2.1 Data generation

The dataset for the neural network is generated on the basis of the classical SAXS theory. The expected scattering intensity ${I}_{\mathrm{abs}}$ in absolute units can be defined using the following^[ Reference Rödel ^{26 ]}:

(1) $$\begin{align}{I}_{\mathrm{abs}}(q)=\varphi \left(1-\varphi \right)\cdot {\left(\Delta {\rho}_{\mathrm{s}}\right)}^2\cdot V\cdot {\left|F(q)\right|}^2\cdot S(q),\end{align}$$

where $q$ is the wavenumber, $\varphi$ is the volume fraction of the nanoparticles, 1– $\varphi$ is the volume fraction of the solution, $V$ is the volume of a single nanoparticle, $\Delta {\rho}_{\mathrm{s}}$ is the total scattering length density contrast between solids and solutions and $F(q)$ and $S(q)$ represent the form factor and structure factor, respectively. When assuming spherical nanoparticles in one case, the form factor $F(q)$ can be expressed as follows:

(2) $$ \begin{align}F(q)& =\displaystyle{\int}_0^R\displaystyle{\int}_0^{2\pi }\displaystyle{\int}_0^{\pi }{e}^{jqr\cos \theta }{r}^2\sin \theta \mathrm{d}\theta \mathrm{d}\varphi \mathrm{d}r\nonumber\\ & =\dfrac{4}{3}\pi {R}^3\cdot 3\dfrac{\sin (qR)- qR\cos (qR)}{(qR)^3},\end{align}$$

where $R$ is the effective radius of the scatterer. By introducing the repellent potential $U(r)$ of the hard-sphere in the Percus–Yevick closure^[ Reference Wertheim ^{27 –} Reference Pedersen ^{30 ]}, the structure factor can be described as follows:

(3) $$\begin{align}S(q)=1+{\rho}_{\mathrm{part}}\int g(r)\cdot {e}^{- iqr}\mathrm{d}V.\end{align}$$

We created a dataset of scattering intensity generated by hard-spheres with an effective particle size range of 1–9 nm and a volume fraction range of 0.1–0.5, which corresponds exactly to our physical conditions^[ Reference Kraus, Vorberger, Pak, Hartley, Fletcher, Frydrych, Galtier, Gamboa, Gericke, Glenzer, Granados, MacDonald, MacKinnon, McBride, Nam, Neumayer, Roth, Saunders, Schuster, Sun, van Driel, Dppner and Falcone ^{13 ,} Reference Schuster, Hartley, Vorberger, Döppner, van Driel, Falcone, Fletcher, Frydrych, Galtier, Gamboa, Gericke, Glenzer, Granados, MacDonald, MacKinnon, McBride, Nam, Neumayer, Pak, Prencipe, Voigt, Saunders, Sun and Kraus ^{15 ,} Reference He, Rödel, Lütgert, Bergermann, Bethkenhagen, Chekrygina, Cowan, Descamps, French, Galtier, Gleason, Glenn, Glenzer, Inubushi, Hartley, Hernandez, Heuser, Humphries, Kamimura, Katagiri, Khaghani, Lee, McBride, Miyanishi, Nagler, Ofori-Okai, Ozaki, Pandolfi, Qu, Ranjan, Redmer, Schoenwaelder, Schuster, Stevenson, Sueda, Togashi, Vinci, Voigt, Vorberger, Yabashi, Yabuuchi, Zinta, Ravasio and Kraus ^{22 ]}. For example, we estimated the size of the generated nanoparticles from shock-compressed polymer samples via other diagnostics (such as X-ray diffraction) to be several nanometers, with a volume fraction between 10% and 50%. Considering the possible distribution range of actual grain size in these experiments, we chose particle size distribution between 0.9 and 9.9 nm as the target labels, represented by 100 equally discrete probability values. When creating a random particle size distribution, we selected the composition of three Shulz distributions with random effective radii as the total probability for different particle sizes. To assess the robustness of the network, we artificially added 0–3% noise to the theoretical SAXS intensity data. This is based on the fact that the error of SAXS lineouts obtained on a high-energy X-ray light source is mainly related to the instrument and light source quality, which causes an intensity error of less than 2%^[ Reference He, Rödel, Lütgert, Bergermann, Bethkenhagen, Chekrygina, Cowan, Descamps, French, Galtier, Gleason, Glenn, Glenzer, Inubushi, Hartley, Hernandez, Heuser, Humphries, Kamimura, Katagiri, Khaghani, Lee, McBride, Miyanishi, Nagler, Ofori-Okai, Ozaki, Pandolfi, Qu, Ranjan, Redmer, Schoenwaelder, Schuster, Stevenson, Sueda, Togashi, Vinci, Voigt, Vorberger, Yabashi, Yabuuchi, Zinta, Ravasio and Kraus ^{22 ]}. ANNs can generalize well to any particle size distribution by training on this simple data. In addition, the $q$ range that can be detected by an SAXS detector from beamline experiments is about 0.4– $1.05\;{\mathrm{nm}}^{-1}$ . Correspondingly, we selected the same $q$ range of scattering intensity in our ANN dataset, and took 1000 equally discrete intensity values within this range as well as the volume fractions (which can be estimated by comparing static experiments with dynamic experiments) as the input set (see Figure 1 for the input and output parameters). The detailed parameter information in the dataset is shown in Table 1. We set 80% of the data in the dataset as the validation set and the remaining 20% as the testing set.

