Full information of system properties inferred from individual particle dynamics

Dusty plasma, also termed as complex plasma, refers to the combination of dust particles and plasma.^1–13 Due to free electrons and ions in plasma, the micrometer-sized dust particles are highly charged to $≈−104 e$⁠.¹⁴ In the laboratory conditions, the electric field in the plasma sheath is able to confine and suspend these charged dust particles to form a single layer, i.e., two-dimensional (2D) dusty plasma.^15–21 The interaction between these dust particles can be described using the Yukawa repulsion $ϕij=Q2 exp(−rij/λD)/4πϵ0rij$⁠, where Q is the particle charge, $rij$ is the distance between the particles i and j, and $λD$ is the Debye length due to the shielding from free electrons and ions.²² Due to their high charges, these dust particles are strongly coupled,²³ resulting in the typical solid- and liquid-like behaviors.^24–33 To characterize properties of 2D dusty plasmas, two global parameters of the screening parameter $κ=a/λD$ and the coupling parameter $Γ=Q2/4πϵ0akBT$ are traditionally used.⁶ Here, T is the kinetic temperature of dust particles and a is the Wigner–Seitz radius of $(nπ)−1/2$ for the areal number density n for 2D systems.⁶ 

One significant advantage of dusty plasma is the experiment diagnostic of individual particle identification and tracking,⁸ so that locations and velocities of all particles can be determined experimentally.^34–37 Because of this diagnostic of individual particles at the kinetic level,⁸ the micro-mechanisms of various fundamental procedures are studied using dusty plasmas, such as transports,¹⁶ phase transition,³ and waves.³⁷ In Ref. 36, from the captured stochastic random motion of particles in 2D dusty plasma, the phonon spectra of longitudinal and transverse waves are obtained using the Fourier transform of the longitudinal and transverse currents for the whole observed system.¹⁹ By comparing the experimentally obtained phonon spectra with the theoretical dispersion relationships of the 2D Yukawa lattice with different parameters, a $χ2$ surface is obtained with the two axes of the varying screening parameter $κ$ and 2D nominal dusty plasma frequency $ωpd=Q2/(2πϵ0ma3)$⁠, similar to Fig. 2 in Ref. 38, Note that m is the mass of one dust particle.⁶ As a result, the best fit of the experimental phonon spectra to the theoretical dispersion relations occurs at the location of the minimum of $χ2$⁠,³⁸ just corresponding to the determined $κ$ and $ωpd$ values, also resulting in the related $Γ$ value.³⁹ Another simpler method is to measure the speeds of longitudinal and transverse waves generated in 2D dusty plasmas and then substitute these measured speeds into their theoretical analytical expressions to determine these global characterization parameters of $Γ$ and $κ$⁠.⁴⁰ In addition, the screening parameter $κ$ and coupling parameter $Γ$ are also able to be determined by comparing the configurational temperature with the standard kinetic temperature.^41,42

Machine learning⁴³ is a powerful tool to discover underlying relations inside massive data, which has been widely exploited in various fields,⁴⁴ such as fluid mechanics,⁴⁵ materials science,⁴⁶ and plasma physics.⁴⁷ In Ref. 48, the material global properties are obtained with the density functional theory accuracy level using machine learning from the connection of atoms in the crystal, and the contributions from local chemical environments to the global properties are achieved. In dusty plasmas, machine learning is also introduced in different investigations,^51–61 such as phase transitions,^52–54 particle tracking,⁵⁷ and experiment diagnostics.⁵⁹ Recently, instead of using the traditional statistics, a new machine learning method is developed to determine the screening parameter $κ$ of 2D dusty plasma purely from either the position fluctuations of each individual particle or the spatial distribution of particles.⁵¹ In Ref. 51, the determined $κ$ value from the 2D dusty plasma experiment using the machine learning method well agrees with the determined $κ$ value from the phonon spectra fitting method.

