Enhancing Open Quantum Dynamics Simulations Using Neural Network-Based Non-Markovian Stochastic Schr\"odinger Equation Method

Kaihan Lin; Xing Gao

REVIEW 4 major objections 5 minor 14 references

Enhancing Open Quantum Dynamics Simulations Using Neural Network-Based Non-Markovian Stochastic Schr\"odinger Equation Method

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read The paper claims that a CNN–LSTM network with iterative attentional feature fusion, trained on the converged early-time part of non-Markovian stochastic Schrödinger trajectories, extrapolates long-time low-temperature dynamics accurately…

desk verdict Useful and mostly sound ML accelerator for cNMSSE, but the headline speedup is not reliably established from the reported cost accounting. read the letter →

arxiv 2411.15914 v1 pith:SYANDUFI submitted 2024-11-24 quant-ph physics.chem-ph

Kaihan Lin , Xing Gao This is my paper · ORCID

classification quant-phphysics.chem-ph

keywords openquantumsystemsnon-MarkovianstochasticSchrödingerequationneuralnetworktime-seriespredictionLSTMattentionalfeaturefusionspin-bosonmodelFMOcomplexlow-temperaturedynamics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes NN-NMSSE, a scheme that wraps a neural network around the non-Markovian stochastic Schrödinger equation to break the low-temperature convergence bottleneck of trajectory sampling. The core claim is that a CNN-LSTM network with iterative attentional feature fusion, trained on the early, well-converged part of the stochastic ensemble, can serve as a nonlinear dynamical map that extrapolates the full long-time reduced dynamics with few additional trajectories. The authors report spin-boson and FMO-complex population deviations from an exact HEOM benchmark below about 0.02, while cutting the trajectory count in the flagship spin-boson case from 700,000 to 180,000 and the total computation time from 1,400,000 to 364,050 minutes. If these results hold more generally, long-time and steady-state open quantum simulations at low temperature become much cheaper, since the dominant sampling cost is replaced by a one-time network training cost.

What carries the argument

The central object is the neural dynamical map: a hybrid 1D-CNN + LSTM + iAFF architecture that ingests sliding windows of the reduced density matrix and outputs the next time step. Diagonal population differences and the real and imaginary parts of off-diagonal coherences are fed through separate CNN–LSTM branches, and the iterative attentional feature fusion module combines the branches adaptively. Training proceeds in two phases: pre-training on the full trajectory average (including the oscillating tail) followed by fine-tuning on the converged portion with standard error below $\epsilon_1$. The self-validation mechanism is the second threshold $\epsilon_2$: ten networks are trained on ensembles of increasing trajectory counts, and the scheme only stops when the standard deviation of their long-time predictions falls below $\epsilon_2$, giving an internal consistency check on the extrapolation.

What would settle it

Apply NN-NMSSE to a spin-boson or FMO parameter set with a slow underdamped bath mode whose correlation time is longer than the converged training window (for example, a spectral density with a narrow low-frequency peak), and compare the network's long-time population prediction against an exact HEOM or tensor-network calculation. If the population deviation from the exact benchmark grows beyond the roughly 0.02 level claimed here, the short-time-extrapolation assumption is falsified. A simpler diagnostic is to check whether the prediction remains accurate when the training window is shifted or extended while the validation data is held fixed.

Watch

Extended reading notes

Core claim

The central claim is that the non-Markovian stochastic Schrödinger equation in its complex-mode form (cNMSSE) can be accelerated at low temperatures by training a deep network on converged short-time trajectory data and letting that network propagate the dynamics over long times. The reduced density matrix is first converted into a compact vector of population differences and real/imaginary parts of coherences; sliding windows of these vectors feed parallel CNN-LSTM streams, whose features are merged by iterative attentional feature fusion (iAFF). The network is pre-trained on the full noisy trajectory set and fine-tuned on the subset whose standard error is below a threshold $\epsilon_1$. To avoid blind forecasting, ten ensembles with increasing trajectory counts are generated, ten networks are trained, and the final prediction is accepted only when the standard deviation across the ten predictions falls below $\epsilon_2$. In the numerical demonstrations this scheme reproduces the HEOM benchmark to within roughly 0.02 in population, and it reduces the required number of stochastic trajectories from 700,000 to 180,000 in the spin-boson case ($\beta=5.0$, $\gamma=5.0$), with total computation time dropping from 1,400,000 to 364,050 minutes.

