Single information dissemination is the basic structure of network public opinion information dissemination, and the process of an individual participating in single information dissemination is also the basis of single information dissemination analysis [15]. The dynamic model of single information spreading based on forwarding is called Susceptible-Forwarding-Immune (SFI) model. Here, the sum of the total number of people in susceptible state ( S ), forwarding state ( F ), and immune state ( I ) remains the same. Therefore, the SFI dynamic model of single information propagation in the form of a differential equation is established as Equations 1-3: d d t S t = − β S t F t (1) d d t F t = p β S t F t − α F t (2) d d t I t = 1 − p β S t F t + α F t (3) where the average contact rate β represents the average rate at which an individual in the susceptible state can access the information, the average forwarding rate p represents the average rate that an individual in the susceptible state forwards the information after being exposed to the information, and the average immune rate α represents the average rate at which an individual changes from the forwarding state to the immune state. In the SFI dynamic model of single information transmission, another variable defined by the researchers is the cumulative number of forwarding users, which is a quantity that can be directly obtained from the network transmission platform and is also a crucial quantity for the analysis of the model. It is defined as Equation 4: C t = ∫ 0 t p β S t F t d t (4)

For the SFI dynamic model with single information dissemination, when fitting the actual case data, the above three key variables in the model will be formed as an unknown parameter vector and estimated to make the cumulative number consistent with the real data. Therefore, to find the best fit for the data is to find the best combination of parameters to minimize the error between the estimated and real values. In general, the least squares method is the most widely used in the fitting of information propagation dynamics research, where other machine learning methods such as the Monte Carlo method are also used. Since the model does not have an analytical formula and its form is very complex, minimizing the sum of squared deviations becomes a nonlinear least squares problem.

According to the universal approximation theorem, the neural network can be regarded as a general nonlinear function approximator, and the modeling process of a differential equation is to find nonlinear functions that meet relevant constraints [8]. Using neural networks to approximate model differential equations has become a research hotspot. The automatic differentiation technology in deep neural networks can be naturally applied to the calculation in differential equations and the constraint conditions of differential form are integrated into the loss function design of neural networks, so as to obtain neural networks with physical model constraints, which is the most basic idea to design embedded physical neural networks [29]. The PINNs model aims to establish a correlation between deep neural networks and various physical phenomena represented as systems of differential equations, thereby enhancing the interpretability of neural networks and expediting the resolution of differential equations. In the common application model of PINNs, the incorporation of physical information is primarily manifested in the loss function. The implementation of PINNs involves the integration of physics principles and neural networks through a well-designed approach, which does not pose significant challenges. First, the neural network is constructed, where the parameters are randomly initialized. The initialized neural network takes in the independent variables of the system of differential equations and produces the solutions that are needed to be optimized for the dependent variable of the system. Secondly, the output value of the dependent variable generated by the neural network fails to provide evidence for the validity of the equation, and this discrepancy constitutes the loss of equations. At the same time, the loss of the data level and the loss of the boundary condition are introduced to combine with the given weights, which become the loss of the whole model. Finally, the gradient descent method and other optimization methods are used to train the model and fit the differential equation.

Before data-driven machine learning made great progress, many physics and engineering fields were physically model-driven. Over the years, these fields had accumulated a wealth of physical models, most of which were described in the form of partial differential equations, such as Navier-Stokes equations in fluid dynamics [30], Maxwell equations in electromagnetic field theory [31] and Schrodinger equation in quantum mechanics [32]. Directly solving the physical model can make accurate predictions, but it faces the problems of too large errors caused by simple physical models, too high solution complexity caused by complex physical models and too large solution errors caused by missing or inaccurate measurement of physical model parameters and initial boundary values. The traditional numerical methods of partial differential equations face great challenges in solving inverse problems, complex geometric regions and high-dimensional space. In contrast, the classical machine learning algorithms are purely data-driven. The task of training a supervised machine learning model is to establish a functional mapping from the input data to the output data, that is, to learn a specific model from the pre-obtained training data and the pre-defined algorithm structure. However, in many physical and engineering fields, these training data often imply part of the prior knowledge, such as the law of conservation of momentum, the law of conservation of mass and so on [29]. PINNs combine the advantages of data-driven machine learning models and physical models. Under the condition of a small amount of training data, physics-based neural networks can train models satisfying physical constraints automatically, have better generalization performance while ensuring accuracy, and predict important physical parameters of the model [33].

