. The hierarchical initialization from noninteracting states and the explicit symmetry/node-count penalties can produce plausible-looking nodal profiles even if the saddle-point equations are not satisfied. Because these excited states feed directly into the spatiotemporal-chaos analysis, the downstream claims (positive Lyapunov exponents, ESS) inherit this unverified input. This is not a dispute about consensus; it is an internal gap between the stated objective and the reported convergence.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a neural-network quantum state (NNQS) framework for computing steady states of the nonlinear Schrödinger equation, specifically the 1D Gross-Pitaevskii equation in a harmonic trap. The wavefunction is represented by a multilayer perceptron and trained by minimizing the composite loss in Eq. (7), which combines the energy functional, a normalization penalty, an overlap penalty for excited-state separation, and an NLSE residual term. The authors report ground states and excited states up to i = 9 for g = 1, 10, and 100, then distill the large network solutions into a compact two-hidden-layer tanh network with 46 parameters. These states are used as initial conditions for real-time evolution to compute Lyapunov exponents and extended self-similarity (ESS) exponents. The central claim is that this is the first NNQS computation of NLSE excited states and that the compact networks provide interpretable analytical approximations enabling studies of spatiotemporal chaos.","tokens_in":11438,"tokens_out":3888,"duration_ms":50593,"significance":"If fully verified, the framework would provide a mesh-free, differentiable route to NLSE excited states and a practical bridge between machine learning and nonlinear wave dynamics. The paper has several strengths: the ground-state results are validated against imaginary time evolution; the compact 46-parameter network reproduces the reported state with MSE 6.1 × 10^-6; Table I provides a systematic scan of Lyapunov exponents over g, α, and excited-state order; and the ESS analysis in Fig. 6 presents a clear comparison with K41 scaling. However, the excited-state verification is currently incomplete. The first-excited-state NLSE residual is admitted to remain O(1), and higher excited states are validated only by an unshown comparison to Ref. [12]. Because these states feed directly into the chaotic dynamics study, the downstream Lyapunov and ESS claims inherit this unverified input. The central claim is therefore plausible but not yet established to the standard required for publication.","major_comments":[{"comment":"The authors state that after 25000 epochs the NLSE residual loss 'oscillates persistently about unity' and that a reduced learning rate 'would be necessary' for complete residual minimization, yet no refined calculation is reported. Since L_NLSE is the only term in Eq. (7) that directly enforces the steady-state equation Eq. (2), an O(1) residual means the reported first excited state is not demonstrated to be a solution of the NLSE to any stated accuracy. The same state is used in Eq. (8) for the j = 0 chaotic initial conditions in Table I, so this gap propagates into the chaos analysis. Please provide the refined run with a smaller learning rate and report the final L_NLSE value, or explicitly restrict the claims to energy-level convergence only.","section":"III.B, Fig. 3(a)"},{"comment":"For the higher excited states i = 2,...,9, the entire validation consists of the sentence 'showing agreement with Ref. [12]'; no comparison plot, error metric, or residual value is given. These states are obtained with symmetry penalties, overlap penalties, and hierarchical initialization that already encode the expected parity and node count, so matching the node structure of the noninteracting case is not an independent verification. Please report quantitative measures for every state and every g, such as the final L_NLSE, the maximum pointwise residual of Eq. (2), or the difference from the reference solutions in Ref. [12].","section":"III.C, Fig. 4"},{"comment":"The compact tanh network is fitted to the large-network solution with MSE 6.1 × 10^-6, but it is not checked against the steady-state equation Eq. (2). Calling this an 'analytical approximation of solutions' requires showing that the compact output itself approximately satisfies the NLSE, not merely that it fits a possibly unconverged state. Please report the steady-state residual of the compact network, or optimize it with the L_NLSE term included.","section":"IV, Fig. 5"},{"comment":"The Lyapunov exponents and ESS exponents are computed from initial states whose accuracy is unquantified, given the residuals discussed above. Positive Lyapunov exponents and K41-like ESS scaling cannot by themselves distinguish accurate dynamics from artifacts of inaccurate initial conditions. The authors should either provide a convergence metric for all input states or show that the reported λ and βp values are stable under the expected initial-state error.","section":"V, Table I and Fig. 6"}],"minor_comments":[{"comment":"The