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

Abstract

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.

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

  $$\begin{aligned} \bar{r}_{\cdot j}=\frac{1}{n} \sum _{i=1}^n r_{i j} \end{aligned}$$

  (A.1)

• Calculate the test statistic Q, which is given by

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

  (A.2)

• 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

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

(3)

$q_\alpha$ 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.