Deep learning for the semi-classical limit of the Schr\"odinger equation

Jizu Huang; Rukang You; Tao Zhou

REVIEW 4 major objections 6 minor 52 references

Deep learning for the semi-classical limit of the Schr\"odinger equation

T0 review · 4 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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

Pith's one-line read This paper shows that rewriting the semiclassical Schrödinger equation as Gaussian wave packet ODEs lets MscaleDNNs stay accurate at ε = 1/6400, while physics-informed DeepONets map initial conditions to solutions without paired training da

desk verdict A clean integration of Gaussian wave packets with MscaleDNNs for harmonic potentials; for non-harmonic potentials the reported errors mix ansatz error with solver error, so the claimed advantage is not yet isolated. read the letter →

arxiv 2509.04453 v1 pith:WXIZWXUP submitted 2025-08-21 physics.comp-ph

Jizu Huang , Rukang You , Tao Zhou This is my paper · ORCID

classification physics.comp-ph

keywords SchrödingerequationsemiclassicallimitGaussianwavepacketsMscaleDNNphysics-informedneuralnetworksDeepONetsoperatorlearningmultiscalePDEs

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 claims that the reason neural networks fail on the Schrödinger equation in the semiclassical regime is the ε-scale oscillation in the wavefunction, not smallness of ε itself. It removes those oscillations by writing the solution as a single Gaussian wave packet whose center, momentum, width, and phase obey a four-equation ODE system; the PDE problem then becomes an ODE problem that multiscale networks can solve. For one initial condition, MscaleDNNs trained on these ODEs reproduce the wavefunction at ε as small as 1/6400 with relative L2 error around 8×10−3 for harmonic potentials, while standard MscaleDNNs already fail at ε = 1/32. For many initial conditions, a physics-informed DeepONet maps the initial packet parameters to their time evolution, with mean relative L2 errors 1.9–3.6×10−2 at ε = 0.01 and inference time almost independent of the number of initial conditions. The whole approach inherits the known limitation of single wave packets: for non-harmonic potentials it is accurate only up to the Ehrenfest time.

What carries the argument

The load-bearing object is the Gaussian wave packet ansatz (2.3), which turns the highly oscillatory PDE into the four-dimensional ODE system (2.4): q and p follow classical Hamiltonian motion, α controls width and chirp, and γ accumulates phase and normalization. Because the wave packet factors out the 1/ε oscillations, the remaining unknowns vary slowly enough for neural networks to learn. MscaleDNNs contribute input scaling embeddings that compress the long-tailed Fourier spectrum of the ODE solutions, while physics-informed DeepONets replace the operator on wavefunctions by a finite-dimensional operator on the packet parameters (q, p, α, γ), reducing the branch input from a sampled field

What would settle it

Run the method on V(x) = 1 − cos(x) with ε = 0.01 over a time horizon well beyond the Ehrenfest time, say T = 5 or 10, while verifying that the ODE system (2.4) is solved accurately by the neural network. If the reconstructed wavefunction's relative L2 error grows sharply despite accurate ODE solutions, the single-wave-packet assumption is the limiting mechanism and the method fails for long-time simulation, exactly as Remark 3.1 concedes.

Watch

Extended reading notes

Core claim

The central claim is that the multiscale difficulty of the semiclassical Schrödinger equation can be moved out of the wavefunction and into a small ODE system. Using the Gaussian wave packet ansatz ψ(x,t) = exp((i/ε)(α(t)(x−q(t))² + p(t)(x−q(t)) + γ(t))), the paper derives ODEs for q, p, α, γ whose solutions reconstruct the original wavefunction. MscaleDNNs solve these ODEs accurately enough that, after the 1/ε amplification inherent in the reconstruction, the final relative L2 error stays around 8×10−3 for harmonic potentials at ε = 1/6400, while direct MscaleDNNs on the original PDE fail for ε ≤ 1/32. For the operator problem, physics-informed DeepONets learn the map from initial packet pa

Load-bearing premise

For non-harmonic potentials, the single Gaussian wave packet remains a good approximation of the true solution over the whole time window [0,1]; the paper states this holds only up to the Ehrenfest time, beyond which the ansatz model error dominates and the network accuracy becomes irrelevant.