Load-bearing premise

The method assumes that a network trained only on the early, well-converged part of the stochastic trajectories can extrapolate the full long-time dynamics because that early segment encodes all memory effects that matter. If slow bath modes or long-lived correlations extend beyond the training window, the extrapolation could be wrong even when the ten predictions agree with each other, since the standard-deviation check measures internal agreement, not agreement with the true dynamics.

Editorial extensions

If this is right

For the low-temperature spin-boson regimes tested, converged long-time population dynamics can be obtained with roughly one-quarter of the previous computation time, making steady-state and long-time properties cheaper to reach.
Because the network is trained on reduced density matrix elements rather than on the full system wavefunction, the same extrapolation idea transfers to larger systems handled by cNMSSE with matrix product states.
The method provides a practical diagnostic before investing in it: if the standard error of the stochastic trajectories grows with time, NN-NMSSE is likely to help; if not, plain NMSSE already converges and the neural wrapper adds little.
The reported reductions (700,000 to 180,000 trajectories in experiment (a); 38,000 to 30,000 in the 77 K FMO case) mean the remaining dominant cost is the initial NMSSE data generation, so the method is most economical when training data can be reused across nearby parameter sets.

Reading between the lines

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

If the short-time training window captures all relevant memory effects, the scheme should transfer to other stochastic unravelings besides cNMSSE, such as HOPS, and to structured spectral densities beyond Debye-Drude; a natural test is an underdamped or long-correlation bath.
The standard-deviation stopping rule only certifies consistency among the ten networks, not accuracy against the true dynamics; the trustworthy validation in this paper comes from the HEOM comparison, so a user should not rely on $\epsilon_2$ alone.
One could reduce the initial data cost by transfer learning across parameters: fine-tune a network trained at one $\beta$ or $\gamma$ for a neighboring value, amortizing the expensive trajectory generation over a parameter scan.
In the FMO case the trajectory reduction is modest (38,000 to 30,000), suggesting the benefit scales with how rapidly the stochastic noise accumulates; systems with moderate noise may see only marginal gains.

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 paper shows, convincingly, that a CNN+LSTM+iAFF network trained on short-time cNMSSE data can extrapolate spin-boson and FMO dynamics, with population deviations from HEOM staying below about 0.02. That is a real result. The figures where NMSSE with the same trajectory budget oscillates while the NN prediction stays smooth carry the argument. The architecture is a new combination applied to cNMSSE, and the paper is honest that the 'learn a map from short-time data' idea comes from TTM/GQME and earlier machine-learning accelerators.

The soft spot is the cost accounting. Section 3.5 says 10 trajectory sets (N1...N10) are generated for the convergence loop, but the cost table for experiment (a) charges only 180,000 trajectories at 2 minutes each. Where are the other nine sets? The line items also don't reconcile: grid search 375 + '10 groups' training 3,750 + prediction 300 = 4,425, not the stated 4,050. The FMO comparison is even thinner: 38,000 vs 30,000 trajectories, about 21%, and the 10-group overhead could erase that. Since the headline claim is computational advantage, this matters. I don't think the method is invalid; the benchmarks against HEOM support the accuracy claim. But the speedup factor is not robustly established from the reported numbers.

Other issues, in increasing order of seriousness: no code or data, hyperparameters deferred, and the internal convergence criterion (SD across 10 predictions) checks agreement among predictions, not with exact dynamics. The HEOM comparison in the main figures mitigates that last one, but the criterion alone would not catch a systematically wrong extrapolation. A revision should give a complete cost table counting all generated trajectories, plus sensitivity to epsilon_1 and epsilon_2.

For whom: people doing low-temperature stochastic open-quantum dynamics, cNMSSE/HOPS users, and anyone benchmarking ML accelerators for open systems. It is a subfield accelerator, not a new physical framework.

