Construction pit deformation measurement technology based on neural network algorithm

3.1 Defects of traditional PSO algorithm

In the course of the rapid development of China’s market economy, the number of deep foundation pit construction projects is also increasing year by year. The deformation of deep foundation pit projects can affect the construction safety and the safety of the surrounding buildings, so the measurement of foundation pit deformation has always been the focus of engineering practice. The BPNN-based pit deformation prediction model has been proven to be effective in previous studies, but the BPNN is not stable enough and easily falls into local optimisation, so the study uses the PSO algorithm to improve the BPNN. The PSO algorithm is a biomimetic intelligent algorithm that can obtain the optimal solution through information transfer between particles and the overall iterative update of the population. Therefore, the PSO algorithm is widely used in the solution of optimisation problems. Let there be a population containing m particles, denoted as x = { x 1 , x 2 , … , x m } ; x i = { x i 1 , x i 2 , … , x i d } denotes the coordinates of the i particle; v i = { v i 1 , v i 2 , … , v i d } denotes the flight speed of the i particle; p i = { p i 1 , p i 2 , … , p i d } denotes the fitness value of the i particle, where the best fitness value is denoted as p best and is considered as the individual extreme value; p g = { p g 1 , p g 2 , … , p g d } is the position of all particles in the population; and the best position is denoted as g best and is considered as the global extreme value. In previous studies, in order to improve the performance of the PSO algorithm, some scholars have introduced inertia weights on the velocity term w , and after obtaining the individual and global extremes, PSO can be updated iteratively according to equation (1) to update the velocity and position of the particle i in the d dimension.

where k represents the number of iterations of the algorithm update; c 1 , c 2 is the learning factor, where c 1 mainly regulates the step of the optimal flight of individual particles and c 2 mainly regulates the step of the global optimal flight; rand ( ⋅ ) is a random number with the value range of [0,1]; v i , d ( k ) represents the velocity of the i th particle in the d th dimension at the k th iteration; x i , d ( k ) represents the position of the i th particle in the d th dimension at the k th iteration; p i , d ( k ) represents the global extreme value of the i th particle in the k th iteration; d p g , d ( k ) represents the global extremum of the population in the d dimension at the k iteration. In equation (1), when the value of w is large, it is good for global optimisation, but it will affect the solving efficiency of the algorithm; when the value of w is small, the solving efficiency of the algorithm is faster, but it is easy to fall into local optimisation. Therefore, the setting of the w value is related to the speed and accuracy of the PSO algorithm. In order to adjust the w value, using a linear decreasing strategy, w can be expressed in the following equation:

where w max and w min are the maximum and minimum values of w and T max is the maximum number of iterations of the algorithm. The basic flow of the PSO algorithm is shown in Figure 1.

Figure 1

Basic flow of the PSO algorithm.

However, the PSO algorithm also has more obvious limitations, such as weak global convergence, the tendency to fall into local optima, and the tendency to converge early. To this end, the study proposes strategies to improve the PSO algorithm in terms of particle flight speed, inertia weights, and learning factors.

3.2 Optimisation strategy for the PSO algorithm

To facilitate the calculation, set the particle dimensions to one dimension, and assume that the position and velocity of all particles in the particle swarm, except for the particle i , do not change, then the subscript d , i can be ignored in the formula. Let φ = φ 1 + φ 2 , where φ 1 and φ 2 are defined as shown in the following equation:

where r 1 and r 2 are two random numbers with values in the range [0,1]. Set a variable ρ , which is calculated as shown in the following equation:

Based on the aforementioned equations, equation (1) can be replaced with the following equation:

From equation (5), the following equation is obtained:

Through equation (6), it can be learned that the concept of speed can be removed from the PSO algorithm, thus being able to dispense with the initialisation of the initial speed when initialising the various parameters, thus reducing the complexity of the algorithm and improving its operational efficiency. Based on the aforementioned fact, it is possible to obtain an iterative update formula for the PSO algorithm, which is deformed by simplification to obtain a first-order differential equation, as shown in the following equation:

It can be seen that the new iterative update formula in equation (7) has fewer parameters compared to equation (1), simplifying the algorithm’s operations and improving efficiency. In traditional PSO algorithm optimisation, the inertia weights are generally used in a linear decreasing strategy to ensure both the global and local search capability of the algorithm. However, this method can lead to weaker flight ability of particles in the late iterative stage due to the small value of w , and thus, it is difficult to jump out of the local extrema for particles caught in the local extrema. To address this drawback, a random inertia weight is proposed to make the w value with some uncertainty, so that the particles have the ability to fly out of the local optimum even in the late iteration. The study combines the stochastic inertia weight and the linear decreasing inertia weight to obtain the following equation:

where W 1 , W 2 , and μ are defined as shown in the following equation:

where W max and W min are the maximum and minimum weight values set, respectively. Drawing on the idea of variation in GA, a variation operation is added to the random inertia weights to make it possible for the particles to have a reverse search to avoid missing the global optimum, and this parameter is represented by r 3 . Combining the aforementioned equations, the expression for the stochastic inertia weights is obtained and is given in the following equation:

There are two learning factors in the PSO algorithm, c 1 , c 2 . c 1 is used to adjust the step size of the particles towards the individual extremes, while c 2 is used to adjust the step size of the particles towards the global extremes. Therefore, a large value of c 1 is needed at the beginning of the iteration to ensure that all particles in the swarm are closer to the optimal position to obtain the individual extremes, thus enhancing the global search capability; a large value of c 2 is needed at the end of the iteration to ensure the convergence of the PSO algorithm and thus obtain the global optimal solution. From the aforementioned fact, we know that the values of c 1 and c 2 are basically inversely proportional. When the value of c 1 is larger, a smaller value of c 2 is required; when the value of c 2 is larger, a smaller value of c 1 is required. For this reason, the study proposes an evolutionary formula for the learning factor, as shown in the following equation:

where c max and c min are the maximum and minimum values of the pre-set learning factors, respectively. According to equation (11), in the early iteration of the PSO algorithm, the value of c 1 is larger and the value of c 2 is smaller, which makes the PSO algorithm have a strong global search capability; in the late iteration, the value of c 2 is larger and the value of c 1 is smaller, which makes the PSO algorithm have a strong local search capability and higher accuracy, which in turn strengthens the convergence performance of the PSO algorithm. In the process of the PSO algorithm particle search, there is a certain possibility of flying out of the feasible domain range, leading to a decrease in the accuracy of the PSO algorithm and the occurrence of invalid solutions. Therefore, certain measures need to be taken to deal with the out-of-bounds particles accordingly. The strategy adopted in the study is as follows: when a particle in the population appears to be out of bounds, a dimension of the particle flies beyond the flight domain, or the particle flies beyond the flight domain, at which point the out-of-bounds particle is reset and a value is regenerated in the flight domain, as shown in the following equation:

Based on the aforementioned fact, the modified particle swarm optimization (MPSO) is constructed to improve and optimise the particle swarm algorithm to increase efficiency and accuracy. The flow of the MPSO algorithm is shown in Figure 2.

Figure 2

MPSO algorithm flow.

3.3 Construction of a pit deformation measurement model based on MPSO–BP

BPNN is a kind of error direction propagation neural network, which is one of the most widely used and mature neural networks.

As can be seen in Figure 3, the BPNN is a three-layer model, consisting of an input layer, an implicit layer, and an output layer. In Figure 3, x j represents the input signal of the node j ; w i j represents the connection weight of the node i in the hidden layer and the node in the input layer; j θ i represents the threshold of the node in the hidden layer; i ϕ ( x ) is the activation function of the neuron in the hidden layer; ψ ( x ) is the activation function of the neuron in the output layer; a k represents the threshold of the node k in the output layer; and o k is the output value of the node k in the output layer. There are two signals in the underlying BPNN. One is called the work signal, which is propagated forward in the BPNN; the other signal is called the error signal, which is propagated backward in the BPNN. During the forward propagation of the working signal, the output value o k of the node k in the output layer is shown in the following equation:

Figure 3

Basic structure of BPNN model.

In the back-propagation process of the error, the error between the output value of the output layer and the desired output value is obtained after the calculation, and this error is back-propagated up to the neurons in the hidden layer, and then, the connection weights and thresholds are adjusted according to the value of this error, and this operation is repeated in the process of iteration until the error is less than the set value or the number of iterations reaches the set value. The training error of the BPNN is calculated as shown in the following equation:

where E denotes the training error of the BPNN and p denotes the number of training samples. BPNNs have important applications in various fields, for example, in deep foundation pit projects, where the extent of pit deformation is predicted and measured by monitoring data. There are generally three types of pit deformation: surface settlement, pit bottom uplift deformation, and enclosure deformation. The historical data obtained from the monitoring points are fed into the BPNN, which is trained to obtain the predicted values of pit deformation and then formulate countermeasures to ensure the safety of the pit project. However, the performance of the BPNN depends on the initial parameters, which is prone to local optimisation and unstable performance, so a global optimisation algorithm is needed to optimise the training of the BPNN. The study uses the MPSO algorithm to optimise the training of BPNNs. Before the MPSO algorithm is used to optimise the BPNN, the connection weights and thresholds of the layers in the BPNN are first encoded. The study uses vector coding to perform the coding operation; in vector coding, all particles are considered as a vector, and each vector represents a neural network parameter that needs to be initialised, such as weights and thresholds. The coding can be expressed in the following equation:

where i denotes the number of particles and i = 1 , 2 , … , m . After the coding is completed, the parameters to be initialised are mapped to particle dimensions and an optimisation search operation is performed to obtain the optimal parameters. After optimising the BPNN using the MPSO algorithm, the MPSO–BP model is constructed to achieve predictive measurements of pit deformation. The basic flow of the MPSO–BP algorithm is shown in Figure 4.

Figure 4

Basic flow of the MPSO-BP algorithm.