Editorial extensions

If this is right

For harmonic potentials, where the wave packet ansatz is exact, MscaleDNNs with Gaussian wave packets reach relative L2 errors below 1e-2 for ε values down to 1/6400, with error growing roughly linearly in 1/ε as the 1/ε amplification predicts.
For non-harmonic potentials, the method's accuracy is floored by the O(√ε) model error of the single wave packet; only below ε ≈ 1/100 does the network accuracy start to matter, and MscaleDNNs then beat PINNs by one to two orders of magnitude.
A single trained physics-informed DeepONet serves thousands to tens of thousands of initial conditions at inference times that stay nearly flat (about 3–5 seconds) while classical RK4 times grow linearly (about 7–46 seconds) at matched accuracy.
The same ODE-reformulation strategy extends to Hagedorn wave packets and Gaussian-beam decompositions, giving one-to-two-order accuracy gains for the former and near-constant evaluation cost as the number of Gaussian beams grows for WKB-type initial data.

Reading between the lines

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

Because the final solution error is the ODE error divided by ε, the practical range of ε is bounded below by how accurately the neural ODE solver can be trained; the 1/6400 result is a demonstration that this bound is reachable, not evidence that the method works for arbitrarily small ε.
The operator reformulation effectively converts an infinite-dimensional initial-condition-to-solution map into a finite-dimensional map on (q, p, α, γ), which is why the branch network sees only six numbers. The same trick should work for any initial data well approximated by a few wave packets, with cost growing with the number of packets.
A natural next test is to learn the residual between the wave-packet prediction and the true solution for non-harmonic potentials, extending accuracy beyond the Ehrenfest time; the paper names this direction in its remark but does not implement it.

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

Quick take before the details: this is a competent, honest paper that does what it says—recast the semiclassical Schrödinger equation as Gaussian wave packet ODEs and solve them with MscaleDNNs or PI-DeepONets. It is not a breakthrough, but it is a useful combination that appears to be new, and the numerical evidence for harmonic potentials is clean.

The best part is the harmonic case. There the wave packet ansatz is exact, so the reported relative L2 error is purely a statement about how well MscaleDNNs solve the ODE system, and the observed E_t/ε amplification matches the theory. The gains over plain PINNs are real (two orders of magnitude), and the behavior down to ε=1/6400 is convincing. The operator-learning section also has something new: PI-DeepONets that map Gaussian initial parameters to the ODE solution, with the expected 1/ε blowup in the final solution error. That is worth reporting.

The soft spot is the non-harmonic cases. The errors in Tables 3 and 4 are computed against the full PDE reference, but the single-Gaussian ansatz itself carries an O(√ε) model error that is never quantified. For V=1−cos(x), the MscaleDNN error is non-monotone in ε (3.4e-3 at 1/1600 vs 8.4e-3 at 1/6400), which suggests the reported number is dominated by something other than the neural solver. Remark 3.1 concedes the Ehrenfest-time limitation, but the paper never separates the ansatz error from the ODE error. Until that decomposition is done, the claim that MscaleDNNs outperform PINNs for the original PDE in non-harmonic potentials is not fully supported. Also, the PI-DeepONet loss enforces only the approximate ODE, so the operator learned is for the Gaussian wave packet dynamics, not the true Schrödinger operator; that is fine as long as it is stated, but it should be.

Minor issues: no code or data, loss weights in (2.1) are unspecified, and the MscaleDNN scaling range is chosen from the Fourier spectrum of the exact solution of the test problem, so the hyperparameter selection is partly benchmark-tuned. The runtime comparison excludes training time, which matters for a neural method.

Who is this for? Anyone building neural solvers for semiclassical Schrödinger or multiscale PDEs. It deserves a serious referee; the harmonic tests alone make it publishable. I would ask the authors to ship code and add an ansatz-error decomposition for the non-harmonic examples.

Referee Report

4 major / 6 minor

Summary. The paper proposes to compute the semiclassical limit of the Schrödinger equation (1.1) by first rewriting the solution as a single Gaussian wave packet (2.3) whose parameters satisfy the ODE system (2.4), then solving that ODE system with PINNs or MscaleDNNs and reconstructing the wave function via (2.6). For a family of Gaussian initial conditions, the paper replaces the PDE solution operator by the parameter-flow operator G': y(0)->y(t), learned in a physics-informed way by DeepONets. Numerical experiments cover 1D, 2D and 4D harmonic and non-harmonic potentials, with epsilon as small as 1/6400, and report that MscaleDNNs with Gaussian wave packets are much more accurate than PINNs and than standard MscaleDNNs applied directly to the PDE. Appendices extend the construction to Hagedorn wave packets and to Gaussian-beam decomposition of WKB initial data.