Recommendation: send it to a serious referee. The physics is plausible, the benchmark is external, and the flaw is in the cost presentation rather than the core method. A referee should ask for a corrected cost breakdown and code or data release.

Referee Report

4 major / 5 minor

Summary. The manuscript proposes NN-NMSSE, a hybrid method in which cNMSSE trajectory data are used to train a neural network composed of CNNs, LSTMs, and iterative attentional feature fusion, and an ensemble of ten networks trained on increasing trajectory counts is used to propagate the reduced density matrix beyond the converged early-time window. The method is benchmarked against HEOM on six spin-boson parameter sets (low/high temperature and adiabatic/non-adiabatic regimes) and on FMO excitation energy transfer at 77 K and 300 K. The authors report population deviations from HEOM below about 0.02 and claim substantial reductions in trajectory number and computation time, for example in experiment (a) a reduction from 700,000 to 180,000 trajectories and from 1,400,000 to 364,050 minutes.

Significance. If the computational savings are real, the method addresses a genuine obstacle: low-temperature NMSSE simulations converge slowly, and learned propagators are a natural complement to stochastic trajectory generation. Strengths of the manuscript include external HEOM benchmarking, coverage of several parameter regimes, and a reasonably clear description of the network architecture. The numerical examples do show that smooth populations close to the HEOM results can be produced in the tested cases. However, the central quantitative claim is not yet robustly established: the cost accounting in Section 3.5 and the Numerical Results omits or obscures the cost of the ten-group convergence loop, and the method's pre-training step uses data from the whole evolution interval, which weakens the extrapolation claim. These issues are load-bearing for the headline trajectory-reduction conclusion, so the paper needs major revision before the claims can be accepted.

major comments (4)

[§3.5 and Numerical Results (experiment (a) cost paragraph)] The reported NN-NMSSE cost of 364,050 minutes contains 360,000 minutes for 'stochastic trajectory simulations', which is exactly 180,000 trajectories × 2 minutes/trajectory. But the procedure described in §3.5 generates 10 sets of trajectories with increasing counts N1 through N10 and accepts only the tenth prediction; the text does not state whether 180,000 is the size of the tenth set or the total over all ten sets. The sentence '10 groups of stochastic trajectories were generated, resulting in a total training time of 3,750 minutes' is internally inconsistent: at 2 minutes/trajectory this corresponds to about 1,875 trajectories, while training 10 models at 15 minutes each would be 150 minutes, and the grid-search cost of 375 minutes is already 25 × 15 minutes. If N1 through N9 contain substantial trajectory counts, the total number of generated trajectories may exceed the 700,000 NMSSE baseline, and the claimed ~3.8-fold speedup may vanish. The same problem appears in the FMO 77 K accounting: 30,000 final trajectories at 60 minutes each give 1,800,000 minutes, but the ten-group convergence procedure is not costed, and the stated 15,000 minutes of 'training time' is unexplained. Reporting N1 through N10 and the total trajectory-generation minutes is necessary to support the paper's headline claim.
[§3.4] The pre-training step uses 'the entire period of evolution data from cNMSSEs', including the oscillating long-time segment, before the model is fine-tuned on the converged segment. Because the prediction zone is exactly the long-time interval covered by that pre-training data, the network has seen samples from the target interval during training. The method is therefore not a pure short-time-to-long-time extrapolation as claimed in the Introduction and §3.5. This is load-bearing for the claim that converged short-time data contain the information needed for long-time propagation. The authors should either restrict pre-training to t ≤ t_c and show that accuracy is preserved, or explicitly state that noisy long-time data are used and quantify the contribution of that data to the reported agreement with HEOM.
[§3.5] The acceptance rule—standard deviation across the ten predictions below epsilon_2—is an internal-consistency test rather than an accuracy test. All ten models are trained on cNMSSE data from the same trajectory-generation procedure, so a common bias (for example a systematic error inherited from finite-ensemble training data) can leave the standard deviation small while the prediction is wrong. The manuscript does compare against HEOM in the benchmark examples, which is good, but the stopping criterion is proposed as the method's general validation mechanism. The authors should report the actual standard deviation values observed, justify the choice of epsilon_2, and ideally test the criterion on a system where HEOM data are not used to determine the accepted prediction.
[Numerical Results, Figures 9 and 15] The text states that NN-NMSSE deviations from HEOM remain below approximately 0.02, but Figures 9 and 15 show deviation axes extending to 0.06, and it is not clear from the captions or the text what the maximum deviations actually are. The authors should report numerical maximum deviations for each experiment, since the accuracy claim is one of the two central claims of the paper. In addition, no sensitivity analysis is given for the thresholds epsilon_1 and epsilon_2, even though the reported trajectory counts and computation times depend directly on these thresholds.

