Atmospheric particulate matter particle size evolution analysis method based on physics-informed neural network

Abstract

The present invention discloses a method for analyzing the particle size evolution of atmospheric particulate matter based on a physics-informed neural network. This method applies the physics-informed neural network to the particle population equation, discretizes the particle population balance equation using Gaussian integration, obtains the particle population balance equation that can be represented in a computer after discretization, and embeds the discretized equation into the loss function of the neural network. The neural network is trained using the selected boundary points and internal points as the data set, realizing the calculation of the particle size distribution evolution of atmospheric particulate matter. The present invention does not require a grid, has high solution accuracy and fast speed.

Description

Technical Field

[0001] The present invention relates to the technical field of particle size evolution of atmospheric particulate matter, and particularly relates to an analysis method for particle size evolution of atmospheric particulate matter. Background Art

[0002] With the rapid development of the economy, environmental problems brought about by industrial production have increasingly become the focus of people's attention. Therefore, studying the particle size evolution of particulate matter is of great significance for solving air pollution problems.

[0003] Aerosols in the atmosphere are a multiphase flow system formed by the coexistence of fine particles suspended in a gas and the gas. A multiphase flow system refers to a system in which substances in different phases are mixed and flow. Most flows in nature are multiphase flows, and the same is true for the change process of atmospheric particulate matter. There are phenomena such as Brownian coagulation during the flow of atmospheric particulate matter. Simulating the Brownian coagulation phenomenon with the particle population equation can obtain a relatively accurate prediction of the particle size distribution evolution, which is of great significance for the analysis of atmospheric particulate matter and the treatment of air pollution.

[0004] In the monitoring of atmospheric particulate matter, it is found that the particle size distribution of particulate matter changes with time. This is because there are interactions between particles, such as aggregation, fragmentation, nucleation, etc. This means that it is necessary to add particle population balance after the common energy conservation and mass conservation to better predict the changes of particles. For the solution methods of common particle population balance models, there are discrete methods, standard moment methods, integral moment methods, etc. However, their common problem is that they require grids, and the quality of the grids will have a huge impact on the calculation accuracy. At the same time, due to problems such as large calculation amount and many limitations, it is difficult to predict the particle distribution.

[0005] In summary, it is particularly important to propose a new analysis method for the particle size evolution of atmospheric particulate matter to improve the speed and accuracy of predicting particle size evolution. Summary of the Invention

[0006] The purpose of the present invention is to provide a calculation method for quickly and accurately predicting the evolution of particle size distribution for the study of atmospheric particulate matter. This method combines the particle population balance model with the particle size distribution evolution based on a physics-informed neural network, and realizes the prediction of the particle size distribution evolution of atmospheric particulate matter for Brownian coagulation phenomenon. This method is fast, accurate, and does not require grids and a large amount of data.

[0007] The technical solution of the present invention is as follows:

[0008] An analysis method for particle size evolution of atmospheric particulate matter based on a physics-informed neural network, the specific steps are as follows:

[0009] Step 1. According to the actual size range [v min , vmax and the length of the evolution time [0, T] to determine the sampling range R[v, t] of the sample points, where ν ∈ [v min , v max , t ∈ [0, T].

[0010] Then, using the Latin hypercube sampling method, the internal points u of R are sampled respectively f and the initial point, i.e., the point u at t = 0 b as the data set.

[0011] Step 2. Discretize the population balance equation of particles using Gaussian integration to obtain the population balance equation of particles that can be represented in a computer after discretization.

[0012] Step 3. Construct a feedforward neural network, embed the equation processed in Step 2 as a loss function into the neural network, and optimize the weights and thresholds of the feedforward neural network using the Adam and L-BFGS optimization algorithms.

[0013] Step 4. Use the internal points u sampled in Step 1 f and the initial point u b as the input of the feedforward neural network, and use the feedforward neural network constructed in Step 3 for training.

[0014] Step 5. Uniformly distribute and sample sample points according to the range determined in Step 1, send them into the trained feedforward neural network, and obtain the predicted evolution of the particle size distribution of atmospheric particles through forward propagation;

[0015] The beneficial effects of the present invention are as follows:

[0016] The present invention applies a physics-informed neural network to the population balance equation of particles. Since the model after neural network training only needs volume and time as inputs to obtain the particle size distribution, it is easy to obtain contents such as the scale spectrum, which is convenient for subsequent analysis of the particle size evolution process of atmospheric particles.

