Interpretable Neural Network Quantum States for Solving the Steady States of the Nonlinear Schr\"odinger Equation

Mingshu Zhao; Zhanyuan Yan

REVIEW 4 major objections 5 minor 62 references

Interpretable Neural Network Quantum States for Solving the Steady States of the Nonlinear Schr\"odinger Equation

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2506.10219 v1 pith:2UJZAPU5 submitted 2025-06-11 nlin.CD

Mingshu Zhao , Zhanyuan Yan This is my paper · ORCID

classification nlin.CD MSC 35Q5537D4568T07

keywords neuralnetworkquantumstatesnonlinearSchrödingerequationGross-PitaevskiiexcitedspatiotemporalchaosLyapunovexponentsextendedself-similarityvariationalenergyminimization

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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $p/3$.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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.

Desk Editor's Note

read the 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.

For 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.

To 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.

Who 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.

Referee Report

4 major / 5 minor

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.

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 (4)

[III.B, Fig. 3(a)] 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.
[III.C, Fig. 4] 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].
[IV, Fig. 5] 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.
[V, Table I and Fig. 6] 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.

minor comments (5)

[IV (heading)] The section heading 'INTERPREBILITY' should be corrected to 'INTERPRETABILITY'.
[II.B, Eq. (7)] 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.
[III.A, Fig. 3(a)] The legend in Fig. 3(a) uses 'NLSE residue' while the text uses 'NLSE residual'; please make the terminology consistent.
[References] Reference [5] contains a garbled author list ('Cornell, and EA' instead of the actual authors); please correct it.
[General] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the NNQS claim rests on the direct variational principle and independent external benchmarks.

full rationale

No circular reduction is present in the derivation chain. The ground state is obtained by minimizing the energy functional E in Eq. (6), and the equivalence between saddle points of E and steady-state solutions of Eq. (2) is derived directly through the Euler-Lagrange equations in Eqs. (3)-(5), not imported from prior work or from the method's own outputs. Excited states are computed by minimizing the composite loss in Eq. (7), which includes an explicit NLSE residual term; the paper's admission in Section III.B that the first-excited residual at g=100 'oscillates persistently about unity' and that a reduced learning rate would be needed is a convergence and accuracy limitation, not a circular step, because the resulting profiles are validated against imaginary time evolution and the independent, external reference [12] by Liu, Xie, and Yuan. Higher excited states are also benchmarked against that same external reference rather than against the paper's own fitted quantities. The compact tanh network in Section IV is explicitly described as 'fitting the solutions' with a reported MSE of 6.1e-6, so the interpretable approximation is presented as a fit, not as an independent prediction. The spatiotemporal-chaos analysis consumes the computed states and is checked by standard diagnostics such as Lyapunov exponents and extended self-similarity, so no fitted parameter is renamed as a prediction. The self-citations (Refs. [6], [7], and [62]) provide contextual and methodological background for the chaos portion and are not load-bearing for the central NNQS claim, which stands on the direct variational principle and external validation. Accordingly, the paper is self-contained with respect to its central claim, and there is no definitional, fitted-input, or self-citation circularity.

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

The central claim rests on hand-chosen penalty weights and thresholds, on the variational saddle-point correspondence, and on domain assumptions about parity, Kohn's theorem, and GPE validity. No new physical entities are introduced.

free parameters (6)

lambda_norm = 100
Normalization penalty weight in Eq. (7), hand-chosen to enforce the normalization constraint; no sensitivity analysis is given.
lambda_excited = 20
Excited-state overlap penalty weight in Eq. (7); hand-chosen with no sensitivity analysis.
lambda_NLSE = 10
Steady-state residual penalty weight in Eq. (7); hand-chosen and described as accelerating convergence.
epsilon = 0.5
Overlap threshold in the excited-state loss; described as nonlinearity-dependent but fixed at 0.5.
learning rate (Adam) = 5e-5
Used for all reported runs; the authors state it is too large for full NLSE residual minimization.
compact network parameters (46) = not provided
Weights of the two-hidden-layer tanh network fitted to psi_9 at g=100; mean squared error is 6.1e-6, but the parameters are not listed.