Significance. If the claims are fully supported, the paper makes a useful contribution: it combines a classical Gaussian-wave-packet reduction with multiscale neural solvers, so that the epsilon-resolution constraint is moved from the PDE to a low-dimensional ODE system, and it offers a clear operator-learning formulation for many initial conditions. The derivation of the ODE system (2.4) is correct, and the harmonic-potential experiments provide a clean setting in which the Gaussian ansatz has zero model error, making the observed 1/epsilon amplification of the ODE error a meaningful diagnostic. This is a strength. However, the paper's load-bearing claims for non-harmonic potentials and for the DeepONet operator are not yet fully supported because the ansatz error itself is never isolated from the neural-network ODE error, and several training hyperparameters (loss weights) are not reported. The contribution is valuable but needs a revision that separates approximation error from optimization error and states the reproducibility-critical settings.

major comments (4)

[Section 4.2.2, Table 3; Remark 3.1] For V=1-cos(x), the reported errors are end-to-end relative L2 errors against a full-PDE reference. The paper never reports the error of the Gaussian wave packet ansatz alone, i.e. the reconstruction (2.6) obtained from a highly accurate solver of the ODE system (2.4). The observed MscaleDNN errors are non-monotone in epsilon (3.408e-3 at epsilon=1/1600 versus 8.404e-3 at epsilon=1/6400), and for epsilon>=1/100 the PINN and MscaleDNN errors are nearly identical. This pattern is consistent with the ansatz model error dominating, as Remark 3.1 concedes that the ansatz is valid only up to the Ehrenfest time. Without a baseline that quantifies the ansatz error at each epsilon, the central claim that MscaleDNNs with Gaussian wave packets accurately solve the non-harmonic Schrödinger equation is not supported. Please add ansatz-only baseline errors (e.g. RK4 solution of (2.4) inserted into (2.
[Section 3.2, Eq. (3.8); Section 5.1] The physics-informed DeepONet loss enforces only the approximate ODE system (2.4), not the original PDE (1.1). Thus for non-harmonic potentials the learned operator is the Gaussian wave packet parameter flow, not the Schrödinger solution operator. The mean relative L2 errors reported in Section 5.1 (2.44e-2 for V=1-cos(x)) are therefore not evidence about operator learning for (1.1) unless the error of the single-Gaussian ansatz itself is quantified on the same test set. Please add a baseline that maps y(0) to the reference solution of (2.4) with a classical ODE solver and reconstructs the wave function, and state explicitly in the abstract/introduction that the operator is learned for the Gaussian wave packet reduction when the potential is non-harmonic.
[Eqs. (2.1)-(2.2) and Section 4 training settings] The loss weights wr, wb, wi in the PINN loss (2.1)-(2.2) are never assigned numerical values, and the ODE-system loss in Section 2.2 uses an analogous weighted combination that is also unspecified. Since the headline quantitative claim is a comparison between PINNs and MscaleDNNs, the comparison is not controlled or reproducible unless the weights and all training settings (learning-rate schedule, mini-batch details, number of random trials, initializations) are reported. Please provide the exact weights used in every experiment, and ideally release code/data so the reported factors of improvement can be verified.
[Section 4.2.1, Figure 7] The claim that 'standard MscaleDNNs fail' for epsilon<=1/32 is based on one configuration (six sub-networks with a_i=2^i). No numerical errors are given for Figure 7, and the comparison is not controlled by the same embedding count or scaling-coefficient tuning as the proposed MscaleDNN. Please report the actual relative L2 errors for the direct MscaleDNN baseline and state whether any attempt was made to optimize its scaling coefficients.

minor comments (6)

[Section 2.2, after Eq. (2.5)] The gamma residual in the loss uses gamma_{theta_3}; from the definition y_theta=(q_{theta_1}, p_{theta_2}, alpha_{theta_3}, gamma_{theta_4}) it should be gamma_{theta_4}.
[Section 2.3, Eq. (2.7)] The reconstruction formula uses the exponent d+1 for z in R^{d+1}. When the network input is only time t, this exponent should be clarified (d=0), otherwise the down-scaling normalization in the experiments is ambiguous.
[Section 4.2.1, text after Table 3] The text says MscaleDNNs improve accuracy by 'exceeding two orders of magnitude', but the ratios in Table 3 are around 50-95, i.e. 1.7-2.0 orders. Please rephrase to 'roughly two orders of magnitude' or report the actual ratios.
[Tables 5, 6, 8] The runtime comparison reports only online inference time for DeepONets, omitting the 200,000-iteration training cost and the hardware/software environment. Training cost should be stated, and the comparison should be labeled as 'inference-time comparison after training'.
[Appendix A] The appendix is described as an extension to 'more general initial conditions using Hagedorn wave packets', but the experiments assume all coefficients c_k(t)=1 and do not project a general initial state onto the Hagedorn basis. Please clarify this limitation in the appendix and in the main text where the appendix is advertised.
[Throughout] Several presentation typos remain: 'strandard variance' in Table 7, 'Physical-Informed' in the Section 3.2 heading, and an apparent duplication of the sentence about comparing PINNs and MscaleDNNs in the Introduction.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the Gaussian-wave-packet ODE reduction is external classical theory, neural solves are validated against independent references, and the only self-citation is non-load-bearing.

