图 1  在$t = 70{T_0}$时刻, (a) He²⁺离子的$x{\text{-}}v$相空间分布, (b) He²⁺离子的能谱分布, (c) C⁶⁺离子的$x{\text{-}}v$相空间分布, (d) C⁶⁺离子的能谱分布. 其中, 激光强度$a = 100$, 靶密度$n = 25{n_{\text{c}}}$, 靶厚 $d = 25{\lambda _0}$; 红色虚线框内为峰值能量所对应的离子, 蓝色虚线框内为截止能量所对应的离子

Fig. 1.  At $t = 70{T_0}$, (a) the $x{\text{-}}v$ phase space diagram of He²⁺, (b) the energy spectrum of He²⁺, (c) the $x{\text{-}}v$ phase space diagram of C⁶⁺, (d) the energy spectrum of C⁶⁺, with $ d = 25{\lambda _0} $, $n = 25{n_{\text{c}}}$ and $a = 100$. The red dashed box circles the ions with peak energy, and the blue dashed box circles the ions with cutoff energy.

图 2  离子峰值能量随输入参数$d$, $n$和 $a$分布的三维散点图　(a) 质子; (b) He²⁺ 离子; (c) C⁶⁺ 离子; (d) O⁸⁺离子

Fig. 2.  Scatter plot depiction of ion peak energy data ensembles as a function of input parameters d, n and a: (a) Proton case; (b) He²⁺, (c) C⁶⁺; (d) O⁸⁺.

图 3  训练所使用的神经网络结构

Fig. 3.  Neural network architecture used in the following training.

图 4  He²⁺离子 (a) 峰值能量${E_{\text{p}}}$与 (b) 截止能量${E_{\text{m}}}$的二维连续映射图, 其中靶厚$d = 15{\lambda _0}$, 参数映射范围为$63.2 \leqslant a \leqslant 173.2$和$10{n_{\text{c}}} \leqslant n \leqslant 40{n_{\text{c}}}$

Fig. 4.  Two-dimensional continuous mapping of peak energy ${E_{\text{p}}}$ (a) and cutoff energy ${E_{\text{m}}}$(b) for He²⁺ over $63.2 \leqslant a \leqslant 173.2$ and $10{n_{\text{c}}} \leqslant n \leqslant 40{n_{\text{c}}}$ with $d = 15{\lambda _0}$.

图 5  (a) He²⁺离子峰值能量${E_{\text{p}}}$关于靶密度$n$的映射曲线, 其中靶厚$d = 15{\lambda _0}$, 激光强度$a = 100$; (b) He²⁺离子峰值能量${E_{\text{p}}}$关于激光强度$a$的映射曲线, 其中靶厚$d = 15{\lambda _0}$, 靶密度$n = 20{n_{\text{c}}}$; (c) He²⁺离子峰值能量${E_{\text{p}}}$关于靶厚$d$的映射曲线, 其中靶密度$n = 20{n_{\text{c}}}$, 激光强度$a = 100$; (d)离子峰值能量${E_{\text{p}}}$关于离子质量数$A$的映射曲线, 其中靶厚$d = 15{\lambda _0}$, 靶密度$n = 20{n_{\text{c}}}$, 激光强度$a = 100$. 图中红色实线为代理模型的映射曲线, 模型标准差用红色色块填充于代理模型曲线两侧. 蓝色虚线给出的是根据HB-RPA机制理论公式所绘出的离子峰值能量随离子密度变化的曲线, 黑色实心数据点为测试集数据点, 参数取值范围与模型预测范围用黑色虚线分隔

Fig. 5.  (a) Parameter scan of He²⁺ peak energy ${E_{\text{p}}}$ over target density $n$ with $d = 15{\lambda _0}$, $a = 100$; (b) parameter scan of He²⁺ peak energy ${E_{\text{p}}}$ over laser intensity $a$ with $d = 15{\lambda _0}$, $n = 20{n_{\text{c}}}$; (c) parameter scan of He²⁺ peak energy ${E_{\text{p}}}$over target thickness $d$ with $a = 100$, $n = 20{n_{\text{c}}}$; (d) parameter scan of ion peak energy ${E_{\text{p}}}$ over ion mass number $A$ with $a = 100$, $n = 20{n_{\text{c}}}$ and $d = 15{\lambda _0}$. The SE mapping curves are drawn with red solid, the theoretical curves are drawn with dashed, the red filled region indicates the standard deviation, and the untrained data from test subset are drawn with black dot.