The information propagation process in the compartment model can be abstracted into a dynamic pattern, as illustrated in Figure 1. Here, u T represents the system state at time t , which is influenced by both the initial value condition a and the boundary condition ∂ X over time, and undergoes transformation according to the propagation law f * . Typically described by differential equations, this propagation law f * is approximated using embedded physical information neural networks to facilitate the solving of the entire dynamics process or other tasks.

Throughout this paper, we make the assumption that the underlying model of single-information network propagation follows the structure depicted in the SFI model, which can be mathematically represented by a system of ordinary differential equations. All propagation dynamics based on compartment models can be described using either systems of partial differential equations or ordinary differential equations. Therefore, our focus lies on the SFI model as it serves as a fundamental framework for studying information dissemination. This foundation allows us to derive various single information dissemination models under different scenarios or influence factors, which share similar and universal sets of differential equations. Taking the SFI model as an illustrative example, it is important to note that the total number of users in different states within the compartment model remains constant throughout its dynamic process, where changes are reflected through mutual transformations between different groups.

In order to fit the real data combined with the platform data, relevant scholars introduce the cumulative forwarding number C t 5 , which is the same as other similar models. In the system of ordinary differential equations with single information propagation such as SFI, the initial values are designed as follows: F 0 = F t t = 0 = C 0 = C t t = 0 = 1 , I 0 = I t t = 0 = 0 . In different information propagation models, whether ordinary differential equations or partial differential equations, the initial value or boundary value is related to the numbers of each state, which can be obtained from the real data. Based on the ordinary differential equations of single information propagation, some scholars introduce the latency period and other factors affecting the propagation, such as opinion, emotion, etc [34–36]. This kind of propagation dynamic equations is similar to the equations of single information propagation, and their solution methods are also universal.

Based on the method of physics-informed neural networks, we introduce a deep learning framework informed by the information propagation dynamic equations that describe the single information propagation processes and their derivatives. Most studies express the information propagation dynamics as ordinary differential equations, and some introduce other factors besides time as independent variables of the equations to construct partial differential equations. Our PINN modeling framework takes into account the two types of equations simultaneously, the only difference between the two ways is that the input of the neural network is one or more.

In our framework shown in Figure 2, the Application Program Interface (API) is used to obtain real propagation data from the social media platform, including the changes of the cumulative forwarding number of a certain news over time. Therefore, the time t and other independent variables are the input of the neural network, and the cumulative forwarding number C t is supervised and studied as real data, which is applied to design the loss functions of our model. In different information propagation dynamic problems, the supervision information we use may also be different. For example, considering the information propagation dynamics driven by the emotion factors, the real data we focus on for fitting can be the cumulative forwarding numbers under different categories of emotion. Moreover, when using the ordinary differential equations to represent the information propagation, the independent variable of the input model is only time t , namely, the input layer of our proposed neural network model has only one variable; When applying the partial differential equations to describe the information propagation, the independent variables of the input model can be time t and social distance x , namely, the input layer of our proposed neural network model has two inputs: time t and social distance x .

A neural network with parameters θ takes time t and other independent variables affecting information propagation as the input and outputs a vector of the state variables as a surrogate of the PDE solution, such as S t , C t and Θ t which represent the other possible variables. We use multiple fully connected layers as the hidden layers of a deep neural network because fully connected neural networks can in some sense be used to approximate arbitrary functions. It should be noted that in the dynamic equations of information propagation, there are often dynamic parameters that need to be fitted, such as β , p and so on, which are directly involved in the calculation of the equations. Therefore, we use independent neurons to represent these dynamic parameters that need to be optimized and use the automatic differentiation mechanism of neural networks to design and optimize the loss function. Finally, this kind of inverse problem is solved.