Figure 1 Schematic of the training process.

Table 1 Parameter ranges during data generation.

In order to prevent the slow convergence of features during optimization^[ Reference Lecun, Bottou, Orr and Muller ^{31 ]}, it is necessary to preprocess and normalize the data, which is an important step of neural network training. The entire dataset was scaled after creating the training-validation databases so that all the input and target features were in the range [0, 1] according to the following:

(4) $$\begin{align}{z}_i^{\prime}=\frac{z_i-\min (z)}{\max (z)-\min (z)},\end{align}$$

where $z$ represents the dataset of a certain feature, ${z}_i$ represents a sample of $z$ and max( $z$ ) and min( $z$ ) are the maximum and minimum values of $z$ , respectively. The scaling is undone afterward.

2.2 Compilation and training of the ANN

The compilation of the neural network and the training were performed using the Keras package running the TensorFlow backend^[ Reference Chollet ^{32 ]}.

We established a fully connected dense feedforward network according to our physical case with a rectified linear unit (ReLU)^[ Reference Agarap ^{33 ]} as the nonlinear activation function. The input layer includes 1000 discrete intensity values as well as one volume fraction value, while the output layer contains 100 discrete probability values representing size distribution. The middle three hidden layers have 128, 64 and 32 neurons, respectively. We chose the mean absolute error (MAE) as the loss function in the training and testing process (see Equation (5)). The backpropagation^[ Reference Rumelhart, Hinton and Williams ^{34 ]} and the stochastic gradient descent optimization algorithm^[ Reference Bottou ^{35 ]} were used to adjust the weights and biases iteratively until the loss function was minimized. In each iteration process, the network calculated the gradient of the batch loss relative to the weight and then updated the weight accordingly:

(5) $$\begin{align}\mathrm{MAE}\left(X,h\right)=\frac{1}{m}\sum \limits_{i=1}^m\left|h\left({x}_i\right)-{y}_i\right|.\end{align}$$

The ANN was trained along with adjusting the memorization capacity (the number of hidden layers or neurons) to find the optimal network structure, avoiding an underfit and overfit. In each training process, 4680 datasets with more than 4 million trainable parameters were trained.

Figure 2 shows the training and validation loss or accuracy as a function of epoch. The validation loss is very close to the training loss, indicating that although the model is complex, it does not overfit the training data, and has good generalization ability in the verification dataset.

Figure 2 Training and validation loss as well as accuracy of the neural network which contains three middle hidden layers with 128, 64 and 32 neurons, respectively.

3.1 Applying the ANN to the theoretical models

Figure 3 depicts the predictions of the trained ANN on several theoretical data. After inputting the theoretical SAXS intensity lineout data with 0–3% noise to the trained ANN, the network can directly give the particle distribution, as shown in the blue histogram in the right-hand column of Figure 3. Aiming at arbitrary particle distribution with various patterns, our predicted particle distribution and the effective radius are in good agreement with the theoretical ones. Conversely, the corresponding SAXS lineout can also be calculated from the predicted distribution, as shown in the left-hand column of Figure 3, which is almost consistent with the original theoretical SAXS lineout in the majority of cases, implying a high reliability in the prediction of arbitrary particle distributions from theoretical data with noise.

Figure 3 Applying the ANN to the theoretical models. Seven arbitrary particle distributions predicted by the ANN (right-hand panel) and their corresponding fitting curves (left-hand panel) compared with the initial theoretical models.

However, for simple distributions with small particle radii, the difficulty of fitting increases due to the tiny variation of scattering intensity in the range of $0.4\hbox{--} 1.05\;{\mathrm{nm}}^{-1}$ , as shown in the first panel in the right-hand column of Figure 3. The more complex the intensity curve and the larger the particle radius, the higher the fitting accuracy.