In this paper, we further develop the machine learning method in Ref. 51 to extend its application in determining both the screening parameter $κ$ and the coupling parameter $Γ$ of 2D dusty plasma simultaneously purely from position fluctuations of individual particles. As a result, the two global characterization parameters of 2D dusty plasma are both determined from the position fluctuation of each individual particle. The rest part of this paper is organized as follows: in Sec. II, we briefly introduce our data acquisition, including both the simulation and experiment data. The 2D dusty plasma simulation data are used for the training, validation, and test of our machine learning method, while our experiment data are used to demonstrate the capability of the machine learning method in accurately determining the $κ$ and $Γ$ values simultaneously. In Sec. III, we present the operation of our developed machine learning method, including the structure of the newly modified convolutional neural network (CNN) model, the dataset packaging using the data from Sec. II, and the training procedure of the CNNs. In Sec. IV, we report the performance of our CNNs in determining the $κ$ and $Γ$ values from position fluctuations of individual particles using both simulation and experiment data. We also discuss the obtained error distributions of the determined $κ$ and $Γ$ values and provide our interpretation of these results. Finally, a summary of our findings is provided in Sec. V.

Since the position fluctuations of individual particles under various conditions from both simulation and experiment data are all used in our current investigation, the spatial and temporal scales change drastically. To compare the dynamics of the individual particles under different conditions of 2D dusty plasmas, the spatial and temporal scales for the particle motion should be normalized to be dimensionless, as in Ref. 18. Following the traditional method,⁶² we use the Wigner–Seitz radius a and the inverse of 2D nominal dusty plasma frequency $ωpd−1$ to normalize the spatial and temporal scales, respectively.⁶ 

To generate simulation data to train our CNN model, we perform 104 independent 2D dusty plasma simulation runs, marked as Set1 in Fig. 1. Within the typical ranges of $κ$ and $Γ$ in 2D dusty plasma experiments, we specify 8  $κ$ values of {0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 1.75, 2.00} and 13  $Γ$ values of $Γm/Γ∈$ {0.01, 0.02, 0.04, 0.08, 0.1, 0.2, 0.4, 0.8, 1.0, 2.0, 4.0, 8.0, 10.0}. Here, $Γm$ is the solid–liquid phase transition point of 2D dusty plasma for the corresponding $κ$ value.^66,67 To obtain sufficient amount of simulation data for training, for each independent simulation run of these 104 different combinations of the specified $κ$ and $Γ$ values of Set1 in Fig. 1, we use a larger 2D simulation box of $243.8 a×211.1 a$⁠, containing $214$ particles, with the periodic boundary conditions.

To generate simulation data to validate and test our CNN model, we perform another 375 independent 2D dusty plasma simulation runs, marked as Set2 in Fig. 1. All 8  $κ$ and 13  $Γ$ values in Set1 are also used as the specified $κ$ and $Γ$ values in Set2. Furthermore, to validate and test the generalization ability of our trained CNNs, as shown in Fig. 1, Set2 also contains the simulation runs with $κ$ and $Γ$ values different from Set1. For any two adjacent specified $κ$ values in Set1, we select a new random value between them as one additional specified $κ$ value in Set2, leading to seven new $κ$ values. Using the same method as for the $κ$ values, we also randomly choose 12 new $Γ$ values based on the existing 13  $Γ$ values, as shown in Fig. 1. In total, we specify 15 different $κ$ values and 25 different $Γ$ values as the simulation parameters for Set2, i.e., 375 different combinations of $κ$ and $Γ$ values, leading to 375 independent 2D dusty plasma simulation runs in Fig. 1. Since the data for the validation and test of our CNN model are not necessarily to be as large as the training data, for Set2, we use a smaller 2D simulation box of $121.9 a×105.6 a$⁠, containing $210$ particles, also with the periodic boundary conditions. Note, for Set2, our specified $κ$ and $Γ$ values are $κ∈$ {0.25, 0.40, 0.50, 0.60, 0.75, 0.90, 1.00, 1.10, 1.25, 1.40, 1.50, 1.60, 1.75, 1.90, 2.00} and $Γm/Γ∈$ {0.01, 0.015, 0.02, 0.03, 0.04, 0.06, 0.08, 0.09, 0.1, 0.15, 0.2, 0.3, 0.4, 0.6, 0.8, 0.9, 1.0, 1.5, 2.0, 3.0, 4.0, 6.0, 8.0, 9.0, 10.0}, respectively.