The next key step is to constrain the neural network to satisfy the scattered observations of C t and its variants as well as the PDE system (including ODEs), which is realized by constructing the loss function considering terms corresponding to the observations and the PDE system. Specifically, we assume that we have the measurements of y 1 , y 2 , · · · , y M at the time t 1 , t 2 , · · · , t N respectively and we enforce the neural network to satisfy the PDE system at the time point τ 1 , τ 2 , · · · , τ N . The times t 1 , t 2 , · · · , t N and   τ 1 , τ 2 , · · · , τ N could be chosen at random. Then, the total loss is defined as a function of both θ and ε , θ refers to all the parameters that need to be trained in the neural network, including the weights and biases of the neurons, and ε represents all the parameters that need to be predicted in PDEs: L θ , ε = L d a t a θ + L e q θ , p + L i c θ (5) L data θ = ∑ m = 1 M   w m data L m data = ∑ m = 1 M   w m data 1 N data ∑ n = 1 N data   y m t n − x ^ s m t n ; θ 2 (6) L eq θ , ε = ∑ s = 1 S   w s eq L s eq = ∑ s = 1 S   w s eq 1 N eq ∑ n = 1 N eq   d x ^ s d t τ n − f s x ^ s τ n ; θ , τ n ; ε 2 (7) L ic θ = ∑ s = 1 S   w s ic L s ic = ∑ s = 1 S   w s ic x s T 0 − x ^ s T 0 ; θ 2 + x s T 1 − x ^ s T 1 ; θ 2 2 (8)

The loss function is designed as shown in Equation 5, where the variable L data represents the association between the data acquisition system’s M sets of observations y and the variable “data” as Equation 6, while L eq enforces the dynamic equations for information propagation as Equation 7. In particular, w m data , w s eq and w s ic represent the weights of the data, the equations and the boundary conditions respectively, m is the number of real data that can be obtained, and s represents the number of PDE equations, in addition, f S represents a single equation in a PDE system, a total of s . We utilize automatic differentiation to analytically compute the derivative of the dependent variable in a PDE system. The third auxiliary loss term, L ic , is introduced as an additional source of information for system identification, which essentially contributes to the data loss component. T 0 represents the boundary conditions of the system at the initial moment, whereas T 1 denotes the boundary conditions at a subsequent arbitrary time point. It should be noted that both L data and L ic represent discrepancies between neural network outputs and measurements, making them supervised losses as Equation 8, whereas L eq , based on PDE systems for information propagation, is unsupervised. The weights are hyperparameters that can be manually adjusted, by default, the weights ​​are set to 1. In our final step, we simultaneously infer both neural network parameters θ and unknown parameters ε of the PDEs by minimizing the loss function using gradient-based optimizers such as Adam optimizer. It is important to note that our proposed method optimizes θ and ε concurrently, distinguishing it from meta-modeling.

The third auxiliary loss term L ic is introduced as an additional source of information for system identification, serving as a crucial component in the data loss. It should be noted that both L data and L ic represent the disparities between neural network outputs and measurements, thus functioning as supervised losses. On the other hand, L eq , which relies on the PDE system for information propagation, operates as an unsupervised loss. In the final step, we simultaneously optimize both the neural network parameters θ and unknown parameters ε of the PDEs by minimizing the loss function using gradient-based optimizers like Adam optimizer. Importantly, our proposed method distinguishes itself from meta-modeling by optimizing θ and ε concurrently as Equation 9 shows. θ * , ε * = arg ⁡ ⁡ ⁡ min θ , p ⁡   L θ , ε (9)

Remark 1

In the actual training process, due to the extensive data requirements for neural network training and the limited data collected from social media platforms, our framework necessitates the sampling of points within the defined domain to acquire a more substantial dataset, which is essential to facilitate the training of our PINN model.

Remark 2

The algorithm is implemented in Python using paddlepaddle. The width and depth of the neural networks depend on the size of the equations and the complexity of the information propagation dynamics. We use the sigmoid activation function except for the last neural network layer which uses sigmoid function to scale the data at different dimensions. For the training, we use a combination of two optimizers, Adam and L-BFGS, to optimize the θ and ε in order to reduce the training time while ensuring accuracy and the learning rate of 0.001, where the training is performed using the full batch of data. Since the total loss consists of two supervised losses and one unsupervised loss, we perform the training using the following two-stage strategy, which is found in our experiments to speed up the network convergence:

Stage 1. The network is initially trained using the two supervised losses L data and L ic for a few iterations, taking into account the fact that supervised training is typically more straightforward than unsupervised training. This enables the network to rapidly align with the observed data points.

