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Predicting Quantum Potentials by Deep Neural Network and Metropolis Sampling

by Rui Hong, Peng-Fei Zhou, Bin Xi, Jie Hu, An-Chun Ji, Shi-Ju Ran

Submission summary

As Contributors: Shi-Ju Ran
Arxiv Link: (pdf)
Code repository:
Data repository:
Date accepted: 2021-09-02
Date submitted: 2021-08-10 04:55
Submitted by: Ran, Shi-Ju
Submitted to: SciPost Physics
Academic field: Physics
  • Quantum Physics
Approach: Theoretical


The hybridizations of machine learning and quantum physics have caused essential impacts to the methodology in both fields. Inspired by quantum potential neural network, we here propose to solve the potential in the Schrodinger equation provided the eigenstate, by combining Metropolis sampling with deep neural network, which we dub as Metropolis potential neural network (MPNN). A loss function is proposed to explicitly involve the energy in the optimization for its accurate evaluation. Benchmarking on the harmonic oscillator and hydrogen atom, MPNN shows excellent accuracy and stability on predicting not just the potential to satisfy the Schrodinger equation, but also the eigen-energy. Our proposal could be potentially applied to the ab-initio simulations, and to inversely solving other partial differential equations in physics and beyond.

Published as SciPost Phys. Core 4, 022 (2021)

Author comments upon resubmission

Thank you very much for forwarding us the report on our manuscript entitled “Predicting Quantum Potentials by Deep Neural Network and Metropolis Sampling”. Firstly, we would like to thank the referee for their efforts on reviewing our work. After thoroughly considering the comments, we now resubmit the manuscript to SciPost physics for consideration.

List of changes

1.The second paragraph in Sec. I “Introduction”, the sentences “These problems are critical … … and designing quantum simulators” were modified.
2.The first paragraph in Sec. III, several sentences were added, which are “The sampling process can be … … analytically or numerically accessible.”
3.Below the Eq. (8), several sentences were added, which are “Since any global constant shift of the potential… … and Uθ is given by the NN.”
4.Below the Eq. (9), several sentences were added, which are “With the loss L → 0., the NN would give … …the constraint is satisfied, i.e., |Uθ(r0) ) V (r0)| → 0, with L → 0.”
5.The second paragraph in Sec. IV, several sentences were added, which are “To compare with QPNN, here we use the … …different from those in the training set.”
6.The Table I(a) were modified to update the values of the error ε by increasing the number of coordinates.
7.Below the Eq. (10), “We evaluate ε by averagely taking … …full play to the advantages of Metropolis sampling.” were modified.
8.The Fig. 2 were added to show the error ε with different numbers of samples N used to optimize NN. The caption was added accordingly.
9.In the captions of Table I and Figs. 1-4, we added some words to specify the numbers of hidden variables in the NN.
10.The Fig. 4 were modified to update the curve of the error ε.
11.The sixth paragraph in Sec. IV “MPNN also shows its advantage on the sampling efficiency… … MPNN achieves a lower error than QPNN.” were added.

Reports on this Submission

Anonymous Report 2 on 2021-8-24 (Invited Report)


I really appreciate the detailed response of the authors to my questions and the changes in the manuscript. However, I feel that the manuscript does not contain sufficiently novel results or approaches compared to previous work that would justify publication in the premium journal SciPost Physics. I would, instead, suggest to transfer it to SciPost Physics Core.

  • validity: high
  • significance: ok
  • originality: ok
  • clarity: high
  • formatting: good
  • grammar: excellent

Anonymous Report 1 on 2021-8-23 (Invited Report)


In the reply of the first round, the authors have throughly addressed all my previous questions. However, on top of the QPNN algorithm proposed in Ref.[31], the novelty of the new algorithm in this work, as well as the improvements on the numerical results are not significant enough to merit publication on SciPost physics. I believe it would be more suitable for other journals such as SciPost Physics Core.

  • validity: good
  • significance: good
  • originality: high
  • clarity: high
  • formatting: good
  • grammar: good

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