Dead or alive: Distinguishing active from passive particles using supervised learning^(a)

Our model system is a two-dimensional (2D) Kob-Andersen mixture [42–44] composed of 650 and 350 particles of type A and B, respectively. We extend the model by turning a fraction $\phi_a$ into active Brownian particles [45–48] (same proportion for each particle type). We have also conducted similar investigations in a three-dimensional (3D) mixture, comprising 800 particles of type A and 200 particles of type B, and obtained similar results.

The self-propelled motion is characterized by a constant active force of magnitude F_a . To develop a model that can be applied to study highly crowded biological systems, such as dense cellular collectives, we have chosen to focus on steady-state configurations with a density of $\rho=1.2$ and temperature of $T=0.5$ (in simulation units) and retrieve 2000 different independent configurations for each studied setting. Moreover, to effectively characterize the large activity regime, we consider activities ranging from 0.5 to 100, as cells primarily move due to active motion. Furthermore, we have verified that similar results are obtained at a temperature of $T=0.6$. Finally, to confirm that the dynamics at high activities are not influenced by finite-size effects, we performed the same analysis on a larger system with 10000 particles, and we obtained similar results. All the details of the simulation model and the data collection are reported in the SM.

We treat the identification of the active particles as a binary classification problem. For this we use LightGBM, which is a gradient-boosting machine-learning method based on decision trees [49]. This algorithm is widely used due to its efficiency, accuracy, and interpretability, and has been successfully employed, e.g., in the context of low-temperature glassy materials [23]. We show in the SM that it outperforms neural networks for our task of distinguishing active from passive particles.

In general, gradient boosting decision trees [50] comprise an ensemble learning method that combines the power of decision trees [51] and gradient boosting. It involves building a sequence of decision trees, where each subsequent tree corrects the errors made by the previous ones, by performing a residual update step. By iteratively adding decision trees and adjusting their weights based on the residuals, the model gradually improves its ability to make accurate predictions.

LightGBM introduces gradient-based one-side sampling and exclusive feature bundling [49] in order to avoid the need to scan all the data to update the residuals at each step. As a result, gradient boosting can be used for large datasets, such as the collection of simulation snapshots used in this work. In the end, after calculating all the relevant structural input features (discussed in the next section), the training of our model takes only several minutes on a standard laptop.

To quantify our ML performance we use the accuracy, defined as the number of correct predictions divided by the total number of predictions. When the fraction of active particles $\phi_a\neq 0.5$ we simply discard a subset of the particles from the data such that the number of active and passive species is equal. This allows us to work with a balanced dataset, which enhances the classifier's generalizability. Since our input features are particle-resolved and additional data collection is quite inexpensive, this presents no practical challenges.

Our main goal is to distinguish the active particles from the passive ones using only instantaneous static information, i.e., a single snapshot. For this, we employ two alternative sets of local, particle-resolved structural properties as input for our machine-learning model. We refer to these as i) the shell-based approach, developed earlier for purely passive systems [16], and ii) the Voronoi approach, which we introduce here to capitalize on the information contained in local anisotropy.

The features of the shell-based approach have been introduced to effectively predict the dynamic propensity of purely passive particles [16,17]. Since the dynamic propensity quantifies the average squared displacement of a particle from its initial condition [52,53], and active particles have an additional self-propelling force, it is natural to expect that active particles have a larger dynamic propensity. We also confirm this observation by measuring the mean-squared displacement [54,55], reported in the SM. Notice that similar indicators also predict localized plastic events [56,57]. This means that if the dynamical information remains encoded in the structure even for active systems, then this approach should be able to pick it up and identify the active particles from their enhanced propensity.