图 6  在$d = 15{\lambda _0}$, $n = 20{n_{\text{c}}}$和$a = 100$下, 离子峰值能量随离子质量的变化

Fig. 6.  Ion peak energy scan with ion mass number $A$ with $d = 15{\lambda _0}$, $n = 20{n_{\text{c}}}$ and $a = 100$.

图 7  (a) He²⁺离子截止能量${E_{\text{m}}}$关于靶密度$n$的映射曲线, 其中靶厚$d = 15{\lambda _0}$, 激光强度$a = 100$; (b) He²⁺离子截止能量${E_{\text{m}}}$关于激光强度$a$的映射曲线, 其中靶厚$d = 15{\lambda _0}$, 靶密度$n = 20{n_{\text{c}}}$; (c) He²⁺离子截止能量${E_{\text{m}}}$关于靶厚$d$的映射曲线, 其中靶密度$n = 20{n_{\text{c}}}$, 激光强度$a = 100$; (d) 离子截止能量${E_{\text{m}}}$关于离子质量数$A$的映射曲线, 其中靶厚$d = 15{\lambda _0}$, 靶密度$n = 20{n_{\text{c}}}$, 激光强度$a = 100$

Fig. 7.  (a) Parameter scan of He²⁺ cutoff energy ${E_{\text{m}}}$ over target density $n$ with $d = 15{\lambda _0}$, $a = 100$; (b) parameter scan of He²⁺ cutoff energy ${E_{\text{m}}}$ over laser intensity $a$ with $d = 15{\lambda _0}$, $n = 20{n_{\text{c}}}$; (c) parameter scan of He²⁺ cutoff energy ${E_{\text{m}}}$ over target thickness $d$ with $a = 100$, $n = 20{n_{\text{c}}}$; (d) parameter scan of ion cutoff energy ${E_{\text{m}}}$ over ion mass number $A$ with $a = 100$, $n = 20{n_{\text{c}}}$ and $d = 15{\lambda _0}$.

图 8  (a) 质子, (b) He²⁺, (c) C⁶⁺和 (d) O⁸⁺的能量比值$k = {E_{\text{m}}}/{E_{\text{p}}}$的二维连续映射图, 其中靶厚$d = 20{\lambda _0}$, 参数映射范围为$63.2 \leqslant a \leqslant 173.2$和$10{n_{\text{c}}} \leqslant n \leqslant 30{n_{\text{c}}}$

Fig. 8.  Two-dimensional continuous mapping of $k = {E_{\text{m}}}/{E_{\text{p}}}$ over $63.2 \leqslant a \leqslant 173.2$ and $10{n_{\text{c}}} \leqslant n \leqslant 30{n_{\text{c}}}$ with $d = 20{\lambda _0}$: (a) Proton case; (b) He²⁺; (c) C⁶⁺; (d) O⁸⁺.

图 9  C⁶⁺离子能量比值$k$随(a)激光强度和(b)靶密度的映射曲线(实线)和拟合公式结果(虚线), 其中参数范围分别为$63.2 \leqslant a \leqslant 173.2$, $10{n_{\text{c}}} \leqslant n \leqslant 30{n_{\text{c}}}$, 靶厚$d = 15{\lambda _0}$

Fig. 9.  Parameter scan (solid) and fitted formula (dashed) of k over (a) $63.2 \leqslant a \leqslant 173.2$ and (b) $10{n_{\text{c}}} \leqslant n \leqslant 30{n_{\text{c}}}$ for C⁶⁺ with $d = 15{\lambda _0}$.