section heading 'INTERPREBILITY' should be corrected to 'INTERPRETABILITY'.","section":"IV (heading)"},{"comment":"The definition of L_NLSE uses the symbol N without a clear definition in the displayed equation; the text says 'N denotes wavefunction normalization,' but the operator to which N is applied is ambiguous. Please clarify, for example by writing the residual with the nonlinear eigenvalue μ expressed through the normalized Rayleigh quotient.","section":"II.B, Eq. (7)"},{"comment":"The legend in Fig. 3(a) uses 'NLSE residue' while the text uses 'NLSE residual'; please make the terminology consistent.","section":"III.A, Fig. 3(a)"},{"comment":"Reference [5] contains a garbled author list ('Cornell, and EA' instead of the actual authors); please correct it.","section":"References"},{"comment":"No data or code availability statement is provided. Given that all results are numerical, including code and hyperparameter details (seeds, optimizer settings, convergence thresholds) would substantially improve reproducibility.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The central verification gap is real and load-bearing: the paper's own text concedes that the first-excited-state residual remains O(1), and the higher states lack any quantitative benchmark. This is fixable within the scope of the manuscript by adding refined runs and residual comparisons, so I recommend major revision rather than rejection. I would also ask the editor to ensure the comparison to Ref. [12] is shown, since the current statement 'showing agreement' is not verifiable from the manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The new thing here is genuine: this is the first NNQS attempt at excited states of the NLSE, and the penalty-based orthogonalization with an overlap threshold plus hierarchical initialization is a sensible way to push beyond ground states. The compact tanh network (46 parameters) fitting ψ9 to MSE 6e-6 is also a nice touch, and the chaos application shows ambition. The authors are transparent about their main numerical problem: for the first excited state at g=100 the NLSE residual loss \"oscillates persistently about unity,\" and they admit a smaller learning rate is needed. That is an O(1) violation of the equation they claim to solve, and they still present that state as a first-excited-state solution. That is not a minor loose end; it is the central verification step, and it is missing.\n\nFor the higher states (i=2,...,9) there are no residuals at all, only a statement that they agree with Ref. [12]. No quantitative comparison, no error bars, no convergence curves. The hierarchical initialization and the explicit symmetry penalty can produce convincing-looking nodal profiles even when the state is not stationary, so the figures alone do not establish that these are solutions. The Lyapunov exponents and ESS analysis inherit this problem: if the input states are not actual stationary solutions, the chaotic dynamics computed from them are suspect. The \"strong ESS across all examined cases\" claim is also only supported by one displayed case (Fig. 6); the positive exponents in Table I look fine but come with no algorithmic detail.\n\nTo be fair, this is not a circular paper: the ground states are checked against imaginary time evolution, and the compact fitting is labeled as fitting, not prediction. The citations to the relevant continuation and Newton methods are appropriate, and the authors do flag the residual issue instead of hiding it. The method could plausibly be made to work—the loss landscape for excited states is hard, and they identify the right bottleneck. But as it stands, the paper promises a tool for computing excited states and does not deliver the verification that would make that claim credible.\n\nWho is this for? Researchers working on neural-network PDE solvers or NLSE numerics might be interested, but they would need to redo the validation themselves. I would send this to peer review because the idea deserves expert scrutiny and the required fixes are concrete: report residuals for every state, rerun the first-excited case with a reduced learning rate, compare higher states quantitatively against Ref. [12], and release code. If the residuals cannot be driven down, the method is not solving the NLSE. Right now it is a solid proposal in need of a demonstration.","headline":"Promising NNQS extension to excited states of the NLSE, but the central claim is unproven because the reported residuals stay O(1) and higher states get no quantitative checks.","tokens_in":11979,"tokens_out":2294,"would_cite":false,"duration_ms":31551,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q55","37D45","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"A neural-network wavefunction, trained by direct energy minimization with overlap and symmetry penalties, computes ground and excited steady states of the nonlinear Schrödinger equation, including states that imaginary time evolution…","keywords":["neural network quantum states","nonlinear Schrödinger equation","Gross-Pitaevskii