full rationale

The paper's derivation chain is self-contained and does not reduce to its inputs. The Gaussian wave packet ansatz (2.3) and the parameter ODE system (2.4) are standard semiclassical results imported from Heller [10] and Leung–Qian [44]; the paper does not derive them from its own claims. The neural networks (PINNs, MscaleDNNs, DeepONets) are trained to satisfy the ODE residual, not fitted to reference solutions. For harmonic potentials the reference ψ_ref is generated by RK4 on the same ODE system, but since the Gaussian wave packet is exact there, the reference is effectively an independent high-accuracy ODE solution and the comparison measures the neural ODE solver's error, amplified by the known 1/ε factor. For non-harmonic potentials the reference is a full fourth-order PDE solve (Chin–Chen scheme), so the reported errors include both the ansatz model error and the neural solver error; the paper explicitly separates these in §4.2.2 and Remark 3.1 concedes the Ehrenfest-time limitation. The single self-citation [36] appears only in a list of MscaleDNN applications and is not load-bearing; the MscaleDNN construction itself is cited to [31]. The choice of MscaleDNN scaling coefficients in §4.1 is informed by the Fourier spectrum of the exact ODE solution, which is hyperparameter tuning on the benchmark rather than fitted-input-called-prediction; it does not force the reported errors by construction. Overall, the central numerical claims are backed by independent comparisons and no prediction is equivalent to an input by definition.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The method rests on the classical Gaussian wave packet reduction, which is exact for harmonic potentials and approximate to O(√ε) for smooth non-harmonic potentials up to the Ehrenfest time. The neural networks introduce no new physical entities, but several hand-chosen hyperparameters (scaling coefficients, embedding count, network widths, iteration counts) are tuned on the benchmark examples.

free parameters (4)

MscaleDNN scaling coefficients a_i = 0.1, 0.2, ..., 10.0 (100 coefficients)
Chosen by hand to cover the frequency range of the solution; varies with the problem and affects accuracy (Fig. 6).
Number of MscaleDNN embeddings = 100
Selected in §4.1 based on the exact solution's Fourier spectrum; more embeddings improved accuracy up to 100, and 100 is used throughout.
DeepONet dimensions (J1...JI, branch/trunk widths) = e.g., I=6, J=100 per output; branch [6,100,100,100,100,600], trunk [1,100,100,100,100,100]
Hyperparameters chosen without a systematic study; the central claim (operator learning works) is demonstrated with these choices.
Training iterations = 100,000 (single IC) and 200,000 (DeepONet)
Ad hoc stopping criterion; no convergence check reported.

assumptions (4)

domain assumption Gaussian wave packet ansatz (2.3) is an exact solution for harmonic potentials and an O(√ε) approximation for smooth non-harmonic potentials.
Invoked in §2.2 to replace the PDE (1.1) with the ODE system (2.4); for non-harmonic potentials this is valid only up to the Ehrenfest time, as stated in Remark 3.1.
domain assumption Error in the final solution ψ from solving the ODE system with error E_t is E_t/ε.
Used in §2.2 and §4 to relate ODE errors to PDE errors; cited from [10,44], not proven in the paper.
domain assumption The operator G': y(0) → y(t) can be approximated by a finite-dimensional DeepONet trained only with ODE residuals and initial conditions.
Assumed in §3.2; relies on universal-approximation-type arguments from the operator learning literature [43].
standard math Solutions of the ODE system (2.4) exist and are unique on [0,1] for the smooth potentials considered.
Standard ODE theory for smooth V(x); implicitly assumed throughout.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Deep learning for the semi-classical limit of the Schr\"odinger equation." pith.science (2026). https://pith.science/paper/WXIZWXUP

@misc{pith2026250904453,
  author       = {Pith},
  title        = {Pith review of: Deep learning for the semi-classical limit of the Schr\"odinger equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WXIZWXUP}},
  note         = {Machine review of arXiv:2509.04453}
}

abstract

In this paper, we integrate neural networks and Gaussian wave packets to numerically solve the Schr\"odinger equation with a smooth potential near the semi-classical limit. Our focus is not only on accurately obtaining solutions when the non-dimensional Planck's constant, $\varepsilon$, is small, but also on constructing an operator that maps initial values to solutions for the Schr\"odinger equation with multiscale properties. Using Gaussian wave packets framework, we first reformulate the Schr\"odinger equation as a system of ordinary differential equations. For a single initial condition, we solve the resulting system using PINNs or MscaleDNNs. Numerical simulations indicate that MscaleDNNs outperform PINNs, improving accuracy by one to two orders of magnitude. When dealing with a set of initial conditions, we adopt an operator-learning approach, such as physics-informed DeepONets. Numerical examples validate the effectiveness of physics-informed DeepONets with Gaussian wave packets in accurately mapping initial conditions to solutions.