Here, we provide other detailed parameters in our simulations. The gas damping rate is always specified as $ν=0.027ωpd$⁠, comparable to this variable in typical 2D dusty plasma experiments.^18,34 For all simulation runs, the integration time step is always specified as $δtωpd=0.001$⁠, small enough to mimic 2D dusty plasmas, as justified in Ref. 68. The locations of all simulated particles are recorded every 600 integration steps, i.e., $ωpddt=0.6$⁠, for the latter data analysis of our machine learning method, as one typical example shown in Fig. 2. The total time duration of each simulation run is always specified as $tωpd=900$⁠. Other simulation details are the same as Ref. 51.

Following Ref. 51, we use the same particle position fluctuation data from the 2D dusty plasma experiment,^49–51 as shown in Fig. 3, to test the performance of our CNN with the experiment data. For this 2D dusty plasma experiment, using the traditional phonon spectra fitting method described in Sec. I, the values of the screening parameter $κ$ and the coupling parameter $Γ$ are both determined, which are $κ≈0.50$ and $Γ≈700$⁠.^49–51 From the previous experience, the nominal dusty plasma frequency $ωpd$ for 2D dusty plasma experiments under similar conditions only varies slightly within the typical range of $70s−1⩽tωpd⩽90s−1$⁠.^49–51 As in Ref. 51, to fully utilize our experiment data, we extract 72 section movies containing the same time duration of $10 s$⁠, with the same delay of $t=0.37 s$ for the starting frames of these section movies, for the frame rate of 55 frames per second of the data recording. The same experiment data processing method in Ref. 51 to remove the overall drift is also used here. Finally, we obtain the position fluctuation data of $≈$ 6000 particles from this 2D dusty plasma experiment.^49–51 

In the previous paper,⁵¹ we report a new method to determine the screening parameter $κ$ of 2D dusty plasma purely from either the individual particle position fluctuations or the particle position distributions using machine learning methods. We design two different CNN models named as model A and model B, leading to three different trained CNNs, named as CNN1, CNN2, and CNN3. First, we apply the position fluctuation data of individual particles with the constant time duration to train model A, leading to CNN1, which has an excellent performance only with the simulation data. Although we know the typical varying range of $ωpd$ in different 2D dusty plasma experiments, the exact value of $ωpd$ is traditionally determined using the fitting of phonon spectra, where the $κ$ and $Γ$ values are also determined simultaneously. For finite data in some experiments, the signal-to-noise ratio in the obtained phonon spectra is too low, so that the corresponding $κ$⁠, $Γ$⁠, and $ωpd$ cannot be accurately determined. Since CNN1 is not able to work with experiment data, we design CNN2 and CNN3 to deal with the experiment data. We use the time-independent position distribution data of individual particles to train model B, and the position fluctuation data of individual particles with the varying value of time duration to train model A, leading to CNN2 and CNN3, respectively. Both CNN2 and CNN3 are able to determine the $κ$ value using the experiment data, while the performance of CNN3 is significantly better than CNN2. As a result, in this paper, we further develop model A in Ref. 51, marked as model A′ here, including increasing the hidden layer number, adding the max pooling layers and the batch normalization layers, as well as removing the dropout layer. As a result, our newly developed machine learning methods, marked as CNN1′ and CNN3′, are able to accurately determine the two global characterization parameters of both $κ$ and $Γ$ simultaneously, using the simulation and experiment data of 2D dusty plasmas. Note, to demonstrate the feasibility our machine learning methed in determining $κ$ and $Γ$ simultaneously with the simulation data, we first focus on the developed machine learning mothed of CNN1′, as presented in detail next.

In this paper, we modify the previous four-step model A in Ref. 51 to achieve model A′, as in Fig. 4. Our modification in model A′ mainly contains two parts. First, in step 1, besides the average pooling layer used in model A, the max pooling layer is also exploited in parallel simultaneously to analyze the input data. Compared with the previous model A, now our current model A′ is able to learn the information at higher frequencies inside the input data, as a result, more complete detailed information is captured by model A′. Second, to enhance the performance, starting from model A,⁵¹ we add a batch normalization layer after each pooling or linear layer in step 1 or 3. Note, to do this successfully, the dropout layer in step 2 of model A⁵¹ has to be removed as in Ref. 69, since this dropout layer leads to the malfunction of all the followed batch normalization layers. Compared with that for model A in Ref. 51, the hidden layer number of our current model A′ is greatly enhanced $≈$ 4 times.