assumptions (5)

standard math Steady states of the NLSE correspond to saddle points of the energy functional E[psi, psi*] with a normalization constraint.
Invoked in Eqs. (3)-(5) to motivate the variational loss.
domain assumption For stationary solutions in a 1D real potential, the wavefunction can be taken real.
Section II.A; neglects phase or current degrees of freedom.
ad hoc to paper Each target excited state has definite parity and the node count of the noninteracting case.
The symmetry penalty L_sym introduced after Fig. 4 enforces |psi(x)| = |psi(-x)| and assumes this selects the physical state.
domain assumption Kohn's theorem decouples center-of-mass and relative motion in the harmonic trap.
Section V.B; used to build the ensemble of density profiles for structure function analysis.
domain assumption The Gross-Pitaevskii equation remains valid up to the Ehrenfest time tau_E = ln(N)/lambda with N = 1e5.
Section V.A; restricts the statistical ensemble to n <= 100 snapshots.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Interpretable Neural Network Quantum States for Solving the Steady States of the Nonlinear Schr\"odinger Equation." pith.science (2026). https://pith.science/paper/2UJZAPU5

@misc{pith2026250610219,
  author       = {Pith},
  title        = {Pith review of: Interpretable Neural Network Quantum States for Solving the Steady States of the Nonlinear Schr\"odinger Equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2UJZAPU5}},
  note         = {Machine review of arXiv:2506.10219}
}

read the original abstract

The nonlinear Schr\"odinger equation (NLSE) underpins nonlinear wave phenomena in optics, Bose-Einstein condensates, and plasma physics, but computing its excited states remains challenging due to nonlinearity-induced non-orthonormality. Traditional methods like imaginary time evolution work for ground states but fail for excited states. We propose a neural network quantum state (NNQS) approach, parameterizing wavefunctions with neural networks to directly minimize the energy functional, enabling computation of both ground and excited states. By designing compact, interpretable network architectures, we obtain analytical approximation of solutions. We apply the solutions to a case of spatiotemporal chaos in the NLSE, demonstrating its capability to study complex chaotic dynamics. This work establishes NNQS as a tool for bridging machine learning and theoretical studies of chaotic wave systems.

Figures

Figures reproduced from arXiv: 2506.10219 by the authors.

**Figure 2.** FIG. 2. Optimization history and ground state solutions. (a) [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Optimization history and first excited state solutions. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. Excited state solutions [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5. Interpretable NNQS architecture and performance. [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 6.** Figure 6: FIG. 6. Statistics of the initial [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

62 extracted references · 49 canonical work pages

[12]

W. Liu, Z. Xie, and Y. Yuan, Journal of Computational Physics 473, 111719 (2023)

work page 2023
[1]

Y. V. Kartashov, B. A. Malomed, and L. Torner, Reviews of Modern Physics83, 247 (2011)

work page 2011
[2]

Becker, S

C. Becker, S. Stellmer, P. Soltan-Panahi, S. Dörscher, M. Baumert, E.-M. Richter, J. Kronjäger, K. Bongs, and K. Sengstock, Nature Physics4, 496 (2008)

work page 2008
[3]

Kuznetsov, A

E. Kuznetsov, A. Rubenchik, and V. E. Zakharov, Physics Reports 142, 103 (1986)

work page 1986
[4]

Pitaevskii and S

L. Pitaevskii and S. Stringari,Bose-Einstein condensa- tion and superfluidity, Vol. 164 (Oxford University Press, 2016)

2016
[5]

Schweikhard, I

V. Schweikhard, I. Coddington, P. Engels, S. Tung, Cor- nell, and EA, Physical review letters93, 210403 (2004)

work page 2004
[6]

Zhao, Physical Review A111, 033320 (2025)

M. Zhao, Physical Review A111, 033320 (2025)

work page 2025
[7]

M. Zhao, J. Tao, and I. Spielman, Physical Review Let- ters 134, 083402 (2025)

work page 2025

Show all 62 references

[8]

Bao, I.-L

W. Bao, I.-L. Chern, and F. Y. Lim, Journal of Compu- tational Physics 219, 836 (2006)

2006
[9]

Chang and C.-S

S.-L. Chang and C.-S. Chien, Computer Physics Com- munications 177, 707 (2007)

2007
[10]

Chen, I.-L

J.-H. Chen, I.-L. Chern, and W. Wang, Journal of Com- putational Physics 230, 2222 (2011)

2011
[11]

Marojević, E

Ž. Marojević, E. Göklü, and C. Lämmerzahl, Computer Physics Communications 184, 1920 (2013)

2013
[13]

M.MedvidovićandJ.R.Moreno,TheEuropeanPhysical Journal Plus 139, 1 (2024)

2024
[14]

H.Lange, A.VandeWalle, A.Abedinnia,andA.Bohrdt, Quantum Science and Technology (2024)

2024
[15]

I. L. Gutiérrez and C. B. Mendl, Quantum6, 627 (2022)

2022
[16]