Figures

Figures reproduced from arXiv: 2509.04453 by the authors.

**Figure 1.** The structure of PINNs. Here PDE loss, BC loss, and IC loss are Lr(θ), Lb (θ), and Li (θ) in equation (2.1), respectively. The networks structure of the general PINNs is presented in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** The Schrödinger equation (1.1): the relative L 2 errors of PINNs corresponding to different ε values. 2.2. PINNs with Gaussian wave packets To simplify the introduction of Gaussian wave packets, we focus on the one-dimensional case (d = 1). This framework can be readily extended to higher-dimensional cases using tensor product grids. By expressing the solution as a sum of Gaussian wave packets, Heller [10] proposed … view at source ↗

**Figure 3.** Physics-Informed DeepONets: The DeepONets architecture [43] consists of two sub-networks: the branch network and the trunk network. The outputs of these sub-networks are combined using a dot product to produce a continuously differentiable representation of the output function. The optimal parameters are obtained by minimizing a loss function defined by the governing PDEs. The optimal parameters θ ∗ of the DeepONets… view at source ↗

Figures from the paper (13 more)

**Figure 4.** Figure 4: Physics-informed DeepONets structure of G′ θ . The branch networks extract the initial values of y(0), while the trunk networks only take the input coordinate t at which the output functions are evaluated. Remark 3.1. Due to the limitations of a single Gaussian wave pa…

**Figure 5.** Figure 5: (a) Distribution of Fourier coefficients Cn for the exact solution q(t). (b) The blue dotted line and the orange solid line represent the distribution of absolute errors in the Fourier coefficients obtained using PINNs and MscaleDNNs, respectively. 12 [PITH_FULL_IMAGE…

**Figure 6.** Figure 6: The relative L 2 errors of q(t) computed using MscaleDNNs with varying numbers of embeddings. Here, 0 corresponds to the results obtained using PINNs. 4.2. 1D examples In the following two 1D examples, we consider the domain Ω × [0, T] = (−π, π) × [0, 1]. For PINNs wit…

**Figure 7.** Figure 7: The comparison of the standard MscaleDNNs and MscaleDNNs with Gaussian wave packets. the relative L 2 error consists of two components: the model error of Gaussian wave packets and the error associated with solving the ODE system (2.4). The model error of Gaussian wave…

**Figure 8.** Figure 8: Schrödinger equation near the semi-classical limit with modified Henon-Heiles potential and ε = 1 25 . (a) Left: the real part of reference solutionψre(x, 1). Mid: the real part of prediction solutionψre(x, 1; θ). Right: absolute error of real part [PITH_FULL_IMAGE:fi…

**Figure 9.** Figure 9: Schrödinger equation near the semi-classical limit with modified Henon-Heiles potential and ε = 1 100 . (a) Left: the real part of reference solutionψre(x, 1). Mid: the real part of prediction solutionψre(x, 1; θ). Right: absolute error of real part [PITH_FULL_IMAGE:f…

**Figure 10.** Figure 10: 1D Schrödinger equation (1.1) with ε = 0.01, torsional potentia V(x) = 1 − cos(x), and q(0) = π 2 , p(0) = 0, α(0) = i, γ(0) = − 1 4 log 2 πε i. (a) Left: the real part of reference solution ψre(x, t). Mid: the real part of prediction solution ψre(x, t; θ). Right: …

**Figure 11.** Figure 11: 1D Schrödinger equation (1.1) with ε = 0.01, torsional potential V(x) = 1 − cos(x), and q(0) = 1, p(0) = 0, α(0) = i, γ(0) = − 1 4 log 2 πε i. (a) Left: the real part of reference solution ψre(x, t). Mid: the real part of prediction solution ψre(x, t; θ). Right: ab…

**Figure 12.** Figure 12: 1D Schrödinger equation (1.1) with ε = 0.01, torsional potential V(x) = 1 − cos(x), and q(0) = 1.5, p(0) = 0.2, α(0) = 0.2 + i, γ(0) = − 1 4 log 2 πε i. (a) Left: the real part of reference solution ψre(x, t). Mid: the real part of prediction solution ψre(x, t; θ).…

**Figure 13.** Figure 13: 1D Schrödinger equation (1.1) with ε = 0.01, harmonic potential V(x) = x 2 2 , and q(0) = 1, p(0) = 2, α(0) = i 2 , γ(0) = − 1 4 log 1 πε i: (a) Left: the real part of reference solution ψre(x, t). Mid: the real part of prediction solution ψre(x, t; θ). Right: abso…