To implement the shell-based approach we follow the definitions of ref. [16]. Briefly, these features consist of n-th–order radial and angular descriptors, denoted as $G_i^{(n)}$ and $q_i^{(n)}$, respectively. The former measures for each particle i the local particle density within a radial shell, while the latter expands the local density within a shell in terms of spherical harmonics; the order n indicates the degree of averaging over the structural features of nearby particles. Here we consider $n=0, 1, 2$. For n = 0, we compute 50 radial descriptors per species with radial distance $r \in [0,5]$ and shell width $\delta=0.1$, and 192 angular descriptors with $r \in [1,2.5]$, $\delta=0.1$, and spherical harmonics of order $l \in [1,12]$. In sum, we use a total of 876 static quantities as input in the shell-based approach.

As an alternative to the shell-based features of ref. [16], we introduce the Voronoi approach, which naturally quantifies the local anisotropy created by the active particles. Briefly, we perform a Voronoi tessellation of all particle positions and compute the features based on the shape of the Voronoi polygon around each particle. Explicitly, we consider the number of neighbors, the polygon area and perimeter, the distance between the center of the polygon and the actual particle position, the maximum and minimum distances between nearest neighbors, the maximum and minimum distances between the vertices of a polygon, the maximum and minimum distances between the particle position and its polygon vertices, and finally we add the particle type (A or B). Overall this amounts to only 11 relatively simple input features. Compared to the shell-based approach, the Voronoi approach is thus significantly cheaper.

We report in fig. 2 the accuracy achieved by our Voronoi ML model in the classification of an active/passive mixture with $\phi_a=0.008$, for different values of the active force F_a . We compare the accuracy that we get when the model is trained and tested at a single specific value of F_a (orange circles), with the accuracy of a model trained when fixing $F_a=50$ (blue squares) or $F_a=100$ (pink triangles). All three curves produce rather similar accuracies, implying that a single ML model trained at a fixed F_a can also produce good predictions for unseen parameter regimes. This shows reasonable generalizability of the model. However, the accuracy in all three cases drops significantly for lower activity. We attribute this to the fact that small values of F_a render it extremely challenging to differentiate between structural signatures induced by either active forces or passive Brownian motion. Hence, any static approach based purely on instantaneous structural properties fails.

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The inset of fig. 2 demonstrates that very similar accuracies can be achieved when training the ML algorithm with the more intricate structural properties $G_i^{(n)}$ and $q_i^{(n)}$ as input (shell-based approach). However, the shell-based approach is considerably more computationally expensive compared to the Voronoi method. The reason is that the shell-based approach employs almost two orders of magnitude more features. Consequently, we can conclude that, for this specific classification task, the Voronoi approach is more efficient. Moreover, we find similar results for both the Voronoi and the shell-based approach in 3D (see the SM).

Figure 3 reports the performance of the Voronoi active/passive classifiers as a function of the percentage of active particles $\phi_a$, while keeping F_a fixed at 100. For $\phi_a \leq 0.1$, our approach demonstrates excellent predictive capability when the ML model is trained and tested at a specific $\phi_a$ value (represented by orange circles). Additionally, we compare the accuracy obtained when training the model with $\phi_a$ fixed at 0.008 (blue squares) or 0.05 (pink triangles). The latter case yields better transferability than when training only on $\phi_a=0.008$, especially in the region $\phi_a\leq 0.1$, where it rivals the performance of the model trained at each $\phi_a$ value separately. Noticeably, however, when the fraction of active particles $\phi_a$ becomes large, the accuracy drops to below 70%, even for our best ML model. This indicates that the instantaneous structural input becomes less informative to differentiate between active and passive particles, which we attribute to the fact that the presence of more active particles increasingly affects the entire structure of the material and diminishes the front-wake asymmetry. Finally, we have also repeated the Voronoi tessellation analysis at a lower density of $\rho=1.0$ with a fraction of active particles $\phi_a=0.008$. Interestingly, the results are qualitatively similar to those obtained at $\rho=1.2$, but with a more pronounced anisotropy observed at a lower activity $(F_a=50)$ compared to the higher-density system (see the SM for more details).