As shown in Fig. 4, the structure of our model A′ is presented in detail with the example input data of one particle's x and y position fluctuations in 512 frames. In step 1, starting from the input of a two channel sequence data $(2×512)$ of the particles' x and y positions, two different sets of hidden layers are applied, leading to the two different outputs, with each containing a 1024 feature vector $(1024×1)$⁠. Each hidden layer set contains nine consecutive subsets, while for one hidden layer set, each subset is composed of one 1D convolutional layer, one max/average pooling layer, and one batch normalization layer.⁷⁰ The kernel size of each 1D convolutional layer is $1×5$⁠, while that of each pooling layer is $1×2$⁠. In step 2, a flatten layer is used to convert all 2048 feature vectors $(2048×1)$ into a new feature vector $(1×2048)$⁠. In step 3, we apply nine consecutive sets of one linear layer combined with one batch normalization layer to analyze the feature vector $(1×2048)$ from step 2, leading to a new feature vector $(1×4)$⁠. In step 4, a linear layer without an activate function is used, resulting in the final output $(1×2)$⁠. We consider that the two elements of the final output are the determined $κNN$ and $αNN=lg(ΓmNN/ΓNN)$ values. Here, $ΓmNN$ is the solid–liquid phase transition point for 2D dusty plasma under $κNN$⁠. Note, in steps 1 and 3, the rectified linear unit (ReLU) function is applied as the activate function of each convolutional or linear layer. Note, to avoid the effect of significant value difference between $κ$ and $Γ$ in our machine learning method, we use $α=lg(Γm/Γ)$ to scale the value of $Γ$ to the same magnitude order of $κ$⁠.

For our CNN1′, we pack the simulation data from Set1 as the training dataset using one step and pack the simulation data from Set2 as the validation and test datasets using two steps. For Set1, we extract the first 512 frames simulation data from each simulation run, i.e., the time duration of the extracted simulation data is $tωpd=307.2$⁠, and then pack all these extracted data as the training dataset. This training dataset contains the position fluctuation data of all $104×214$ particles with the time duration of $tωpd=307.2$⁠. For Set2, first, we also extract 512 frames simulation data from each simulation run. Second, we randomly select $29$ particles for each simulation run from the previous step and then pack all the extracted simulation data of these selected particles as the validation dataset. The extracted simulation data of the rest particles are packed as the test dataset. Thus, both the validation and test datasets contain the position fluctuation data of $375×29$ particles with the time duration of $tωpd=307.2$⁠.

For the training of our CNN3′, we pack the simulation data from Set1 as the training dataset using two steps. We specify the time duration of our simulation data to match the typical value in 2D dusty plasma experiments, which is $≈10$ s.^16–18 For 2D dusty plasmas under similar experiment conditions, the typical range of $ωpd$ is $70 s−1⩽ωpd⩽90 s−1$⁠,^16–18 leading to $700⩽tωpd⩽900$⁠. For Set1, first, we choose five different values of time duration within the range of $700⩽tωpd⩽900$ and then extract the simulation data with these five different values of time duration from each simulation run, the same as the operation of CNN3 in Ref. 51. Second, we use the down-sampling technique to reduce the frame number of these extracted simulation data to 512 and pack all obtained simulation data as the training dataset, i.e., the position fluctuation data of all $104×214$ particles with five different values of time duration. Note, for Set1, the five different values of time duration are $tωpd∈{700,750,800,850,900}$⁠, corresponding to 1167, 1250, 1333, 1417, 1500 frames in our original simulation data.

For validation and test of our CNN3′, we pack the simulation data from Set2 as the validation and test datasets using three steps. First, besides the five values of time duration chosen for the training above, we also choose another four different values of time duration within $700⩽tωpd⩽900$⁠. We extract the simulation data with these nine different values of time duration from each simulation run. Second, we also use the down-sampling technique to reduce the frame number of these extracted simulation data to 512. Third, after randomly selecting $29$ particles from each simulation run in Set2, we pack all down-sampling data of these selected particles as the validation dataset, while packing the left down-sampling data of the rest particles as the test dataset. These three steps are the same as the operation steps of CNN3 in Ref. 51. Both the validation and test datasets contain the position fluctuation data of all $375×29$ particles with nine different vales of time duration. Note, for Set2, the chosen nine different values of time duration are $tωpd∈{700,725,750,775,800,825,850,875,900}$⁠, corresponding to 1167, 1208, 1250, 1292, 1333, 1375, 1417, 1458, 1500 frames in our original simulation data.