W. Bao, Z. Chang, and X. Zhao, Journal of Computa- tional Physics 520, 113486 (2025)

2025
[17]

S.Poudel, X.Wang,andS.Lee,EngineeringApplications of Artificial Intelligence133, 108313 (2024)

2024
[18]

K. W. Morton and D. F. Mayers, Numerical solution of partial differential equations: an introduction(Cam- bridge university press, 2005)

2005
[19]

Johnson, Numerical solution of partial differential equations by the finite element method(Courier Corpo- ration, 2009)

C. Johnson, Numerical solution of partial differential equations by the finite element method(Courier Corpo- ration, 2009)

2009
[20]

Galerkin, Vestnik inzhenerov1, 897 (1915)

B. Galerkin, Vestnik inzhenerov1, 897 (1915)

1915
[21]

Bland, N

T. Bland, N. G. Parker, N. P. Proukakis, and B. A. Mal- omed, Journal of Physics B: Atomic, Molecular and Op- tical Physics 51, 205303 (2018)

2018
[22]

E. V. Podivilov, D. S. Kharenko, V. Gonta, K. Krupa, O. S. Sidelnikov, S. Turitsyn, M. P. Fedoruk, S. A. Babin, and S. Wabnitz, Physical review letters 122, 103902 (2019)

2019
[23]

D. E. Rumelhart, G. E. Hinton, and R. J. Williams, na- ture 323, 533 (1986)

1986
[24]

L. Bottou, inProceedings of COMPSTAT’2010: 19th In- ternational Conference on Computational StatisticsParis France, August 22-27, 2010 Keynote, Invited and Con- tributed Papers(Springer, 2010) pp. 177–186

2010
[25]

D. P. Kingma and J. Ba, arXiv preprint arXiv:1412.6980 (2014)

2014 arXiv
[26]

A. G. Baydin, B. A. Pearlmutter, A. A. Radul, and J. M. Siskind, Journal of machine learning research18, 1 (2018)

2018
[27]

Stokes, J

J. Stokes, J. Izaac, N. Killoran, and G. Carleo, Quantum 4, 269 (2020)

2020
[28]

Amari, Information geometry and its applications, Vol

S.-i. Amari, Information geometry and its applications, Vol. 194 (Springer, 2016)

2016
[29]

Martens and R

J. Martens and R. Grosse, inInternational conference on machine learning(PMLR, 2015) pp. 2408–2417

2015
[30]

Paszke, S

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, (2017)

2017
[31]

Hendrycks and K

D. Hendrycks and K. Gimpel, arXiv preprint arXiv:1606.08415 (2016)

2016 arXiv
[32]

Zwiebach,Mastering quantum mechanics: essentials, theory, and applications(MIT Press, 2022)

B. Zwiebach,Mastering quantum mechanics: essentials, theory, and applications(MIT Press, 2022)

2022
[33]

Cranmer, arXiv preprint arXiv:2305.01582 (2023)

M. Cranmer, arXiv preprint arXiv:2305.01582 (2023)

2023 arXiv
[34]

Makke and S

N. Makke and S. Chawla, Artificial Intelligence Review 57, 2 (2024)

2024
[35]

Cranmer, A

M. Cranmer, A. Sanchez Gonzalez, P. Battaglia, R. Xu, K. Cranmer, D. Spergel, and S. Ho, Advances in neural information processing systems33, 17429 (2020)

2020
[36]

Z. Liu, Y. Wang, S. Vaidya, F. Ruehle, J. Halverson, M. Soljačić, T. Y. Hou, and M. Tegmark, arXiv preprint arXiv:2404.19756 (2024)

2024 arXiv
[37]

Z. Liu, P. Ma, Y. Wang, W. Matusik, and M. Tegmark, arXiv preprint arXiv:2408.10205 (2024)

2024 arXiv
[38]

A. N. Kolmogorov,On the representation of continuous functions of several variables by superpositions of contin- uous functions of a smaller number of variables(Ameri- can Mathematical Society, 1961)

1961
[39]

De Boor and C

C. De Boor and C. De Boor,A practical guide to splines, Vol. 27 (springer New York, 1978)

1978
[40]

D. A. Takahashi, M. Kobayashi, and M. Nitta, Physical Review B 91, 184501 (2015)

2015
[41]

J. O. Indekeu, C.-Y. Lin, N. Van Thu, B. Van Schaey- broeck, and T. H. Phat, Physical Review A91, 033615 (2015)

2015
[42]