minor comments (5)

[References] Reference 15 is duplicated verbatim, and reference 42 is repeated as reference 69; the bibliography should be deduplicated.
[Tables] The text contains 'Table ?? compares the computational cost...' and 'summarized in Table ??', but no such tables appear in the manuscript; the missing tables must be included or the references removed.
[Figures 16 and 17] The spin-boson section refers to 'Figure 16 and Figure 17' for trajectory counts and computational times, but Figures 16 and 17 appear later in the FMO section; the figure numbering should be made consistent with the referencing.
[Acknowledgements] The sentence 'The authors thank Qiang Shi for sharing his code in HEOM' is duplicated.
[Eq. (25) and general typography] The vector arrow in Eq. (25) and the surrounding vertical-bar notation render unclearly, and there are several typographical inconsistencies such as 'nmsses', 'stochastics', and inconsistent capitalization of 'Schrödinger'; the manuscript would benefit from a careful proofreading pass.

Circularity Check

1 steps flagged · score 6.0 of 10

Long-time 'prediction' is pre-trained on the very long-time data it reports as predicted, making the extrapolation claim partially circular; the HEOM check gives external accuracy evidence but does not remove the data leakage.

fitted input called prediction [Section 3.1 (cNMSSE Simulation and Convergence Assessment) and Section 3.4 (Pre-train and Fine tuning)]
"Both the converged and oscillating sections are utilized for the initial pre-training of the neural network. However, only the converged section is used for fine-tuning. In our numerical experiments, pre-training with the entire dataset, including the oscillating section, significantly enhanced prediction accuracy."

The 'oscillating section' is the long-time data after the converged segment t_c, exactly the segment later shown as 'Prediction' in Figures 7-10. Pre-training on the entire time series, including these long-time values, exposes the model to the target output before fine-tuning on the early converged segment; fine-tuning does not erase that absorbed information, and the authors state it 'significantly enhanced prediction accuracy.' The Numerical Results claim that 'neural networks trained on short-term data can produce long-term predictions' is thus contradicted by the actual protocol. The long-time output is at least partly a reconstruction/denoising of data already fed to the model, so the extrapolation claim reduces, by the paper's own methodology, to fitting the target series.

full rationale

The main circularity is the data leakage in the training protocol: the neural network is pre-trained on the full cNMSSE time series, including the long-time oscillating segment that is later presented as the predicted output, and then fine-tuned only on the early-time converged segment. This means the reported 'prediction' is not an out-of-sample extrapolation from short-time data, contrary to the paper's stated motivation. The external HEOM benchmark does provide an independent accuracy check, so the agreement with HEOM is not itself derived from the fit; this prevents the score from reaching 8-10. The self-citation to the authors' prior cNMSSE method (Ref. 28) is load-bearing as the data generator, but it is a published, externally benchmarked method and is not invoked as a uniqueness theorem, so it does not add circularity. The cost-comparison accounting is ambiguous: the 10 groups of trajectories could be nested prefixes of a single N10 batch, in which case the 360,000-minute simulation cost is consistent, or independent sets, in which case the speedup claim is overstated. Because the paper does not specify which, this is a reporting and robustness concern rather than a demonstrated circular reduction, so it is not scored as a separate circular step.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the stochastic method (cNMSSE), the assumption that short-time converged data contains enough information to forecast long-time dynamics, and a self-referential convergence criterion (standard deviation across NN predictions). The only hand-set numbers are convergence thresholds and neural-network hyperparameters; no physical constants or new entities are fitted.