3.2 Applying the ANN to the experiments

In order to evaluate the performance of the ANN on the experimental data, we assess the ANN with the real-time in situ SAXS experimental data measured in dynamic experiments. Five SAXS intensity data of shock-compressed $100\;\mu \mathrm{m}$ polyethylene terephthalate (PET)^[ Reference Lütgert, Vorberger, Hartley, Voigt, Rödel, Schuster, Benuzzi-Mounaix, Brown, Cowan, Cunningham, Döppner, Falcone, Fletcher, Galtier, Glenzer, Garcia, Gericke, Heimann, Lee, McBride, Pelka, Prencipe, Saunders, Schölmerich, Schörner, Sun, Vinci, Ravasio and Kraus ^{36 ]} at various probing times under approximately 100 GPa were obtained at the Linac Coherent Light Source (LCLS) laser facility. The details of the experiment can be found in previous work^[ Reference He, Rödel, Lütgert, Bergermann, Bethkenhagen, Chekrygina, Cowan, Descamps, French, Galtier, Gleason, Glenn, Glenzer, Inubushi, Hartley, Hernandez, Heuser, Humphries, Kamimura, Katagiri, Khaghani, Lee, McBride, Miyanishi, Nagler, Ofori-Okai, Ozaki, Pandolfi, Qu, Ranjan, Redmer, Schoenwaelder, Schuster, Stevenson, Sueda, Togashi, Vinci, Voigt, Vorberger, Yabashi, Yabuuchi, Zinta, Ravasio and Kraus ^{22 ]}. The in situ SAXS lineout data were used as input of the network, then the particle distribution can be predicted by the trained ANN. As shown in Figure 4, the predicted distributions at various probing times are compared with those obtained from an analytical method (applying the non-negative least squares method)^[ Reference Rödel ^{26 ]} and the Monte Carlo method^[ Reference Pauw, Pedersen, Tardif, Takata and Iversen ^{19 ,} Reference Breßler, Pauw and Thünemann ^{20 ]}. The left-hand panel reveals the particle radius distribution obtained by the three methods, while the right-hand panel exhibits the corresponding fitting lineouts. Interestingly, the three methods consistently point out that the mean particle size gradually increases with time in the shock-compressed PET. It can be seen that the mean size predicted by the ANN is relatively close to that of the Monte Carlo method, and both of these methods can obtain arbitrary size distribution and find a more discrete distribution at the last two probing times. As an analytical model, a Schultz distribution^[ Reference Kotlarchyk and Chen ^{37 ,} Reference Als-Nielsen and McMorrow ^{38 ]} is assumed with the polydispersity $p=\sigma$ / $R\approx 0.1$ , resulting in systematically larger effective radius at latter times than that of the other two methods. In fact, due to the complexity of the dynamic compression process, the nanoparticles generated from the sample are usually too discrete and too arbitrary to be simply represented by a certain distribution. Therefore, the acquisition of arbitrary distribution from raw SAXS data will better reflect the actual situation, as predicted by the ANN and Monte Carlo method. Of course, in the actual experimental process, one still needs to azimuthally integrate the original 2D scattering pattern as well as subtract the backgrounds, which requires batch processing on written codes and usually takes a few minutes to complete. Afterwards, among the three methods, the ANN can invert the particle distribution from the scattering data with a few milliseconds of computing time on a standard laptop, while the Monte Carlo method takes 10–30 minutes and the analytical method requires even more time per shot because it often requires manual input to achieve good convergence, and therefore poses challenges when scaled to high-repetition-rate experiments or online analysis. Comparatively, in the experimental process of compact beam time from raw images to size distribution, the ANN is very suitable for in situ data analysis, which is of great significance for the promotion of physical experiments and timely parameter adjustment. In further research, the process parameters of the Monte Carlo method, such as the local volume fraction and scattering length density, can serve as intermediate inputs for the ANN to finely tune the network. This kind of physics-informed neural network (PINN) is very promisingly beneficial for improving the accuracy of size distribution inversion from experimental data.

Figure 4 The nanoparticle distributions generated from shock-compressed PET obtained by the ANN, Monte Carlo method and analytical model (left-hand panel) and their corresponding SAXS fitting curves compared with the experimental data (right-hand panel). The red dots indicate the resulting mean particle radius from the three methods. The color bar represents the various probing times.

Other aspects of actual experiments, such as the bandwidth and the high harmonics of the probe light spectrum, can also affect the size distribution inversion from SAXS^[ Reference Chen and Luo ^{39 ]}. For the shock-compression phase transition described above, the accuracy of the experiment is not that high and the influence of the spectral bandwidth and high harmonics can be ignored. Of course, in some more precise SAXS inversion problems, the influence of the spectrum cannot be ignored. In these cases, the ANN can be constructed through the following methods: directly input the scattering spectrum and particle size distribution obtained from the specified spectrum into the ANN for training, or add parameters representing the spectral features to the ANN for training.