equation","excited states","spatiotemporal chaos","Lyapunov exponents","extended self-similarity","variational energy minimization"],"falsifier":"Run the same optimization with the overlap threshold changed to, say, 0.2 or 0.8, or with the symmetry penalty removed, and compare the resulting states and energies against a high-accuracy continuation or spectral solver; if the states change identity or the residuals do not also vanish, the claim that the loss landscape supports distinct excited states is false. A simpler check is to compute the raw NLSE residual of the reported 46-parameter tanh approximation on a fine grid and see whether it decreases as the grid is refined.","tokens_in":10985,"feed_emoji":"🌊","tokens_out":10170,"duration_ms":111609,"temperature":0.7,"pith_summary":"This paper tries to establish that neural network quantum states, wavefunctions parameterized by multilayer perceptrons and trained by direct minimization of a composite energy loss, can compute both ground and excited steady states of the nonlinear Schrödinger equation. Imaginary time evolution reaches the ground state but fails for excited states, so a variational neural ansatz would fill a real gap. The paper demonstrates this on the one-dimensional Gross-Pitaevskii equation in a harmonic trap for interaction strengths g equal to 1, 10, and 100, obtaining states up to the ninth excited level. It then distills a hard solution into a 46-parameter tanh network, giving a continuous analytic approximation, and uses the excited states as initial data to show that the trapped condensate exhibits spatiotemporal chaos with positive Lyapunov exponents and extended self-similarity consistent with Kolmogorov scaling. If the claim holds, NNQS becomes a mesh-free, differentiable route to nonlinear excited states and a bridge from machine learning to chaotic wave studies.","feed_headline":"Compute excited states of a nonlinear wave equation with neural nets","feed_subtitle":"A neural wavefunction ansatz reaches ground and excited states, then seeds chaotic dynamics with Kolmogorov-like scaling.","key_machinery":"The load-bearing object is the composite loss functional of Eq. (7), which combines the real-wavefunction energy functional $E[\\psi]$, the normalization penalty $L_{\\mathrm{norm}} = (\\int \\psi^2\\,dr - 1)^2$, a conditional overlap penalty $L_{\\mathrm{excited}}$ for separating the current state from lower-energy states, and an NLSE residual term $L_{\\mathrm{NLSE}}$ that accelerates convergence. For states above the first excited level, an additional symmetry penalty $L_{\\mathrm{sym}} = \\mathrm{mean}\\big((|\\psi(x)| - |\\psi(-x)|)^2\\big)$ enforces parity. A hierarchical initialization, beginning from noninteracting eigenstates at $g=1$ and then reusing the solution for $g=10$ and $g=100$, guides the optimizer into high-lying states that would otherwise be difficult to reach. The interpretability machinery is a compact network with two hidden layers of five tanh nodes, whose 46 parameters give a closed analytical form suitable for further analysis.","core_discovery":"The paper's central claim is that direct minimization of a composite loss, $$L = E[\\psi] + \\lambda_{\\mathrm{norm}} L_{\\mathrm{norm}} + \\lambda_{\\mathrm{excited}} L_{\\mathrm{excited}} + \\lambda_{\\mathrm{NLSE}} L_{\\mathrm{NLSE}},$$ with the energy functional $E[\\psi]$ of Eq. (6), locates saddle points corresponding to excited states, not just the ground-state minimum. The overlap penalty $L_{\\mathrm{excited}}$ is applied only when the overlap $O_i$ with a previously found state exceeds the threshold $\\epsilon = 0.5$, which respects the non-orthonormality of nonlinear eigenstates. The authors report that the wavefunctions match imaginary time evolution for the ground and first excited states, and agree with a continuation-based solver for states 2 through 9; node counts match the noninteracting case even at $g=100$, with nodes compressed toward the trap center. They further claim that a compact network with two hidden layers of five tanh nodes, only 46 parameters, reproduces $\\psi_9$ at $g=100$ with mean squared error $6.1 \\times 10^{-6}$. Finally, superpositions of the ground state with odd excited states produce spatiotemporal chaos in all tested regimes, with structure functions showing extended self-similarity and scaling exponents near the Kolmogorov value $p/3$.","pith_inferences":["Inference: because the loss is differentiable and grid-free in principle, the same framework could be run in higher dimensions or with multi-component wavefunctions; the paper only demonstrates the one-dimensional scalar case, so that extension is untested.","Inference: the non-monotonic Lyapunov spectrum, with the largest value at $\\psi_5$ mixing for $\\alpha=1$, hints that a particular excited mode resonates with the cascade; a systematic scan over $\\alpha$ and state index could turn that hint into a testable resonance condition.","Inference: the Kolmogorov-like ESS scaling inside a harmonic trap may be tied to the Kohn-theorem decoupling of center-of-mass motion; repeating the structure-function