To demonstrate the capability of our CNN3′ with the experiment data, we use four steps to pack the experiment data described in Sec. II. First, we calculate the mean drift velocity of each particle and remove the overall drift motion of the experiment data. Second, for each particle, we move the origin point of the reference frame to its own equilibrium position within the observation time. Third, for each particle, we calculate the standard deviation of its motion and extract the position fluctuation data of all particles within the standard deviation of $<a/4$⁠, leading to the position fluctuation data of $≈$ 6000 particles from the studied 2D dusty plasma experiment.^49–51 Note, as shown in Fig. 3, for our used experiment data, the 2D dusty plasma is in a highly ordered triangular lattice without any noticeable disturbances so that the distribution of the particle position fluctuations should be much smaller than $a/4$⁠. Finally, we apply the down-sampling technique to reduce the frame number of all the experiment data to 512 and then pack them for the data analysis of our CNN3′ next. Note, these operation steps with the experiment data are the same as those for CNN3 in Ref. 51.

We use nine steps to train our model A′ with our training dataset packed above to achieve CNN1′ and CNN3′. In step 1, after initializing all learnable parameters in model A′ with random values, we randomly divide the train dataset into several batches, each of which includes the position fluctuation data of 1024 different particles. In step 2, after the input of the one batch data into model A′, for each particle information in this batch, model A′ is able to generate two values, which are just the determined $κNN$ and $αNN$⁠, as mentioned in Sec. III A. In step 3, for each particle in this batch, using the Smooth L1 loss function,⁷¹ we quantify the difference between the determined $κNN$ value and the specified $κ$ value. The difference between the determined $αNN$ and the specified $α=lg(Γm/Γ)$ is also quantified similarly. After taking the mean of these two quantitative differences for each particle, we calculate the average of this mean for all particles in this batch, called as the loss. In step 4, we calculate the gradient between the obtained loss and each learnable parameter in model A′. In step 5, based on these calculated gradients in step 4, we apply the Adam optimizer⁷² with the learn rate of $4×10−6$ to update all learnable parameters in model A′, where a L2 regularization⁷³ is added with a weight decay of $10−4$ to prevent over-fitting. After this step, we just complete one training procedure of model A′. In step 6, the cumulative number of training procedure of model A′ is counted. If this cumulative number is an integer multiple of 1664, then we perform step 7 at once. Otherwise, we skip step 7 to perform step 8 directly to reduce the training time. In step 7, using the same method in step 3, we calculate the loss of the validation dataset from model A′ to validate the performance of model A′ during training, as shown in Fig. 5. In step 8, we repeat all the steps from 2 to 7 until all batches are used. In step 9, according to our specified epoch number of 1000, we repeat all of the eight steps above 1000 times. Note, during the training procedure, the sum memory consumption for CNN1′ is $≈$ 8 GB, while this value is much more for CNN3′, which is $≈$ 40 GB.

After training our model A′ as described in Sec. III C, we achieve CNN1′ and CNN3′. In Fig. 5, we present the losses of the training and validation datasets for $κ$ and $α$ during the training procedure of CNN1′ and CNN3′. From Fig. 5, clearly, for both CNN1′ and CNN3′, the training and validation losses gradually decrease with the epoch number to nearly zero. The results in Fig. 5 clearly indicate that our CNN1′ and CNN3′ both converge well without over-fitting, i.e., they both have the excellent generalization ability. Furthermore, the losses of $κ$ and $α$ of the validation dataset decay synchronously, indicating that our CNNs learn enough features from the position fluctuations of individual particles to accurately determine the $κ$ and $Γ$ values simultaneously. Our analysis results from these two trained CNNs with the simulation and experiment data are presented in detail next. Note, the downward tendency to significant small values of the validation losses^74,75 of both $κ$ and $Γ$ clearly indicates that our CNNs are able to accurately determining the $κ$ and $Γ$ values.