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A full overview of the Voronoi ML model accuracy in the $(F_a, \phi_a)\text{-plane}$ is reported in fig. 4. This figure reveals two distinct regions: a red region representing a "distinguishable" area and a blue region representing an "indistinguishable" area. In the red region, characterized by low $\phi_a$ and high F_a values, the algorithm achieves an accuracy greater than 0.75, enabling it to differentiate adequately between active and passive particles based on a single snapshot. In this region, the structure surrounding the active particle significantly differs from that of the passive particle. As a result, even a shell-based approach, which does not consider the directional dependence of passive particle density around the active particle, still yields accurate results (inset of fig. 2).

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In the blue region, characterized by either high $\phi_a$ or low F_a , any static approach (either shell-based or Voronoi) is insufficient to effectively differentiate between active and passive particles. When F_a  < 40, the activity is not high enough to induce noticeable spatial inhomogeneities. On the other hand, when $\phi_a$ is too high, the interaction among active particles complicates the classification task. Particularly, in cases where both F_a and $\phi_a$ are high, the system can undergo motility-induced phase separation (MIPS) [37]. In view of the poor overall performance of our ML method for both high $\phi_a$ and low F_a , we conclude that if a local structure-dynamics relationship exists in active systems, it must be significantly different from that observed in passive systems, as indicated by the failure of the shell-based approach. Additionally, this relationship is not hidden in the anisotropy, as indicated by the failure of the Voronoi approach.

In order to still achieve good predictability in the regime where a purely static ML approach fails (blue region in fig. 4), we have developed a so-called pseudo-static method. This approach, detailed in the SM, is able to identify active particles at the cost of more input information. Briefly, while not requiring a fully time-resolved dynamical trajectory, it requires some knowledge of the averaged statistical fluctuations of the local structure. It is possible to obtain these statistics from a collection of snapshots that do not have to be time-ordered, but in which each particle has to be tracked. This tracking information makes the method naturally more computationally expensive. Overall, in the pseudo-static approach, the information about the local structure is complemented by statistical measurements of the same features at different times. We have verified that using this information on the structural fluctuations, an accuracy of 95%–100% can be achieved (see the SM).

Let us return to the parameter regime where our Voronoi ML model is most successful in distinguishing active from passive particles, i.e., small $\phi_a$ and large F_a . In this regime we can interpret the decisions made by the ML model. To do so, we calculate the model explanations using SHapley Additive exPlanation (SHAP) analysis [58]. Briefly, this method, based on cooperative game theory, calculates Shapley values that distribute the model's prediction among the features while considering their individual contributions to the output. We apply this analysis to the Voronoi model trained on $F_a=100$ and $\phi_a=0.008$.

Figure 5 shows the SHAP beeswarm plot, which indicates the six most important features and how the values of these features influence the model's predictions. The first six features are, in order of importance: the distance between the center of the polygon and the particle (d(c, p)), the perimeter of the polygon, the maximum distance between the particle and its neighbors $(\max(d(n,p)))$, the maximum distance between the vertices of the polygon $(\max(d(v,v)))$, the area of the polygon, and the maximum distance between the particle and the vertices of the polygon $(\max(d(v,p)))$. The colors show that the model interprets low values of d(c, p) as a passive particle and high values of d(c, p) as an active particle. Considering that passive particles tend to accumulate in front of active particles while creating a void behind them, we anticipate the polygons around active particles to exhibit elongation. Consequently, due to this anisotropy, we expect the active particle to be situated at a different position compared to the center of the polygon, as we confirm in representative polygons shown at the top of fig. 5. The SHAP analysis clearly validates this expectation. Hence, rather than using the ML model solely as a black box, we infer that the model captures the correct physical picture and confirms the importance of spatial anisotropy induced by activity. Lastly, the SHAP analysis has also been performed in 3D, and the corresponding results can be found in the SM.

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