analysis in an anharmonic trap would show whether the scaling is a universal feature of the chaos or a property of harmonic confinement."],"forward_implications":["Excited states of the NLSE become accessible through a purely variational neural ansatz, so the same loss construction should transfer to other nonlinear wave equations whose excited states are currently hard to reach.","The 46-parameter tanh approximation provides a continuous, differentiable surrogate for the wavefunction, which can be used directly in Galerkin projections or Bogoliubov–de Gennes stability analyses without grid interpolation.","Superpositions of the ground state with odd excited states in a harmonic trap display spatiotemporal chaos, evidenced by positive Lyapunov exponents across all tested interaction strengths and mixing ratios.","The observed extended self-similarity with scaling exponents near $p/3$ suggests Kolmogorov-like inter-order scaling in trapped-condensate density fluctuations, extending earlier findings for ground-plus-first-excited initial data to higher excited states."],"supporting_citations":[{"why":"Establishes the neural network quantum state paradigm of parameterizing wavefunctions and minimizing energy functionals, which this work adapts to the nonlinear Schrödinger equation.","marker":"[13, 14]"},{"why":"Extends NNQS to the Gross-Pitaevskii ground state; it is the prior nonlinear result this paper builds on and extends to excited states.","marker":"[16]"},{"why":"Supplies the continuation-method excited states used as the benchmark that the neural solutions are compared against.","marker":"[12]"},{"why":"Documents why imaginary time evolution fails for excited states without symmetry restrictions, motivating the new loss-based approach.","marker":"[8]"},{"why":"Defines the harmonically trapped BEC spatiotemporal chaos scenario and its ESS statistics, which the paper uses as the testbed for its excited states.","marker":"[6]"},{"why":"Introduces extended self-similarity, the scaling diagnostic used to show Kolmogorov-like inter-order scaling in the chaotic density fluctuations.","marker":"[43]"},{"why":"Provides the Gross-Pitaevskii and Bogoliubov–de Gennes framework and Thomas-Fermi interpretation used to analyze the computed states.","marker":"[4]"}],"fun_headline_variants":["Neural quantum states crack nonlinear Schrödinger excited states","Neural wavefunctions map out nonlinear Schrödinger steady states","Interpretable nets solve nonlinear Schrödinger ground and excited states","Neural ansatz finds nonlinear Schrödinger excited states, enabling chaos study"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method rests on the assumption that minimizing the composite loss, with the chosen penalties and the fixed overlap threshold, reliably produces the intended excited state for every interaction strength and state index up to nine, rather than drifting to some other stationary solution.","fun_headline_variants_meta":{"raw":{"variants":["Neural quantum states crack nonlinear Schrödinger excited states","Neural wavefunctions map out nonlinear Schrödinger steady states","Interpretable nets solve nonlinear Schrödinger ground and excited states","Neural ansatz finds nonlinear Schrödinger excited states, enabling chaos study"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000662,"raw_usage":{"total_tokens":3045,"prompt_tokens":983,"completion_tokens":2062,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":599,"completion_tokens_details":{"reasoning_tokens":1993}},"tokens_in":599,"tokens_out":2062,"duration_ms":16283,"temperature":1.0,"reasoning_tokens":1993,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:32:45.440364+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same optimization with the overlap threshold changed to, say, 0.2 or 0.8, or with the symmetry penalty removed, and compare the resulting states and energies against a high-accuracy continuation or spectral solver; if the states change identity or the residuals do not also vanish, the claim that the loss landscape supports distinct excited states is false. A simpler check is to compute the raw NLSE residual of the reported 46-parameter tanh approximation on a fine grid and see whether it decreases as the grid is refined.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Extends NNQS to the Gross-Pitaevskii ground state; it is the prior nonlinear result this paper builds on and extends to excited states."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the continuation-method excited states used as the benchmark that the neural solutions are compared against."},{"cited_title":"Bao, I.-L","cited_arxiv_id":null,"evidence_quote":"Documents why imaginary time evolution fails for excited states without symmetry restrictions, motivating the new loss-based approach."},{"cited_title":"Zhao, Physical Review A111, 033320 (2025)","cited_arxiv_id":null,"evidence_quote":"Defines the harmonically trapped BEC spatiotemporal chaos scenario and its ESS statistics, which the paper uses as the testbed for its excited states."}],"review_version":1}