To quantify the performance of our CNN1′, we calculate the mean relative errors (MREs) of the determined $κNN$ and $ΓNN$ of the test dataset from CNN1′. As mentioned in Sec. III A, for each particle, our CNN1′ generates two values from its position fluctuation information, which are the determined $κNN$ and $αNN$ values. For each particle, after determining the $κNN$ value, we use two steps to determine the $ΓNN$ value. First, from the determined $κNN$ value, we obtain the corresponding solid–liquid phase transition point of $ΓmNN$ using the conclusion in Ref. 66. Then, we determine the $ΓNN$ value for each particle from the corresponding $αNN$ and $ΓmNN$ using the equation of $αNN=lg(ΓmNN/ΓNN)$⁠. Since the specified $κ$ and $Γ$ values for each simulation run are known, we calculate the MREs of the determined $κNN$ and $ΓNN$ values for each simulation run. Our obtained MRE results are presented in Fig. 6. Note, we calculate the root mean square error (RMSE) of the determined $κNN$ from CNN1′, which is 0.102.

From Fig. 6, our trained CNN1′ is able to accurately determine the $κ$ and $Γ$ values simultaneously purely from the position fluctuations of individual particles in 2D dusty plasma simulations. As shown in Fig. 6, the MRE of the determined $κNN$ and $ΓNN$ is generally <30%, indicating the excellent performance of our CNN1′. In fact, the absolute error of the determined $κNN$ is mostly <0.1, with only a small portion slightly larger within $≲0.2$⁠. In Fig. 6(a), the average MRE is only $≈10 %$⁠, while the maximum MRE occurs when $κ=0.25$⁠, corresponding to the absolute error of only $≈0.2$ in the determined $κNN$⁠. For the determined $ΓNN$ in Fig. 6(b), the obtained MRE varies from $≈1.26 %$ to $≈30.83 %$ at most, and the average MRE is only $≈11 %$⁠. The performance of our CNN1′ clearly demonstrates its excellent generalization ability, i.e., our developed CNN1′ is universally applicable to determine the $κ$ and $Γ$ values simultaneously from 2D dusty plasma simulations.

The MRE distributions of the determined $κNN$ and $ΓNN$ exhibit nearly regular stripes, as shown in Fig. 6. For the MRE distribution of the determined $κNN$ values in Fig. 6(a), there are nearly regular vertical stripes, especially in the lower-right portion. These vertical stripes mean that the MRE is always lower or larger for some specified values of $κ$⁠, no matter how $Γm/Γ$ varies. In Fig. 6(b), there are regular horizontal stripes in the MRE distribution. Similarly, these horizontal stripes mean that the MRE in Fig. 6(b) is always lower or larger for some specified $Γm/Γ$ values, not related to the $κ$ value much.

To clearly present the nearly regular stripes described above, in Fig. 7, we plot two typical horizontal and vertical profiles corresponding to the cross section views from the two dashed lines in Fig. 6. In Fig. 7(a), the MRE of the determined $κNN$ varies with $κ$ up and down regularly, i.e., a higher MRE is always accompanied by two adjacent lower MREs and vice versa. We use “type A” to mark all conditions with the $κ$ values already in the training dataset. While, for those conditions with the $κ$ values beyond the training dataset, we use “type B” to mark them. From Fig. 7(a), clearly, the MRE of the determined $κNN$ values under the “type A” conditions is significantly lower than that under the “type B” conditions. This difference in the MRE under the “type A” and “type B” conditions is reasonable, because the training dataset contains the “type A” conditions, i.e., our CNN1′ has learned features of 2D dusty plasmas under the “type A” conditions. However, our CNN1′ has not learned features under the “type B” conditions, because they are not included in the training dataset. As a result, the corresponding MRE under the “type B” conditions is reasonably larger. Similar results are also presented in Ref. 51, where the distribution of the determined $κNN$ values exhibits one prominent peak around the specified $κ$ value of the “type A” conditions. However, under the “type B” conditions, the distribution exhibits two distinctive peaks around two adjacent $κ$ values of the “type A” conditions.⁵¹ Another feature in Fig. 7(a) is the MRE diminishes gradually as $κ$ increases, which can be attributed to the relative error. In fact, the absolute error of our determined $κNN$ values varies randomly between 0.02 and 0.09, without so significant general diminishing feature as MRE. Clearly, for either “type A” or “type B” conditions, our CNN1′ is always able to accurately determine the corresponding $κ$ value, well demonstrating the generalization ability of our CNN1′.