**Figure 14.** Figure 14: 1D Schrödinger equation (1.1) with ε = 0.01, harmonic potential V(x) = x 2 2 , and q(0) = 0.8, p(0) = 2.2, α(0) = 2i 5 , γ(0) = − 1 4 log 4 5πε i: (a) Left: the real part of reference solution ψre(x, t). Mid: the real part of prediction solution ψre(x, t; θ). Right…

**Figure 15.** Figure 15: 2D Schrödinger equation with ε = 0.01, harmonic potential V(x) = x 2 1 +x 2 2 2 , and q(0) = (0.5, 0.5)T , p(0) = (1.0, 0.5)T , A(0) = idiag(1.0, 0.8), γ(0) = 0.75. (a) Left: the real part of reference solution ψre(x, 1). Mid: the real part of prediction solution ψre(…

**Figure 16.** Figure 16: 2D Schrödinger equation with ε = 0.01, harmonic potential V(x) = x 2 1 +x 2 2 2 , and q(0) = (0.8, 0.5)T , p(0) = (0.8, 0.5)T , A(0) = idiag(0.7, 0.7), γ(0) = 0.75. (a) Left: the real part of reference solution ψre(x, 1). Mid: the real part of prediction solution ψre(…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

52 extracted references · 48 canonical work pages

[1]

Guillot, J

J. Guillot, J. Ralston, E. Trubowitz, Semi-classical asymptotics in solid state physics, Communications in Mathematical Physics, 116 (1988), 401–415. 29

work page 1988
[2]

Cingolani, S

S. Cingolani, S. Secchi, Semiclassical limit for nonlinear Schrödinger equations with electromagnetic fields, Journal of Mathematical Analysis and Applications, 275 (2002), 108–130

work page 2002
[3]

Garashchuk, V

S. Garashchuk, V . Rassolov, O. Prezhdo, 6 semiclassical bohmian dynamics, Reviews in Computational Chemistry, 27 (2011), 287

work page 2011
[4]

Lasser, C

C. Lasser, C. Lubich, Computing quantum dynamics in the semiclassical regime, Acta Numerica, 29 (2020), 229–401

work page 2020
[5]

W. Bao, S. Jin, P . A. Markowich, On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, Journal of Computational Physics, 175 (2002), 487–524

work page 2002
[6]

Pathria, J

D. Pathria, J. L. Morris, Pseudo-spectral solution of nonlinear Schrödinger equations, Journal of Computational Physics, 87 (1990), 108–125

work page 1990
[7]

Descombes, M

S. Descombes, M. Thalhammer, An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi- classical regime, BIT Numerical Mathematics, 50 (2010), 729–749

work page 2010
[8]

E. Faou, C. Lubich, A poisson integrator for Gaussian wavepacket dynamics, Computing and Visual- ization in Science, 9 (2006), 45–55

work page 2006

Show all 52 references

[9]

G. A. Hagedorn, Semiclassical quantum mechanics: I. the ℏ→ 0 limit for coherent states, Communi- cations in Mathematical Physics, 71 (1980), 77–93

1980
[10]

E. J. Heller, Time-dependent approach to semiclassical dynamics, The Journal of Chemical Physics, 62 (1975), 1544–1555

1975
[11]

S. Jin, H. Wu, X. Yang, Gaussian beam methods for the Schrödinger equation in the semi-classical regime: Lagrangian and Eulerian formulations, Communications in Mathematical Sciences, 6 (2008), 995–1020

2008
[12]

LeCun, Y

Y. LeCun, Y. Bengio, G. Hinton, Deep learning, Nature, 521 (2015), 436–444

2015
[13]

C. M. Bishop, N. M. Nasrabadi, Pattern recognition and machine learning, volume 4, Springer, (2006)

2006
[14]

J. Li, Y. Chen, Solving second-order nonlinear evolution partial di fferential equations using deep learning, Communications in Theoretical Physics, 72 (2020), 105005

2020
[15]

Krizhevsky, I

A. Krizhevsky, I. Sutskever, G. E. Hinton, Imagenet classification with deep convolutional neural networks, Advances in Neural Information Processing Systems 25 (2012)

2012
[16]

B. M. Lake, R. Salakhutdinov, J. B. Tenenbaum, Human-level concept learning through probabilistic program induction, Science, 350 (2015), 1332–1338

2015
[17]

Raissi, P

M. Raissi, P . Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning frame- work for solving forward and inverse problems involving nonlinear partial di fferential equations, Journal of Computational Physics 378 (2019), 686–707

2019
[18]