Kolmogorov, Turbulence: the legacy of AN Kol- mogorov

A. Kolmogorov, Turbulence: the legacy of AN Kol- mogorov
[43]

Benzi, S

R. Benzi, S. Ciliberto, R. Tripiccione, C. Baudet, F. Mas- saioli, and S. Succi, Physical Review E48, R29 (1993)

1993
[44]

S. H. Strogatz,Nonlinear dynamics and chaos: with ap- plications to physics, biology, chemistry, and engineering 9 (CRC press, 2018)

2018
[45]

Březinová, A

I. Březinová, A. U. Lode, A. I. Streltsov, O. E. Alon, L. S. Cederbaum, and J. Burgdörfer, Physical Review A 86, 013630 (2012)

2012
[46]

Ehrenfest, Zeitschrift für physik45, 455 (1927)

P. Ehrenfest, Zeitschrift für physik45, 455 (1927)

1927
[47]

G. P. Berman and G. M. Zaslavsky, Physica A: Statistical Mechanics and its Applications91, 450 (1978)

1978
[48]

Han and B

X. Han and B. Wu, Physical Review A93, 023621 (2016)

2016
[49]

Rammensee, J

J. Rammensee, J. D. Urbina, and K. Richter, Physical Review Letters 121, 124101 (2018)

2018
[50]

Wanzenböck, S

R. Wanzenböck, S. Donsa, H. Hofstätter, O. Koch, P. Schlagheck, and I. Březinová, Physical Review A103, 023336 (2021)

2021
[51]

Kohn, Physical Review123, 1242 (1961)

W. Kohn, Physical Review123, 1242 (1961)

1961
[52]

L. Brey, N. Johnson, and B. Halperin, Physical Review B 40, 10647 (1989)

1989
[53]

J. F. Dobson, Physical Review Letters73, 2244 (1994)

1994
[54]

Reynolds, Philosophical transactions of the royal so- ciety of london.(a.) , 123 (1895)

O. Reynolds, Philosophical transactions of the royal so- ciety of london.(a.) , 123 (1895)
[55]

A. N. Kolmogorov, inDokl. Akad. Nauk SSSR, Vol. 31 (1941) pp. 538–540

1941
[56]

A. N. Kolmogorov, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 434, 9 (1991)

1991
[57]

A. N. Kolmogorov, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 434, 15 (1991)

1991
[58]

P. J. Schmid, Journal of fluid mechanics656, 5 (2010)

2010
[59]

P. J. Schmid, Annual Review of Fluid Mechanics54, 225 (2022)

2022
[60]

K. F. Thomas, M. J. Davis, and K. V. Kheruntsyan, Physical Review A103, 023315 (2021)

2021
[61]

Orlova, A

E. Orlova, A. Ustimenko, R. Jiang, P. Y. Lu, and R. Wil- lett, arXiv preprint arXiv:2305.19685 (2023)

2023 arXiv
[62]

Y. Geng, J. Tao, M. Zhao, S. Mukherjee, S. Eckel, G. K. Campbell, and I. B. Spielman, arXiv preprint arXiv:2411.19807 (2024)

2024 arXiv

Pith tools

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

[1] [12]

W. Liu, Z. Xie, and Y. Yuan, Journal of Computational Physics 473, 111719 (2023)

work page 2023

[2] [1]

Y. V. Kartashov, B. A. Malomed, and L. Torner, Reviews of Modern Physics83, 247 (2011)

work page 2011

[3] [2]

Becker, S

C. Becker, S. Stellmer, P. Soltan-Panahi, S. Dörscher, M. Baumert, E.-M. Richter, J. Kronjäger, K. Bongs, and K. Sengstock, Nature Physics4, 496 (2008)

work page 2008

[4] [3]

Kuznetsov, A

E. Kuznetsov, A. Rubenchik, and V. E. Zakharov, Physics Reports 142, 103 (1986)

work page 1986

[5] [4]

Pitaevskii and S

L. Pitaevskii and S. Stringari,Bose-Einstein condensa- tion and superfluidity, Vol. 164 (Oxford University Press, 2016)

2016

[6] [5]

Schweikhard, I

V. Schweikhard, I. Coddington, P. Engels, S. Tung, Cor- nell, and EA, Physical review letters93, 210403 (2004)

work page 2004

[7] [6]

Zhao, Physical Review A111, 033320 (2025)

M. Zhao, Physical Review A111, 033320 (2025)

work page 2025

[8] [7]

M. Zhao, J. Tao, and I. Spielman, Physical Review Let- ters 134, 083402 (2025)

work page 2025