In Fig. 7(b), the MRE results also exhibit the similar up and down feature in the determined $ΓNN$ values, due to the similar “type A” and “type B” conditions. Here, the “type A” and “type B” conditions refer to the conditions of the $Γm/Γ$ values inside and beyond the training dataset, respectively. From Fig. 7(b), under the “type A” conditions, the MREs of the determined $ΓNN$ are generally lower than those under the “type B” conditions, because our CNN1′ learns all features of the simulation data under the “type A” conditions, but not for the “type B” conditions yet. Note that, in Fig. 7(b), for the specified values of $lg(Γm/Γ)=−1.05$⁠, $−0.05$⁠, and 0.95 under the “type B” conditions, the MRE results of the determined $ΓNN$ are pretty low, comparable to the MRE results of the “type A” conditions. In fact, these three specified $Γm/Γ$ values are all too close to their adjacent $Γm/Γ$ values in the “type A” conditions, as shown in Fig. 1, so that the corresponding MREs are lower. These three lower MRE results under the “type B” conditions in Fig. 7(b) clearly indicate that the performance of our CNN1′ can be significantly improved by using more training data under more conditions with less intervals between data points.

We use the trained CNN3′ to determine the $κ$ and $Γ$ values simultaneously from the 2D dusty plasma experiment, as well as the corresponding simulations. Based on the position fluctuation data of individual particles within a constant time duration, the performance of our CNN1′ in determining the $κ$ and $Γ$ values from the simulation data is excellent. However, since the time duration of the experiment data in the unit of $ωpd−1$ is actually unknown, our CNN1′ cannot be directly used with the experiment data. To solve this problem, as described in Sec. III B and also used in Ref. 51, we use the simulation data with the varying values of time duration corresponding to the typical temporal range of the 2D dusty plasma experiment to train model A′, leading to the trained CNN3′.

Our CNN3′ is also able to accurately determine the $κ$ and $Γ$ values simultaneously purely from the position fluctuations of individual particles using 2D dusty plasma simulations. To quantify the performance of our CNN3′, as in Sec. IV A, we calculate the MREs of the determined $κNN$ and $ΓNN$ values from the test dataset using our CNN3′, as presented in Fig. 8. The average MRE of the determined $κNN$ values is $≈17 %$ and that of the determined $ΓNN$ values is $≈13 %$⁠. Clearly, compared with the results from our CNN1′ in Fig. 6, the MRE from our CNN3′ in Fig. 8 is slightly higher, due to the varying value of time duration of the simulation data. Nevertheless, most MRE results of the determined $κNN$ and $ΓNN$ values from our CNN3′ are generally $<30 %$⁠, clearly indicating the high-quality performance of our CNN3′. Note, for $κ<1.00$⁠, the absolute error of the determined $κNN$ value does not vary so significantly as MRE in Fig. 8, due to the lower values of $κ$⁠. In fact, the absolute error of the determined $κNN$ values mainly varies randomly from $≈0.1$ to $≈0.2$⁠, with only a few data points beyond this range and still $≲0.3$⁠. Similarly to those in Fig. 6, the MRE results of the determined $κNN$ and $ΓNN$ values from CNN3′ also exhibit nearly regular stripes in Fig. 8, due to the same reason discussed above. Note, the higher MRE conceals the stripe feature in Fig. 8(a), while the regular stripes in Fig. 8(b) are much more prominent. Note, the RMSE of the determined $κNN$ from CNN3′ is 0.172.