As mentioned in the second section of this paper, the SFI model [15], a classic compartmental model in the field of network information dissemination, serves as the foundational framework for numerous works. Consequently, we employed the proposed framework for simulating the dynamic process of single information propagation to compare with the traditional solution method in the SFI model, encompassing the fitting of population quantities for various states and the prediction of the propagation dynamic parameters. Similarly, we employed the same forwarding data collected from a hot topic on the Chinese Sina-microblog as the foundational dataset for our framework. To expedite the convergence of the neural network, we also applied data scaling and subsequently calculated the Mean Squared Error for the fitted results. The ultimate fitting performance is illustrated in Figure 3:

The total number N of individuals in the SFI model remains constant, because the system operates on a compartmental framework. During simulation, we utilize the variables S t and F t to minimize computational complexity, with I t being calculated based on these variables. Therefore, our proposed framework outputs results for S t , F t , and C t . To facilitate comparison, we have employed data from the SFI model. In our framework, the input layer of neural network consists of a single neuron representing time t , while the output layer comprises three neurons corresponding to S t , F t and C t , The neural network employs six hidden layers with 32 neurons in each layer. However, due to insufficient data points (only 50), it is not feasible to train a deep-learning model. Therefore, we obtain additional training data by sampling at intervals along independent variables until we have collected 2000 points. The resulting fitting outcomes are depicted in Figure 3, demonstrating that our proposed framework exhibits superior capability in accurately capturing cumulative forwarding data with a final MSE value of 15.41. Additionally, for enhanced visualization purposes, we have scaled down the neural network’s output by a factor of 2 × 10 5 for S t , 3 × 10 2 for F t , and 10 4 for C t .

The emotion-based susceptible-forwarding-immune (E-SFI) [37] propagation dynamic model incorporates the categorization of emotions into positive, neutral, and negative, aiming to describe the process of emotional choices made by users in various states and investigate the information propagation that influences public sentiment. To accommodate this model in our framework, we have to increase the dimensions of the output layer in the neural network and expand the data for supervising model training. Consequently, we conducted simulations and training using original data from event one in the E-SFI model. The training process still utilized an 8-layer fully connected neural network with initial values determined by real data. The sampling and scaling approaches adopted in this part are similar to those in section 4.1, but with a greater number of neurons in the output layer to generate cumulative forwarding numbers under the three emotional states and produce propagation dynamics parameters. The fitting results in Figure 4 demonstrate that our framework outperforms the E-SFI model in accurately fitting real data including all the emotion types.

The two-stage rumor propagation dynamic model aims to design effective strategies for controlling rumors, where the first stage of rumor propagation is characterized by the susceptible/educated-infected-recovered (SO-S/EIR) dynamics and the second stage is characterized by the susceptible/educated-infected-denied-recovered (C-S/EIDR) dynamics [38]. The conventional least-squares fitting method is inadequate for modeling the two-stage rumor propagation dynamics discussed in this study, making it become necessary to separately fit each stage individually. The advantage of our framework lies in the robust fitting capability of neural networks, which enables us to accurately fit the data without the need for data splitting. Based on data and theory from the original paper, we applied the PINN framework for data fitting, of which notable results for both stages are depicted in Figure 5. Compared with the results in the original model, the fitting effect of our model is not satisfactory for the mutation in the two stages, but it performs well at other locations. Furthermore, our proposed framework demonstrates its capability to fit the curves from two distinct stages, thereby enhancing the efficiency of data fitting.