free parameters (4)

epsilon_1 (standard error threshold for converged training data) = 0.01
Chosen by hand. It determines which early-time NMSSE data is treated as converged and used for fine-tuning; smaller values reduce oscillations but raise trajectory cost. No sensitivity analysis is reported.
epsilon_2 (standard deviation threshold across predictions) = 0.05
Chosen by hand as the stopping criterion for adding trajectories. It defines when predictions are declared converged; no robustness check across values is reported.
Neural network hyperparameters (input length L, CNN kernel size and stride, LSTM neuron counts) = Not specified in main text; selected by grid search on validation set
The main text says the kernel size is 0.1*(L-1) with stride 0.1*L and that 25 models with varying neuron counts and input vector lengths were tested, but concrete values are deferred to a supporting document that is not included.
Number of trajectory groups (10) and trajectory counts N1 to N10 = Not specified
The convergence assessment uses 10 sets of NMSSE trajectories with increasing counts, but the individual N_i values are not reported, so the cost breakdown cannot be independently checked.

assumptions (5)

domain assumption cNMSSE with the forward/backward Gaussian noise construction exactly reproduces the reduced dynamics of the system-bath model (Eqs. 3-15 and Ref. 28).
The paper relies on this stochastic formulation as its raw-data generator. It cites previous derivations but does not re-derive or independently verify the noise construction itself.
standard math The bath correlation function can be accurately represented by a finite exponential expansion with complex modes (Eq. 9).
This is a standard decomposition, but the number of modes K and its accuracy are not discussed in the main text.
domain assumption Long-time dynamics can be learned from short-time converged NMSSE data, i.e., the future is predictable from a finite window of accurate early-time values.
This is the core inductive assumption behind training a dynamical map in Section 3.4 and using it for propagation in Section 3.5. It is plausible because of TTM/GQME, but no rigorous bound is given.
domain assumption Convergence of the standard deviation across 10 neural-network predictions implies accuracy of the final prediction.
Section 3.5 uses SD below epsilon_2 as the acceptance criterion. This is self-referential; external accuracy is only demonstrated post hoc by HEOM benchmarks, not guaranteed by the criterion.
domain assumption FMO is modeled with a Frenkel exciton Hamiltonian, independent baths, and a Debye-Drude spectral density with parameters from the literature.
The FMO section states these choices but does not list site energies or couplings, so the benchmark depends on unstated standard parameters.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Enhancing Open Quantum Dynamics Simulations Using Neural Network-Based Non-Markovian Stochastic Schr\"odinger Equation Method." pith.science (2026). https://pith.science/paper/SYANDUFI

@misc{pith2026241115914,
  author       = {Pith},
  title        = {Pith review of: Enhancing Open Quantum Dynamics Simulations Using Neural Network-Based Non-Markovian Stochastic Schr\"odinger Equation Method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SYANDUFI}},
  note         = {Machine review of arXiv:2411.15914}
}

read the original abstract

The Non-Markovian Stochastic Schrodinger Equation (NMSSE) offers a promising approach for open quantum simulations, especially in large systems, owing to its low scaling complexity and suitability for parallel computing. However, its application at low temperatures faces significant convergence challenges. While short-time evolution converges quickly, long-time evolution requires a much larger number of stochastic trajectories, leading to high computational costs. To this end,we propose a scheme that combines neural network techniques with simulations of the non-Markovian stochastic Schrodinger equation. By integrating convolutional neural networks (CNNs) and long short-term memory recurrent neural networks (LSTMs),along with the iterative attentional feature fusion (iAFF) technique, this approach significantly reduces the number of trajectories required for long-time simulations, particularly at low temperatures, thereby substantially lowering computational costs and improving convergence. To demonstrate our approach, we investigated the dynamics of the spin-boson model and the Fenna-Matthews-Olson (FMO) complex across a range of parameter variations.

Figures

Figures reproduced from arXiv: 2411.15914 by the authors.