[0017] After appropriate training rounds, the error between the predicted value and the particle size distribution of atmospheric particles in the numerical simulation of the present invention can reach the order of 10e-4, with small error and fast calculation speed. Description of the Drawings

[0018] Figure 1 Basic structure diagram of the physics-informed neural network;

[0019] Figure 2 Network structure diagram for the population equation of particles;

[0020] Figure 3 Loss curve graph;

[0021] Figure 4The scale spectrum obtained based on the network prediction value. Detailed implementation manner

[0022] The present invention will be further described below in conjunction with the accompanying drawings and the implementation manner.

[0023] As Figure 1 shown, the structure of the neural network includes a left DNN which is a feedforward neural network, a right PDE which is a loss function embedding physical information, IC is the initial condition, BC is the boundary condition. Through the neural network loop, the error between the predicted value and the true value becomes smaller and smaller until it stabilizes, and finally the prediction of the particle size distribution evolution is realized.

[0024] This embodiment includes the following steps:

[0025] Step 1. According to the particle size volume range [ν min , ν max of atmospheric particulate matter under actual conditions and the time length [0, T] to be predicted, determine the range R[v, t] for sampling as sample points, where v ∈ [v min , ν max , t ∈ [0, T]. In the given example, v ∈ (0, 10], t ∈ [0, 1]. Within this range, using the Latin hypercube sampling method, respectively extract the internal points u f and the initial point, that is, the point u b at t = 0 as the data set.

[0026] Specifically extracting the initial point is to improve the training accuracy of the initial boundary. The forms of the internal points u f and u b extracted here are respectively:

[0027] [[0.12441, 1.34567], [3.56656, 0.243265]…] and [[0.72343, 0], [2.49234, 0]…].

[0028] The number of samples extracted is adjusted according to the required range size. u f extracts 6000 in this example, and u b extracts 500 in the example; here the Latin hypercube sampling is a stratified random sampling different from the Monte Carlo sampling method to ensure uniform and random sampling and improve the generalization ability of the trained neural network; after completion, a data sample of 6500×2 size is obtained.

[0029] Step 2. Discretize the population balance equation of particles by using Gaussian integration to obtain a discretized population balance equation of particles that can be represented in a computer.

[0030] The population balance equation has the form shown in Equation (1). The first term after the equality is called B coag and is processed using the Gauss-Legendre quadrature formula. The second term after the equality is called D coag and is processed using the Gauss-Laguerre integration:

[0031]

[0032] For B coag , by changing the integration limits, it has the form shown in Equation (2):

[0033]

[0034] Using the Gauss-Legendre integration to process it, its discrete form is shown in Equation (3):

[0035]

[0036] For D coag , adjusting the function form gives Equation (4):

[0037]

[0038] Using the Gauss-Laguerre integration to get its discrete form as shown in Equation (5):

[0039]

[0040] Through the above processing process, the finally obtained discrete form is as shown in Equation (6):

[0041]

[0042] After processing, 5 integration points are selected to ensure that the algebraic accuracy meets the equation accuracy requirements.

[0043] Step 3. Construct a feedforward neural network and embed the equation processed in Step 2 into the loss function of the neural network to obtain the objective function to be optimized. At this time, the loss function is Loss = Loss1 + Loss2, where the loss function represented by Loss1 has the form shown in Equation (7):

[0044]

[0045] where M u represents the number of internal sampling points, and the B coag and D coag terms respectively represent the first and second terms after the equality of the discretized population balance equation.

[0046] Loss2 represents the loss function generated by the initial conditions and has the form shown in Equation (8):

[0047]

[0048] where M f represents the number of initial boundary points, and n(v, t) represents the particle density function to be obtained.

[0049] The boundary conditions of the equation are not required here, so there are only Loss1 and Loss2. Its structure is as Figure 2 shown, where IC represents the initial condition of the equation, and dt represents the loss function formed by the particle swarm equation. The weights and thresholds of the feedforward neural network are optimized by combining the Adam and L-BFGS optimization algorithms; the reason for choosing to combine the Adam optimization algorithm with the L-BFGS algorithm here is that Adam is less likely to fall into local minima during the optimization process compared to L-BFGS, while the convergence speed and accuracy of L-BFGS are better than those of Adam, so the two are combined for use. The effect is as Figure 3 shown. It can be seen that the Adam optimization algorithm is used to train for 2000 rounds at the initial stage of training, and the error drops rapidly. Then the speed gradually slows down and converges gradually. At this time, the L-BFGS is used to train for 1000 rounds, and the error further drops.

[0050] Step 4. Use the internal points u f extracted in Step 1 and the initial points u b as the inputs of the feedforward neural network, and use the feedforward neural network constructed in Step 3 for training. The feedforward neural network constructed here is a neural network with two inputs and one output, and its input form is an n u+f ×2 tensor, and the output is the corresponding probability density.