In addition to the time variable t , other independent variables such as distance can also be incorporated into the equation system governing network information propagation dynamics. Traditional methods often encounter challenges in solving partial differential equations compared to ordinary differential equations, and they may struggle to accurately fit real data. However, our framework offers a solution by avoiding complex partial differential equation solving altogether. Instead, we simply adjust the number of input layers in our neural network model to include both time t and other independent variables for finding the equations’ solution. Notably, there is no need to modify the equations in loss function. Consequently, our framework significantly simplifies the resolution of high-dimensional systems involving partial differential equations. To validate its effectiveness in tasks involving multiple independent inputs, we collect data on the trending event “The arrest of the driver involved in the freight lesbian” from Sina-microblog platform, with a sampling time interval of 1 h. The social distance variable is defined based on the group that retweeted the initial message. For example, if user A posts a message and user B forwards it, the social distance between user A and user B is 1. If user C forwards the message forwarded by user B, the social distance between user C and user A is 2, and so on. As social distance increases, there is a decrease in the cumulative number of retweeters for the corresponding message over time. Figure 6 presents the fitting results based on the PSFI model: The fitting results demonstrate that our model effectively captures the variations in curves across various social distances.

The aforementioned experiments have demonstrated the effectiveness of the proposed framework in the context of information dissemination dynamics. Additionally, we seek to validate the robustness of model by adjusting the weights in three key aspects in the SFI model: data, equations and initial conditions and corresponding results are depicted in Figure 7. The values after “Data”, “Eq”, and “Ic” represent the respective proportions of data loss, equation loss, and initial condition loss shown in Figure 7. There is significant fluctuation in results when altering the weights of equations, indicating their crucial role throughout the entire fitting process. Conversely, alterations of the curves are not prominently evident under the changes of initial value conditions. In traditional methods, simulation results are often heavily influenced by initial values, leading that only an appropriate initial value can obtain a reasonable fit. Our proposed framework reduces the susceptibility to variations in initial values while ensuring both fitting effectiveness and model robustness. Furthermore, the overall propagation trends of propagation populations remain unchanged, highlighting the intrinsic mechanisms inherent within the SFI model.

The loss results of each model are computed after 100,000 iterations in Table 1. From the results, it is evident that the PSFI model outperforms both the SFI and E-SFI models in terms of training outcomes. Despite its more intricate structure and resulting complex system of differential equations, the PSFI model benefits from a larger amount of supervised signal data, enabling superior training. In contrast, the two-stage rumor propagation dynamic model based on the SFI model exhibits more complexity due to its lack of sufficient supervised signal data, leading to poorer fitting performance. Ultimately, simulation results based on partial differential equations slightly surpass those based on ordinary differential equations, with only minor differences observed in terms of independent variables in the input model. Thus, to some extent, incorporating social distance variable into the input model significantly impacts propagation dynamics fitting. Moreover, in our proposed framework, data loss often constitutes a substantial portion of the overall loss function, since losses at data level tend to be numerically greater than those incurred by the other two components.

Our framework integrates the forward and inverse problems in the context of information dissemination dynamics. In contrast to conventional approaches, which solve the model parameters and then conduct forward numerical simulations based on these parameters, our approach offers greater efficiency. However, it should be noted that the values of parameters obtained from our framework cannot be directly compared to those obtained through inverse problem-solving methods. This is because the parameters within the neural network in our framework are also an integral part of the overall system, despite its inherent complexity. Our framework primarily focuses on data fitting and predicting the numbers of user in each propagation state. While various modules within the framework interact and depend on each other, minor adjustments are still necessary to align with real-world scenarios. Among these adjustments, scaling pertaining to data handling is crucial way in our framework. Firstly, in propagation dynamic models, there can be substantial numerical disparities in representing different states or groups, ranging from a few to thousands or more, resulting in normalizing these numerical values becoming essential. In our framework, except for the final layer which lacks an activation function, the activation functions in the hidden layers of the neural network are chosen as Tan h functions to aid in numerical normalization. Additionally, certain parameters involved in equations expressing information propagation may also exhibit significant differences. For instance, parameters like the average contact rate β and the average forwarding probability p both fall in the range of 0 , 1 , but only β should be operated on an order of 10 − 4 , which is scaled down β by a factor of 10 4 to facilitate neural network training. Furthermore, for the sake of visualization, we also apply scaling during the plotting process.