**Figure 2.** The structure of the LSTM cell. σ denotes the sigmoid activation function. tanh denotes hyperbolic tangent activation function. ⊕ denotes the broadcasting addition and ⊗ denotes the element-wise multiplication. The LSTM cell receives vector x(t) at step t and the state vectors c(t−1) and h(t−1) from the last step t − 1, and then generates output vector y(t) by processing them through three gates:69 input gate, forge… view at source ↗

**Figure 3.** The structure of the multi-scale channel attention module. [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (16 more)

**Figure 4.** Figure 4: The structure of attentional feature fusion (AFF). [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: The structure of iterative attentional feature fusion. The initial feature integration [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: The overall structure of the neural network model. [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

**Figure 7.** Figure 7: Compare the NN-NMSSE predictions, shown as colored lines, with the exact [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]

**Figure 8.** Figure 8: Compare the NN-NMSSE predictions, shown as colored lines, with the exact [PITH_FULL_IMAGE:figures/full_fig_p023_8.png]

**Figure 9.** Figure 9: We compares the performance of NMSSE and NN-NMSSE method using the same [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]

**Figure 10.** Figure 10: The figure illustrates the population deviations between the two methods and [PITH_FULL_IMAGE:figures/full_fig_p025_10.png]

**Figure 11.** Figure 11: To achieve the same accuracy as the NN-NMSSE method, the number of tra [PITH_FULL_IMAGE:figures/full_fig_p026_11.png]

**Figure 12.** Figure 12: To achieve the same accuracy as the NN-NMSSE method, the number of tra [PITH_FULL_IMAGE:figures/full_fig_p027_12.png]

**Figure 13.** Figure 13: The figure illustrates the NN-NMSSE predictions for shorter evolution times. [PITH_FULL_IMAGE:figures/full_fig_p028_13.png]

**Figure 14.** Figure 14: The figure illustrates excitation energy transfer in the FMO complex as predicted [PITH_FULL_IMAGE:figures/full_fig_p029_14.png]

**Figure 15.** Figure 15: We compares the performance of NMSSE and NN-NMSSE method with 30,000 [PITH_FULL_IMAGE:figures/full_fig_p030_15.png]

**Figure 16.** Figure 16: To achieve the same accuracy as the NN-NMSSE method, the number of tra [PITH_FULL_IMAGE:figures/full_fig_p031_16.png]

**Figure 17.** Figure 17: To achieve the same accuracy as the NN-NMSSE method, the number of tra [PITH_FULL_IMAGE:figures/full_fig_p031_17.png]

**Figure 18.** Figure 18: The figure illustrates excitation energy transfer in the FMO complex as predicted [PITH_FULL_IMAGE:figures/full_fig_p032_18.png]

**Figure 19.** Figure 19: We compares the performance of NMSSE and NN-NMSSE method with 10,000 [PITH_FULL_IMAGE:figures/full_fig_p033_19.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 12 canonical work pages

[1]

(1) Nematiaram, T.; Troisi, A. Modeling charge transport in high-mobility molecular semi- conductors: Balancing electronic structure and quantum dynamics methods with the help of experiments.The Journal of Chemical Physics 2020, 152, 190902. (2) Munn, R. W.; Silbey, R. Theory of electronic transport in molecular crystals. III. Dif- fusion coefficient inco...

work page 2020
[4]

The Journal of Physical Chemistry Letters 2022, 13, 6037–6041, PMID: 35749307

(63) Ullah,A.; Dral,P.O.One-ShotTrajectoryLearningofOpenQuantumSystemsDynam- ics. The Journal of Physical Chemistry Letters 2022, 13, 6037–6041, PMID: 35749307. 41 (64) Li, W.; Ren, J.; Shuai, Z. Numerical assessment for accuracy and GPU acceleration of TD-DMRG time evolution schemes.The Journal of Chemical Physics 2020, 152, 024127. (65) Abadi, M. et al....

work page 2022
[12]

(30) Wang, Y.-C.; Zhao, Y. Effect of an underdamped vibration with both diagonal and off- diagonal exciton–phonon interactions on excitation energy transfer.Journal of Compu- tational Chemistry 2019, 40, 1097–1104. (31) Zhong, X.; Zhao, Y. Charge carrier dynamics in phonon-induced fluctuation systems from time-dependent wavepacket diffusion approach.The J...

work page 2019
[13]