Yu, et al., The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems, Communications in Mathematics and Statistics, 6 (2018), 1–12

B. Yu, et al., The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems, Communications in Mathematics and Statistics, 6 (2018), 1–12

2018
[19]

Y. Zang, G. Bao, X. Ye, H. Zhou, Weak adversarial networks for high-dimensional partial di fferential equations, Journal of Computational Physics, 411 (2020), 109409

2020
[20]

Y. L. Ming, et al., Deep Nitsche method: Deep Ritz method with essential boundary conditions, Communications in Computational Physics, 29 (2021), 1365–1384

2021
[21]

J. Han, A. Jentzen, et al., Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Communications in Mathematics and Statistics, 5 (2017), 349–380. 30

2017
[22]

J. Han, L. Zhang, R. Car, et al., Deep potential: A general representation of a many-body potential energy surface, Communications in Computational Physics, 23 (2018), 629

2018
[23]

J. Han, A. Jentzen, W. E, Solving high-dimensional partial di fferential equations using deep learning, Proceedings of the National Academy of Sciences, 115 (2018), 8505–8510

2018
[24]

J. He, L. Li, J. Xu, C. Zheng, Relu deep neural networks and linear finite elements, Journal of Computational Mathematics, 38 (2020), 502–527

2020
[25]

C. M. Strofer, J.-L. Wu, H. Xiao, E. Paterson, Data-driven, physics-based feature extraction from fluid flow fields using convolutional neural networks, Communications in Computational Physics, 25 (2019), 625–650

2019
[26]

Z. Wang, Z. Zhang, A mesh-free method for interface problems using the deep learning approach, Journal of Computational Physics, 400 (2020), 108963

2020
[27]

Rahaman, A

N. Rahaman, A. Baratin, D. Arpit, F. Draxler, M. Lin, F. Hamprecht, Y. Bengio, A. Courville, On the spectral bias of neural networks, in: International Conference on Machine Learning, PMLR, pp. 5301–5310
[28]

Z.-Q. J. Xu, Frequency principle: Fourier analysis sheds light on deep neural networks, Communications in Computational Physics 28 (2020), 1746–1767

2020
[29]

Zhang, Z.-Q

Y. Zhang, Z.-Q. J. Xu, T. Luo, Z. Ma, Explicitizing an implicit bias of the frequency principle in two-layer neural networks, arXiv preprint arXiv:1905.10264 (2019)

1905 arXiv
[30]

Z. J. Xu, Understanding training and generalization in deep learning by Fourier analysis, arXiv preprint arXiv:1808.04295 (2018)

2018 arXiv
[31]

Z. Liu, W. Cai, Z.-Q. J. Xu, Multi-scale deep neural network (MscaleDNN) for solving Poisson- Boltzmann equation in complex domains, Communications in Computational Physics, 28 (2020), 1970–2001

2020
[32]

Cai, Z.-Q

W. Cai, Z.-Q. J. Xu, Multi-scale deep neural networks for solving high dimensional PDEs, arXiv preprint arXiv:1910.11710 (2019)

1910 arXiv
[33]

B. Wang, W. Zhang, W. Cai, Multi-scale deep neural network (MscaleDNN) methods for oscillatory Stokes flows in complex domains, Communications in Computational Physics, 28 (2020), 2139–2157

2020
[34]

Li, Z.-Q

X.-A. Li, Z.-Q. J. Xu, L. Zhang, A multi-scale DNN algorithm for nonlinear elliptic equations with multiple scales, Communications in Computational Physics, 28 (2020), 1886–1906

2020
[35]

Li, Z.-Q

X.-A. Li, Z.-Q. J. Xu, L. Zhang, Subspace decomposition based DNN algorithm for elliptic type multi- scale PDEs, Journal of Computational Physics, 488 (2023), 112242

2023
[36]

Huang, R

J. Huang, R. You, T. Zhou, Frequency-adaptive multi-scale deep neural networks, Computer Methods in Applied Mechanics and Engineering, 437 (2025), 117751

2025
[37]

S. Jin, Z. Ma, K. Wu, Asymptotic-preserving neural networks for multiscale time-dependent linear transport equations, Journal of Scientific Computing, 94 (2023), 57

2023
[38]

S. Jin, K. Wu, et al., Asymptotic-preserving neural networks for multiscale kinetic equations, Commu- nications in Computational Physics, 35 (2024), 693–723

2024
[39]

S. Jin, Z. Ma, T.-a. Zhang, Asymptotic-preserving neural networks for multiscale Vlasov–Poisson– Fokker–Planck system in the high-field regime, Journal of Scientific Computing, 99 (2024), 61

2024
[40]

