Skip to main content

Gradient-based adaptive sampling framework and application in the laser-driven ion acceleration


Physical model optimisation has frequently been complemented by experimental design in scientific research. However, it can be time consuming to perform real-world experiments and difficult to find affordable experimental designs. Bayesian optimisation based on the Gaussian process model has attracted extensive attention in the field of experimental design because it can build a good surrogate model and generate a sequential design simultaneously. However, it can create problems if researchers have a weak understanding of the system’s overall trend. This study introduces gradient information and proposes a new framework for constructing surrogate models: GRAdient-enhanced SEquential SUrrogate MOdelling (GRASE-SUMO). First-order gradient information is utilised as a guidance for selecting sampling space, and second-order gradient information is then adopted as an objective function in Bayesian optimisation. GRASE-SUMO is designed to mimic system changes and allows general system trends to be easily identified without a high level of prior knowledge. Experiments were conducted to verify the accuracy and stability of GRASE-SUMO, which works especially well in dealing with plate-shaped or valley-shaped response surfaces. When applied to laser-proton acceleration, GRASE-SUMO succeeded in rectifying and expanding the suitable conditions for optimal acceleration using only 30 samples, while the conventional sampling method requires about 10\(^{2-3}\) samples with only three variables.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Data availability

The datasets generated and analysed in study will be publicly available on GitHub after the paper is published

Code availability

All code generated and analysed during the current study will be publicly available on GitHub after the paper is published The authors will provide essential help for the readers if they want to reproduce and use the results of this study.


Download references


This study was supported by the National Natural Science Foundation of China [Nos. 12101608, 11771450, 11875319] and the National Key R &D Program of China [No. 2018YFA0404802].

Author information

Authors and Affiliations



BW contributed towards conceptualisation, methodology, software, formal analysis, data Curation, and visualisation. RS contributed towards data Curation, visualisation, and writing—original draft LY contributed towards conceptualisation, methodology, validation, writing—original draft, and funding acquisition TY contributed towards conceptualisation, supervision, and funding acquisition XD contributed towards conceptualisation, investigation, resources, supervision, project administration, and funding acquisition

Corresponding author

Correspondence to Xiaojun Duan.

Ethics declarations

Competing interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Replication of results

After publication, all materials used for replication will be uploaded to The author’s assistance in replication can be obtained upon reasonable request.

Ethics approval

When carrying out the study, we followed all applicable ethical requirements.

Consent to participate

All authors have approved this study’s publication.

Consent for publication

We agree with all publication policies of the journal.

Additional information

Responsible Editor: D Youn

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: A statistical test for comparing the performance of surrogate models

Appendix A: A statistical test for comparing the performance of surrogate models

To compare the performance of different SUMOs, a two-step statistical test was performed in this study: Friedman’s rank-sum test was used to judge whether there were differences in SUMOs across multiple test results. The Nemenyi post hoc test was then used to carry out pairwise comparisons.

Friedman’s rank-sum test is a non-parametric statistical test used to detect differences between algorithms (Friedman 1940). Assuming there are n test conditions and k kinds of SUMOs, the performance of each was recorded as \(\left\{ x_{i j}\right\} _{n \times k}\). The data in the same row are calculated using the same test function. Data in the same column belong to the performance of the same algorithm on different test functions. The calculation process is given as follows:

  • The ranks within each row of \(\left\{ x_{i j}\right\} _{n \times k}\) were calculated. If there were tied values, we adopted the average of the ranks when there were no ties. Then, a matrix of rank \(\left\{ r_{i j}\right\} _{n \times k}\) was obtained, where \(r_{i j}\) is the rank of \(x_{i j}\) within the i-th row.

  • Calculate \(\bar{r}_{\cdot j}\), which is given by

    $$\begin{aligned} \bar{r}_{\cdot j}=\frac{1}{n} \sum _{i=1}^n r_{i j} \end{aligned}$$
  • Calculate the test statistic Q, which is given by

    $$\begin{aligned} {Q=\frac{12 n}{k(k+1)} \sum _{j=1}^k\left( \bar{r}_{\cdot j}-\frac{k+1}{2}\right) ^2} \end{aligned}$$
  • Calculate the p value

    In this study, there were 10 test functions and 20 different sample sizes, \(n= 10 \times 20 =200, n \gg 15\), and the probability distribution of Q was approximated through \(\chi ^2\) distribution. The p value was given by \(\textbf{P}\left( \chi _{k-1}^2 \ge Q\right) \).

If the p value was significant, there were differences in SUMOs across multiple test results. We used the Nemenyi test as the post hoc test, which carried out the pairwise tests. The Nemenyi test calculated the critical difference (CD), which is given by

$$\begin{aligned} {C D=q_\alpha \sqrt{\frac{k(k+1)}{6 N}}} \end{aligned}$$

\(q_\alpha \) can be obtained by querying (Demšar 2006). If the rank value difference of any two SUMOs was greater than CD, the performances of the two SUMOs were determined to be different.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, B., Sha, R., Yan, L. et al. Gradient-based adaptive sampling framework and application in the laser-driven ion acceleration. Struct Multidisc Optim 66, 217 (2023).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: