Simulation setups

The scattering angle data were provided by a Geant4 simulation incorporating the Cosmic-Ray Shower (CRY) Library [40], which generates muons with the energy and angular distribution of cosmic muons at sea level. The dimension of the object is 10 cm × 10 cm × 10 cm; it may be coated on all six sides with a 1 cm thick layer, resulting in an overall size of 12 cm × 12 cm × 12 cm for the coated object. Four position-sensitive detectors of size 30 cm × 30 cm are placed above and below the object to measure the trajectories of the incident and emergent muons for subsequent angle calculations. A spacing of 20 cm is placed between the second and third detectors to place the tested materials at the center of this gap. The distance between detectors 1 and 2 and detectors 3 and 4 was the same (35 mm). It is worth noting that the detector used in the simulation is an ideal detector, and detector characteristics such as detector noise are not considered. This setup is shown in Fig. 2.

Fig. 2Full-screen View

(Color online) Schematic of the Geant4 simulation setup

First, we set up a muon source with an energy of 1 GeV and conducted scattering angle simulations for nine bare materials (Mg, Al, Ti, Fe, Cu, Zn, W, Pb, and U) to verify the reliability of the simulated data. Figure 3 presents the probability distribution statistics of the simulated scattering angle data using the kernel density estimation (KDE), categorized by both Z classes and different materials. The results indicated significant differences in the scattering angle distributions among different Z-class materials and among materials within the same Z-class. In the formal experiment, we performed simulations based on the energy and angular distributions of cosmic-ray muons at the sea level. For each material in different scenarios (bare, Al-coated, or PE-coated), 500,000 muon scattering angle data were simulated. The scattering angle in our simulation is the included angle between the incoming and outgoing muon tracks in 3D space using the following formula: (1)

Fig. 3Full-screen View

(Color online) Scattering angle distribution of 1 GeV muon with materials: Mg, Al, Ti, Fe, Cu, Zn, W, Pb, U. a categorized by Z-classes. b categorized by each materials

Inverse CDF sampling

As shown in Fig. 3, the primary distinguishing feature of muon scattering through different materials lies in the variations of their scattering angle distributions. Additionally, high-quality training samples contribute to the training of more effective neural networks [41, 42]. When employing a neural network as a mapping function between scattering angle data and material properties, it is desirable for the model to learn features that facilitate differentiation between various materials based on their respective scattering characteristics. To effectively capture the distribution characteristics of the simulation data, a sampling method called the inverse cumulative distribution function was introduced. First, we uniformly selected a set of quantile points within the range of the overall scattering angle distribution for a given material. Non-parametric methods are used to estimate the probability density function (PDF) of the overall data at these quantile points. The corresponding CDF values were then computed based on these PDF values. Finally, we obtained the training samples through inverse CDF sampling. The complete sampling process is shown in Fig. 4, and the process pseudocode is referred to Appendix 1. Compared with the traditional random sampling (RS) method, the samples generated using this method shared the same probability distribution as the overall simulation data.

Fig. 4Full-screen View

(Color online) Schematic of inverse CDF sampling. From left to right: estimated discrete PDF values using KDE or HE; Interpolated cumulative distribution function (CDF), and sampling data from the same distribution via inverse CDF sampling. Axes represent radians (x-axis) and corresponding densities or cumulative probabilities (y-axis)

We employed two non-parametric estimation methods: KDE [43] and histogram estimation (HE), to compute the probability density values for selected quantile points within the overall data. The fundamental idea of the KDE is to use a smooth kernel function to perform weighted averaging of the total data points, thereby obtaining an estimate of the probability density. In other words, the weighted contribution of data points at a given quantile point is calculated. Selecting specific quantile points rather than drawing the entire data ensures computational efficiency and accurate distribution estimation, mitigating the impact of fine-grained details that could disrupt the smoothness of the overall probability distribution function. For a selected quantile point Xq, the kernel density estimation of the PDF is given by: (2)where is the kernel function, h is the bandwidth parameter controlling the smoothness of the estimation, and Xi is the true data value in the overall dataset. We employed a Gaussian function as the kernel function, along with the Silverman method for automatic bandwidth adjustment.

In contrast, the HE method is more straightforward and intuitive. It uniformly divides the overall data range into b bins and counts the number of data points that fall within each bin: (3)where h is the bin width, n is the total number of samples. The expression represents an indicator function that indicates whether the data point Xi falls within the bin Bj. If so, the function takes the value of 1; otherwise, it takes the value of 0. The KDE method provides a smoother estimation of the PDF values, enabling the subsequent acquisition of higher-quality samples. However, for sparse data or in the presence of extreme values, the KDE may lose some accuracy. In contrast, the HE method is more stable, but improper bin selection can lead to overfitting or underfitting of the data distribution. Given the peak and long-tailed distribution characteristics of the simulation data, both methods were incorporated into the sampling process.

To convert discrete PDF values into continuous CDF values, the compound trapezoidal rule (Eq. 4) and cubic spline interpolation (Eq. 5) are employed during the converting process. The integral interval is divided into multiple sub-intervals, and the trapezoidal rule is applied within each sub-interval to improve the integration accuracy. (4)where f(Xq) is obtained from Eq. (2) or Eq. (3), depending on which method of calculating PDF is taken. A smooth and continuous CDF is obtained using cubic spline interpolation between discrete CDF values, and the parameters aj, bj, and cj are determined by zero-, one-, and two-order boundary conditions for different pairs of CDF values: (5)Based on the CDF values obtained from the above calculations and conversions, Ns rounds of sampling were conducted for the simulation data of various materials under different experimental scenarios. In each iteration, n random numbers un that satisfy the uniform distribution U~(0, 1) are generated, and the inverse CDF method is utilized to obtain n scattering angle data points : (6)To prevent overfitting and a loss of robustness owing to the high similarity between samples, a similarity check mechanism was incorporated. By calculating the Euclidean distance between the samples generated in each iteration, those with excessive similarity are filtered out, ultimately resulting in Ns diverse samples that align with the overall simulation data distribution. The parameters related to the sampling process are listed in Tabel 1. The sampled scattering angle data were stored as features, while the corresponding material Z-class served as the label (0, 1, and 2 corresponding to low-, mid-, and high-Z), forming samples with feature-label pairs.

In different scenarios (bare, Al, or PE coated), 1,000 samples were acquired for each material. While in the specific training and prediction phases, a consistent train-test split was applied across different materials. The specific data partitioning ratios for the various phases are detailed in the corresponding parameter tables. In addition, the traditional random sampling method, which is directly sampled from the raw simulated scattering angle data (denoted as RS), was also applied in this study. The sample size generated by the RS method was the same as that of the inverse CDF sampling. The superiority of the inverse CDF sampling method can be demonstrated by comparing the training accuracies.

Pre-training and Fine-tuned transfer

Fine-tuning is an essential technique for transferring neural network tasks and is widely applied in transfer learning because of its low computational cost and high training efficiency under limited target domain data conditions. In this study, we constructed a unified lightweight neural network with two hidden layers for both the pre-training and fine-tuning processes (P&F model). The scattering angle sample data were first received by the input layer, then processed through two hidden layers for feature extraction, and finally classified by the output layer. The detailed network structure is illustrated in Fig. 5.

Fig. 5Full-screen View

(Color online) Structure of pre-train & fine-tune model. The layers with a blue background indicate free/frozen layers. During pre-training, the layers are in a free state, with the source sample as the input. During fine-tuning, the layers are frozen, only the parameters between the last hidden layer and the output layer are updated using target domain samples

Because the P&F model is designed for multi-classification tasks, we used cross-entropy as the loss function [44]: (7)where N is the batch size, and C is the number of classes (low, mid, and high). Given a sample i, the ground-truth class label yi,j is encoded in a One-Hot format and automatically converted by PyTorch. The corresponding logits represent the raw predictions of the neural network, which are normalized by the Softmax activation function. The model was trained on the training set, and its prediction accuracy on the test set was recorded to evaluate its generalization performance and robustness. The specific parameter settings are listed in Table 2.

For the pre-training process, we trained the P&F model with feature-label pairs (xs, ys) samples obtained from the source domain using three different sampling methods. The training process is illustrated in Fig. 6. The goal was to learn the mapping between the scattering angle data of bare materials and their Z categories. After pre-training, the hidden layers, which serve as the key structures for feature extraction, have their parameters optimized to effectively extract high-order features from the original scattering-angle data. In the subsequent fine-tuning process, the parameters of the input and hidden layers are frozen, while only the parameters between the last hidden layer and the output layer are trained using a smaller number of target domain training samples.

Fig. 6Full-screen View

(Color online) Training process of pre-train model on source domain test dataset. Training and testing loss (left y-axis) and accuracy for different Z-classes (right y-axis) over epochs. a Training with RS samples. b Training with KDE samples. c Training with HE samples

In the pre-training stage, we aimed at nine different materials from the high-, mid-, and low-Z categories and conducted supervised training on the training dataset. The training results in Table 3 indicate that the pre-trained model achieves high prediction accuracy on the test dataset, confirming the effectiveness of neural networks in learning the mapping between scattering angle data and Z categories. The accuracy metric is the ratio of correct predictions to the total sample number in their respective scenarios.

Before applying the fine-tuned transfer learning method, we first directly evaluated the pre-trained model on the two target domains, where Al and PE served as coating materials. The test results are shown in Table 4 (Pre-training). When trained with samples obtained using two inverse CDF sampling methods (KDE and HE), the classification accuracy in the Al-coated target domain decreased by approximately 12%, with the most significant drop occurring in the prediction accuracy of low-Z materials (approximately 30%). This phenomenon can be attributed to the fact that, as a low-Z material, the relative influence of the Al coating on the scattering angle distribution of the coated material is inversely proportional to the intrinsic Z value of the coated material. Meanwhile, in the PE-coated target domain, the prediction accuracy of the pre-trained model remained almost unchanged (a decrease of only approximately 1%). Because PE, as a hydrocarbon compound, can be considered an ultra-low-Z material, its coating has a minimal impact on the scattering angle distribution of metallic materials. In contrast, when the model was trained with RS data, the randomness of the sampling process hindered the prediction results from following the physically consistent patterns observed in the inverse CDF-based training.

In our P&F model architecture, the first hidden layer was designed to extract fundamental features reflecting low-level scattering physics. These features are largely invariant across domains, given the shared nature of the input data structures between the source and target domains. Therefore, freezing the first hidden layer preserves these transferable and physically meaningful representations. Regarding the second hidden layer, although it captures more domain-specific cues, our empirical studies showed that fine-tuning this layer, while the first remains frozen, often leads to overfitting owing to the limited size of the target dataset. In contrast, freezing both hidden layers stabilizes performance by retaining shared representations and reducing the influence of domain-specific noise. Additionally, freezing both layers reduces the number of trainable parameters, which is especially beneficial in low-cost applications. For the fine-tuning process, we freeze the corresponding parameters and fine-tune the P&F network with fewer target domain training samples (x_t, y_t). This enables the pretrained model to be adapted efficiently to the target domain tasks. For the two different target domains of Al-coated and PE-coated materials, the classification accuracy of the fine-tuned model on the target test dataset is presented in Table 4 (fine-tuning). Benefiting from the well-optimized parameters obtained during pre-training, the fine-tuned P&F model demonstrated excellent predictive performance across both tasks. The detailed training process is illustrated in Fig. 7.

Fig. 7Full-screen View

(Color online) Training process of fine-tune model on target dataset. Training and testing loss (left y-axis) and accuracy for different Z-classes (right y-axis) over epochs. a-c Results of training with RS, KDE, and HS samples separately on the Al target domain. d-f Results of training with RS, KDE, and HS samples separately on the PE target domain

It should be noted that, owing to the globally shared parameters of the neural network, the fine-tuning process aims to enhance the overall classification accuracy rather than optimize each class individually. As the model improves its discriminative ability for certain categories, the performance of others may deteriorate slightly, resulting in parameter competition and trade-offs. This phenomenon may cause a minor decrease in the prediction accuracy of specific classes. However, the overall classification performance of the target task improved. Moreover, the observation that the training loss exceeds the testing loss, with the training curve showing greater fluctuation while the test curve remains smooth, can be primarily attributed to the limited size of the training dataset. Because only 30% of the total data were used for training, the model became more sensitive to outliers and difficult samples, leading to a higher average loss and increased variance during optimization. In contrast, the larger testing set (70%) provides a more reliable and representative performance estimate, effectively averaging out the anomalies and resulting in a smoother and lower loss curve. Furthermore, because parameter updates occur solely on the training set, its loss is expected to fluctuate across batch iterations, whereas the test loss remains relatively stable.

Transfer learning with DANN model

The identification of the Z-class of an unknown coated material, as an unlabeled target domain problem, presents a more significant challenge. However, by identifying common scattering angle features between the coated material and its bare counterpart, we can train a neural network to achieve superior classification performance in an unknown domain. Traditional domain alignment methods typically compute specific mathematical relationships between the source and target domains and incorporate them into the training process as part of the loss function [45, 46]. Given that different transfer learning tasks exhibit distinct data characteristics, determining the optimal mathematical relationship as a training objective remains highly challenging.

The concept of adversaries in neural networks was first introduced in Ref. [47], where adversarial models generated adversarial samples to enhance model robustness. The DANN extends adversarial training to transfer learning by incorporating a domain discriminator that enforces feature distribution alignment between the source and target domains through adversarial training, thereby facilitating unsupervised learning in the target domain. This approach fully exploits the fitting capabilities of neural networks and enables effective feature alignment without requiring an explicit definition of the feature relationships between a specific source and target domains.

Our DANN model is illustrated in Fig. 8, consists of three main components: a feature extractor, a classifier, and a domain discriminator. During training, the labeled scattering angle data of the bare materials and unlabeled scattering angle data of the coated materials were both fed into the feature extractor. The total extracted features are then passed into the domain discriminator, and the extracted source domain features and corresponding labels serve as inputs to the classifier. First, the domain discriminator determines whether the input features originate from the source or target domain and simultaneously computes the loss function to optimize itself, aiming to improve the domain feature discrimination. Because the domain discriminator only classifies whether the received features belong to the source or target domain, we use binary cross-entropy as the loss function: (8)the ground-truth label indicates the source or target domain class of the sample. The model outputs logits zi represent the raw predictions before applying the Sigmoid activation function. Unlike the default definition of Softmax in cross-entropy, the binary cross-entropy loss requires an explicit definition of the Sigmoid function as the activation function in the output layer. Second, when receiving a sample F(x_s), y_s from the source domain, the classifier computes the loss function for the conventional multi-classification task, similar to the pretraining and fine-tuning approaches. Consequently, the loss function is also defined as the cross-entropy, with the same expression as in Eq. 7. Finally, the overall loss function (9)is used to train the extractor, where is responsible for improving the prediction accuracy of the extractor and classifier, while the reversal discrimination loss is used to train the extractor to gradually extract the shared features between the source and target domains, thereby achieving domain alignment. Gradient reversal ensures that the extractor and discriminator form an adversarial relationship during training and eventually reach equilibrium after iterative optimization. The gradient reversal parameter λ is employed to balance the weight distribution between feature alignment and classification accuracy improvement in the network model. In this study, feature alignment was more challenging than classification. Moreover, owing to the minimal feature distribution discrepancy between the source and target domains, remains relatively low. Based on this analysis, we assigned a higher weight to λ in our training process to enhance the training performance. After this training stage, the extractor and classifier can extract common features from both the source and target domains and perform effective classification. The detailed neural network hyperparameter settings are presented in Table 5.

Fig. 8Full-screen View

(Color online) Structure of DANN model. The legend explains the activation function, the network connection mode, and the direction of information propagation

The training process and final results of DANN are presented in Fig. 9 and Table 6. The training results indicate that when trained with inverse CDF sampled data, the DANN exhibits slightly lower prediction accuracy for low-Z materials than for mid-Z and high-Z materials. This phenomenon is consistent with the results obtained when the pretrained model performs inference directly in the target domain, which can be attributed to the significant change in the scattering angle distribution of low-Z materials after being coated with Al, making feature alignment more challenging than that for mid-Z and high-Z materials. Meanwhile, the introduction of a large gradient reversal coefficient -λ causes the overall training loss to become negative because it includes the adversarial loss from the domain discriminator. However, this does not affect the evaluation of the test set, where the domain discriminator and gradient reversal are not involved. Therefore, the test loss, which was calculated solely based on the standard cross-entropy loss, remained positive. This behavior is expected and does not affect the assessment of the model performance on the target classification task.

Fig. 9Full-screen View

(Color online) Training process of DANN model on target dataset. Training and testing loss (left y-axis) and accuracy for different Z-classes (right y-axis) over epochs. a-c: Results of training with RS, KDE, and HS samples separately on the Al target domain. d-f: Results of training with RS, KDE, and HS samples separately on the PE target domain