[0051] The training process of this network is introduced separately here:

[0052] (1) Initialize the network. The activation function commonly used for fitting curves is the tanh activation function. Corresponding to it is to initialize the weights and thresholds of the network with Xavier. Compared with the common Sigmoid function and relu function, the tanh activation function is symmetric about the origin, adapts to the situation when the input is less than 0, and at the same time the curve is smooth, which can change the linear network into a non-linear network, but there is a problem of gradient disappearance. The Xavier initialization method can alleviate the problem of gradient disappearance existing in tanh.

[0053] (2) Forward propagation. Send the internal points and initial points extracted in Step 1 into the network in the form of tensors. Through the forward propagation process, that is, according to the network structure, the data are respectively operated with the weights, thresholds, activation functions, etc. of each layer, and the partial derivatives of each corresponding neuron are retained. Using the chain rule, calculate the output of the network at this time, and the output and Substitute it into the loss function LOSS, and then use the Adam optimization algorithm and the L-BFGS optimization algorithm to record the gradient at this time.

[0054] (3) Backpropagation. Using the gradient obtained from the forward propagation and the set learning rate, here the learning rate is set to 0.01, perform backpropagation, and according to the chain rule, correct the weights and thresholds existing in each neuron. Through continuous training, correct the weights and thresholds to make the loss function continuously decrease and tend to be stable, and finally achieve the desired accuracy.

[0055] Step 5. According to the volume and time ranges determined in Step 1, sample points are drawn according to a uniform distribution and fed into the trained feedforward neural network, and the particle size distribution of the required atmospheric particulate matter is obtained through forward propagation.

[0056] As Figure 4 shown, the error between the particle size distribution evolution predicted by the present invention and the particle size distribution of the corresponding analytical solution is small. Here, the analytical solution was given by Ramabhadran in 1976, and its initial conditions are as shown in formula (9):

[0057]

[0058] The basic form of the analytical solution is as shown in formula (10):

[0059]

[0060] where θ = KN0t / (2 + KN0t), and here N0 = 1, v0 = 1, K = 1 are taken.

[0061] Table 1

[0062]

[0063] From Table 1, the root mean square error RMSE can be obtained as 0.0004828, and the average training time is 279.91 s.

Claims

1. An analysis method for the particle size evolution of atmospheric particulate matter based on a physics-informed neural network, characterized in that It includes the following steps: Step 1. Determine the range for sampling points according to the size range of atmospheric particulate matter under actual conditions and the time length to be predicted; Using the Latin hypercube sampling method, respectively extract the internal points and initial points within the sampling point range as the data set; Step 2. Discretize the population balance equation of particles using Gaussian integration to obtain the population balance equation of particles that can be represented in a computer after discretization; Step 3. Construct a feedforward neural network, embed the equation processed in Step 2 as the loss function into the neural network, and optimize the weights and thresholds of the feedforward neural network by combining the Adam and L-BFGS optimization algorithms; Step 4. Use the internal points and initial points extracted in Step 1 as the input of the feedforward neural network, and train the feedforward neural network constructed in Step 3; Step 5. Uniformly distribute and extract sampling points according to the range determined in Step 1, send them into the trained feedforward neural network, and obtain the predicted evolution of the particle size distribution of atmospheric particulate matter through forward propagation; Among them, the discretization of the population balance equation in step 2 using Gaussian integration specifically refers to the first term B after the equality of the population balance equation coag processed by the Gauss-Legendre quadrature formula, and the second term D after the equality coag processed by the Gauss-Laguerre integration, specifically as follows: For B coag , by changing the integration limits, it is obtained in the form shown in Formula (2): The discrete form is obtained by using Gauss-Legendre integration as shown in Formula (3): For D coag , adjust the function form to obtain formula (4): Use Gaussian-Laguerre integration to obtain its discrete form as shown in formula (5): Through the above processing process, finally obtain the discrete form as shown in formula (6): Select 5 integration points after processing to ensure that the algebraic accuracy meets the accuracy requirements of the equation; Among them, when the loss function in Step 3 is embedded into the neural network, the loss function Loss at this time is set as: Loss = Loss1 + Loss2 Where Loss1 represents the loss function generated by the population balance equation of particles, and Loss2 represents the loss function generated by the initial conditions; where M u represents the number of internal sampling points, B coag and D coag terms respectively represent the first and second terms after the equal sign of the discretized population balance equation; Loss2 represents the loss function generated by the initial conditions, and its form is as shown in formula (8): where M f represents the number of initial boundary points, and n(v,t) represents the particle density function to be obtained.

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