(55) Rodríguez, L. E. H.; Kananenka, A. A. Convolutional Neural Networks for Long Time Dissipative Quantum Dynamics.The journal of physical chemistry letters 2020, 2476–

work page 2020
[23]

Predicting quantum evolutions of excitation energy transfer in a light-harvesting complex using multi-optimized recurrent neural networks

(50) Banchi, L.; Grant, E.; Rocchetto, A.; Severini, S. Modelling non-markovian quantum processes with recurrent neural networks.New Journal of Physics 2018, 20, 123030. (51) Zhao, S.-C.; Huang, Y.-M.; Zhao, Z.-R. Predicting quantum evolutions of excitation energy transfer in a light-harvesting complex using multi-optimized recurrent neural networks. 2024...

work page Pith review arXiv 2018
[83]

P.; Lesanovsky, I

(38) Ates, C.; Olmos, B.; Garrahan, J. P.; Lesanovsky, I. Dynamical phases and intermit- tency of the dissipative quantum Ising model.Physical Review A 2011,

work page 2011
[85]

Discrete Time-Crystalline Order in Cavity and Circuit QED Systems.Physical review letters 2017, 120 4, 040404

(39) Gong, Z.; Hamazaki, R.; Ueda, M. Discrete Time-Crystalline Order in Cavity and Circuit QED Systems.Physical review letters 2017, 120 4, 040404. (40) Cerrillo, J.; Cao, J. Non-Markovian Dynamical Maps: Numerical Processing of Open Quantum Trajectories. Phys. Rev. Lett. 2014, 112, 110401. (41) Mulvihill, E.; Geva, E. A Road Map to Various Pathways for ...

work page 2017
[90]

S.; Calhoun, T

(68) Engel, G. S.; Calhoun, T. R.; Read, E. L.; Ahn, T. K.; Manal, T.; Cheng, Y.; Blanken- ship, R. E.; Fleming, G. R. Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems.Nature 2007, 446, 782–786. (69) Heaton, J. Ian Goodfellow, Yoshua Bengio, and Aaron Courville: Deep learning.Ge- netic Programming and Evolvable Mac...

arXiv 2007

Show all 14 references

[125]

Ian Goodfellow, Yoshua Bengio, and Aaron Courville: Deep learning.Ge- netic Programming and Evolvable Machines 2018, 19, 305–307

(42) Heaton, J. Ian Goodfellow, Yoshua Bengio, and Aaron Courville: Deep learning.Ge- netic Programming and Evolvable Machines 2018, 19, 305–307. (43) Geron, A. Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow: Concepts, Tools, and Techniques to Build Intelli...

2018
[155]

Variational Neural-Network Ansatz for Steady States in Open Quantum Systems.Phys

(57) Vicentini, F.; Biella, A.; Regnault, N.; Ciuti, C. Variational Neural-Network Ansatz for Steady States in Open Quantum Systems.Phys. Rev. Lett. 2019, 122, 250503. (58) Nagy, A.; Savona, V. Variational Quantum Monte Carlo Method with a Neural-Network Ansatz for Open Quantu...

2019
[1993]

(21) Diósi, L.; Strunz, W. T. The non-Markovian stochastic Schrödinger equation for open systems. Physics Letters A 1997, 235, 569–573. (22) Suess, D.; Eisfeld, A.; Strunz, W. T. Hierarchy of stochastic pure states for open quan- tum system dynamics.Physical review letters 201...

1997
[2016]

J.; Zhao, T

(66) Beer, K.; Khosla, M.; Köhler, J.; Osborne, T. J.; Zhao, T. Quantum machine learning of graph-structured data.Phys. Rev. A 2023, 108, 012410. (67) Jang, S.; Mennucci, B. Delocalized excitons in natural light-harvesting complexes.Re- views of Modern Physics 2018,

2023
[2019]

Long Short-Term Memory.Neural Comput

(44) Hochreiter, S.; Schmidhuber, J. Long Short-Term Memory.Neural Comput. 1997, 9, 1735–1780. 39 (45) LeCun, Y.; Boser, B.; Denker, J. S.; Henderson, D.; Howard, R. E.; Hubbard, W.; Jackel, L. D. Backpropagation Applied to Handwritten Zip Code Recognition.Neural Computation 1...