Using the position fluctuations of individual particles in the 2D dusty plasma experiment,^49–51 our CNN3′ is able to accurately determine the $κ$ and $Γ$ values simultaneously, as presented in Fig. 9. As described in Sec. III B, we obtain the position fluctuation data of $≈6000$ particles from the 2D dusty plasma experiment.^49–51 Using the position fluctuation of each particle, one pair data of the $κNN$ and $ΓNN$ values are determined from CNN3′. As a result, $≈6000$ determined values of both $κNN$ and $ΓNN$ are achieved, with their distributions presented in Fig. 9. From Fig. 9(a), clearly, the distribution of the determined $κNN$ values exhibits one prominent peak at $κNN≈0.50$⁠. The distribution of the determined $ΓNN$ values in Fig. 9(b) also exhibits one prominent peak at $ΓNN≈780$⁠. The two peaks in Figs. 9(a) and 9(b) just correspond to the most probable distribution of $κNN$ and $ΓNN$⁠, so that we choose the locations of these two peaks, $κNN≈0.50$ and $ΓNN≈780$⁠, as the determined $κ$ and $Γ$ values from our CNN3′ with the experiment data. These two values of $κNN≈0.50$ and $ΓNN≈780$ well agree with the determined values of $κ≈0.50$ and $Γ≈700$ from the widely accepted phonon spectra fitting method.^49–51 This agreement from our CNN3′ and the phonon spectra fitting method in determining $κ$ and $Γ$ values from the same experiment clearly demonstrates the applicability of our CNN3′ in accurately determining the two global parameters of $κ$ and $Γ$ simultaneously with the experiment data.

From above, using CNN3′ with both the experiment and simulation data of 2D dusty plasmas, the two global characterization parameters $κ$ and $Γ$ are both determined simultaneously, purely from position fluctuation data of individual particles. This operation indicate the full information of 2D dusty plasmas properties is able to be inferred from the individual particle dynamics. The excellent performance of CNN1′ and CNN3′ in determining the $κ$ and $Γ$ values simultaneously described above clearly indicates that the machine learning method is able to seize the critical information of the position fluctuations containing enough physic information to characterize the whole system. Note, our current trained CNN3′ is only able to deal with the position fluctuations of individual particles in 2D dusty plasmas; however, if one trains CNN3′ using suitable datasets in a different physical system, then the newly trained CNN3′ should be applicable to this system also. In the supplementary material, we provide our trained CNN1′ and CNN3′, as well as some of our experimental data analyzed here.

To summarize, by further developing the machine learning method in Ref. 51, our obtained CNN1′ and CNN3′ are able to accurately determine the $κ$ and $Γ$ values of 2D dusty plasmas simultaneously purely from position fluctuations of individual particles using both the simulation and experiment data. To obtain particle position fluctuation data for the training, validation, and test of our CNN1′ and CNN3′, we perform independent Langevin dynamical simulations with the different specified $κ$ and $Γ$ values. From our test results with the simulation data, our CNN1′ and CNN3′ are both able to accurately determine the $κ$ and $Γ$ values simultaneously from the simulation data. The averaged MSEs in the determined $κNN$ and $ΓNN$ values from our CNN1′ are $≈10 %$ and $11 %$⁠, respectively, while those from our CNN3′ slightly increases to $≈17 %$ and $13 %$⁠, respectively. We also use our trained CNN3′ to analyze the position fluctuation data of individual particles from a 2D dusty plasmas experiment. The obtained distribution of the determined $κNN$ or $ΓNN$ values from CNN3′ always exhibits one prominent peak, at $κNN≈0.50$ and $ΓNN≈780$⁠, respectively, well agreeing with $κ≈0.50$ and $Γ≈700$ determined from the traditional phonon spectra fitting method. Our obtained results above clearly indicate that, using machine learning methods, the two global characterization parameters $κ$ and $Γ$ in 2D dusty plasma experiments are able to be accurately determined simultaneously purely from the position fluctuations of local individual particles. Note, as compared with the previous study,⁵¹ our new CNNs have much higher efficiency with nearly the same accuracy.

See the supplementary material for our trained CNNs (CNN1′ and CNN3′) and a portion of our experimental data analyzed in the main text (data.txt). In addition, this supplementary material also contains the structure of our model A′ (model.py) and an example demonstration of using our trained CNN3′ in determining the screening parameter and coupling parameter from the position fluctuations of five individual particles (Main.py), as well as a text file to describe these supplementary material files (README.txt).

This work was supported by the National Natural Science Foundation of China under Contract No. 12175159, the 1000 Youth Talents Plan, and the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions.

The authors have no conflicts to disclose.

Chen Liang: Data curation (lead); Formal analysis (lead); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (lead); Writing – original draft (supporting). Dong Huang: Formal analysis (supporting); Investigation (supporting); Validation (supporting). Shaoyu Lu: Formal analysis (supporting); Investigation (supporting); Validation (supporting). Yan Feng: Conceptualization (lead); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Resources (lead); Software (equal); Supervision (lead); Validation (equal); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.