Russo, P

G. Russo, P . Smereka, The Gaussian wave packet transform: E fficient computation of the semi- classical limit of the Schrödinger equation. part 1–formulation and the one dimensional case, Journal of Computational Physics, 233 (2013), 192–209. 31

2013
[41]

E. Faou, V . Gradinaru, C. Lubich, Computing semiclassical quantum dynamics with Hagedorn wavepackets, SIAM Journal on Scientific Computing, 31 (2009), 3027–3041

2009
[42]

Wu, X.-B

K. Wu, X.-B. Yan, S. Jin, Z. Ma, Capturing the diffusive behavior of the multiscale linear transport equa- tions by asymptotic-preserving convolutional DeepONets, Computer Methods in Applied Mechanics and Engineering, 418 (2024), 116531

2024
[43]

L. Lu, P . Jin, G. Pang, Z. Zhang, G. E. Karniadakis, Learning nonlinear operators via DeepONets based on the universal approximation theorem of operators, Nature Machine Intelligence, 3 (2021), 218–229

2021
[44]

Leung, J

S. Leung, J. Qian, Eulerian gaussian beams for Schrödinger equations in the semi-classical regime, Journal of Computational Physics, 228 (2009), 2951–2977

2009
[45]

S. Wang, H. Wang, P . Perdikaris, Learning the solution operator of parametric partial di fferential equations with physics-informed DeepONets, Science Advances 7 (2021), eabi8605

2021
[46]

S. Wang, P . Perdikaris, Long-time integration of parametric evolution equations with physics-informed DeepONets, Journal of Computational Physics, 475 (2023), 111855

2023
[47]

Russo, P

G. Russo, P . Smereka, The Gaussian wave packet transform: efficient computation of the semi-classical limit of the Schrödinger equation. part 2. multidimensional case, Journal of Computational Physics, 257 (2014), 1022–1038

2014
[48]

S. A. Chin, C.-R. Chen, Fourth order gradient symplectic integrator methods for solving the time- dependent Schrödinger equation, The Journal of Chemical Physics, 114 (2001), 7338–7341

2001
[49]

Glorot, Y

X. Glorot, Y. Bengio, Understanding the di fficulty of training deep feedforward neural networks, in: Proceedings of the thirteenth international conference on artificial intelligence and statistics, JMLR Workshop and Conference Proceedings, pp. 249–256
[50]

D. P . Kingma, J. Ba, Adam: A method for stochastic optimization, in: Y. Bengio, Y. LeCun (Eds.), 3rd International Conference on Learning Representations, ICLR (2015), San Diego, CA, USA, May 7-9, Conference Track Proceedings

2015
[51]

Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, A. Anandkumar, Fourier neural operator for parametric partial di fferential equations, in: Advances in Neural Information Processing Systems, volume 33, Curran Associates, Inc., 2020, pp. 9460–9471

2020
[52]

Molinaro, Y

R. Molinaro, Y. Yang, B. Engquist, S. Mishra, Neural inverse operators for solving PDE inverse problems, in: Proceedings of the 40th International Conference on Machine Learning, volume 202 of Proceedings of Machine Learning Research. 32

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Guillot, J

J. Guillot, J. Ralston, E. Trubowitz, Semi-classical asymptotics in solid state physics, Communications in Mathematical Physics, 116 (1988), 401–415. 29

work page 1988

[2] [2]

Cingolani, S

S. Cingolani, S. Secchi, Semiclassical limit for nonlinear Schrödinger equations with electromagnetic fields, Journal of Mathematical Analysis and Applications, 275 (2002), 108–130

work page 2002

[3] [3]

Garashchuk, V

S. Garashchuk, V . Rassolov, O. Prezhdo, 6 semiclassical bohmian dynamics, Reviews in Computational Chemistry, 27 (2011), 287

work page 2011

[4] [4]

Lasser, C

C. Lasser, C. Lubich, Computing quantum dynamics in the semiclassical regime, Acta Numerica, 29 (2020), 229–401

work page 2020

[5] [5]

W. Bao, S. Jin, P . A. Markowich, On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, Journal of Computational Physics, 175 (2002), 487–524

work page 2002

[6] [6]

Pathria, J

D. Pathria, J. L. Morris, Pseudo-spectral solution of nonlinear Schrödinger equations, Journal of Computational Physics, 87 (1990), 108–125

work page 1990

[7] [7]

Descombes, M

S. Descombes, M. Thalhammer, An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi- classical regime, BIT Numerical Mathematics, 50 (2010), 729–749

work page 2010

[8] [8]

E. Faou, C. Lubich, A poisson integrator for Gaussian wavepacket dynamics, Computing and Visual- ization in Science, 9 (2006), 45–55

work page 2006