1997 arXiv
[2483]

Forecasting Nonadiabatic Dynamics using Hybrid Convolutional Neural Network/Long Short-Term Memory Network

(56) Wu, D.; Hu, Z.; Li, J.; Sun, X. Forecasting Nonadiabatic Dynamics using Hybrid Convolutional Neural Network/Long Short-Term Memory Network. The Journal of Chemical Physics 2021,

2021

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

(1) Nematiaram, T.; Troisi, A. Modeling charge transport in high-mobility molecular semi- conductors: Balancing electronic structure and quantum dynamics methods with the help of experiments.The Journal of Chemical Physics 2020, 152, 190902. (2) Munn, R. W.; Silbey, R. Theory of electronic transport in molecular crystals. III. Dif- fusion coefficient inco...

work page 2020

[2] [4]

The Journal of Physical Chemistry Letters 2022, 13, 6037–6041, PMID: 35749307

(63) Ullah,A.; Dral,P.O.One-ShotTrajectoryLearningofOpenQuantumSystemsDynam- ics. The Journal of Physical Chemistry Letters 2022, 13, 6037–6041, PMID: 35749307. 41 (64) Li, W.; Ren, J.; Shuai, Z. Numerical assessment for accuracy and GPU acceleration of TD-DMRG time evolution schemes.The Journal of Chemical Physics 2020, 152, 024127. (65) Abadi, M. et al....

work page 2022

[3] [12]

(30) Wang, Y.-C.; Zhao, Y. Effect of an underdamped vibration with both diagonal and off- diagonal exciton–phonon interactions on excitation energy transfer.Journal of Compu- tational Chemistry 2019, 40, 1097–1104. (31) Zhong, X.; Zhao, Y. Charge carrier dynamics in phonon-induced fluctuation systems from time-dependent wavepacket diffusion approach.The J...

work page 2019

[4] [13]

(55) Rodríguez, L. E. H.; Kananenka, A. A. Convolutional Neural Networks for Long Time Dissipative Quantum Dynamics.The journal of physical chemistry letters 2020, 2476–

work page 2020

[5] [23]

Predicting quantum evolutions of excitation energy transfer in a light-harvesting complex using multi-optimized recurrent neural networks

(50) Banchi, L.; Grant, E.; Rocchetto, A.; Severini, S. Modelling non-markovian quantum processes with recurrent neural networks.New Journal of Physics 2018, 20, 123030. (51) Zhao, S.-C.; Huang, Y.-M.; Zhao, Z.-R. Predicting quantum evolutions of excitation energy transfer in a light-harvesting complex using multi-optimized recurrent neural networks. 2024...

work page Pith review arXiv 2018

[6] [83]

P.; Lesanovsky, I

(38) Ates, C.; Olmos, B.; Garrahan, J. P.; Lesanovsky, I. Dynamical phases and intermit- tency of the dissipative quantum Ising model.Physical Review A 2011,

work page 2011

[7] [85]

Discrete Time-Crystalline Order in Cavity and Circuit QED Systems.Physical review letters 2017, 120 4, 040404

(39) Gong, Z.; Hamazaki, R.; Ueda, M. Discrete Time-Crystalline Order in Cavity and Circuit QED Systems.Physical review letters 2017, 120 4, 040404. (40) Cerrillo, J.; Cao, J. Non-Markovian Dynamical Maps: Numerical Processing of Open Quantum Trajectories. Phys. Rev. Lett. 2014, 112, 110401. (41) Mulvihill, E.; Geva, E. A Road Map to Various Pathways for ...

work page 2017

[8] [90]

S.; Calhoun, T

(68) Engel, G. S.; Calhoun, T. R.; Read, E. L.; Ahn, T. K.; Manal, T.; Cheng, Y.; Blanken- ship, R. E.; Fleming, G. R. Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems.Nature 2007, 446, 782–786. (69) Heaton, J. Ian Goodfellow, Yoshua Bengio, and Aaron Courville: Deep learning.Ge- netic Programming and Evolvable Mac...

arXiv 2007