# 基于交替方向乘子法的球磨机负荷分布式随机权值神经网络模型Distributed Random Weight Neural Network Model for Ball Mill Load Based on Alternating Direction Multiplier Method

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When the traditional centralized machine learning algorithms deal with the large-scale data, there exists high communication overhead, low computational efficiency and large space complexity. A distributed random weights neural network modeling method is used to build ball mill load model based on Alternate Direction Multiplier Method (ADMM). Local network nodes are built using Random Vector Functional-Link (RVFL) network with regularized random weights, and the parameters of global distributed ball mill load model are optimized iteratively to update the solution by using the ADMM method. The experimental results show that the ADMM-RVFL-based ball mill load model has comparative advantages in terms of speed and accuracy.

1. 引言

2. 预备知识

$\begin{array}{l}\underset{u,v}{\mathrm{min}}f\left(u\right)+g\left(v\right)\\ \text{s}\text{.t}\text{.}\text{\hspace{0.17em}}Au+Bv=c\end{array}$ (1)

${\text{L}}_{P}\left(u,v,\lambda \right)=f\left(u\right)+g\left(v\right)+{\lambda }^{\text{T}}\left(Au+Bv-c\right)+\frac{\rho }{2}{‖Au+Bv-c‖}_{2}^{2}$ (2)

${u}_{k+1}:=\underset{u}{\mathrm{arg}\mathrm{min}}{\text{L}}_{P}\left(u,{v}_{k},{\lambda }_{k}\right)$ (3)

${v}_{k+1}:=\underset{v}{\mathrm{arg}\mathrm{min}}{\text{L}}_{P}\left({u}_{k+1},v,{\lambda }_{k}\right)$ (4)

${\lambda }_{k+1}:={\lambda }_{k}+\rho \left(A{u}_{k+1}+B{v}_{k+1}-c\right)$ (5)

ADMM算法类似于增广拉格朗日乘子法，实际上，也可以通过采用增广拉格朗日乘子法进行迭代求解 $u,v,\lambda$ 。如果已知道 $k$ 时刻 $\left({u}_{k},{v}_{k},{\lambda }_{k}\right)$ 数值大小，则求得：

$\left({u}_{k+1},{v}_{k+1}\right)：=\underset{u,v}{\mathrm{arg}\mathrm{min}}{\text{L}}_{P}\left(u,v,{\lambda }_{k}\right)$ (6)

${\lambda }_{k+1}:={\lambda }_{k}+\rho \left(A{u}_{k+1}+B{v}_{k+1}-c\right)$ (7)

2.2. RVFL神经网络模型

$f\left(x\right)={\sum }_{m=1}^{L}{\beta }_{m}{h}_{m}\left(x;{w}_{m}\right)={\beta }^{\text{T}}h\left(x;{w}_{1},\cdots ,{w}_{L}\right)$ (8)

$m$ 个变换通过向量 ${w}_{m}$ 进行参数化，输入是一个d维的实向量 $x\in {R}^{d}$ 。每一个 $h:x\to R$ 称为一个基，隐函数，函数链接。在仿真中使用sigmoid基础函数：

$h\left(x;w,b\right)=\frac{1}{1+\mathrm{exp}\left\{-{w}^{\text{T}}x+b\right\}}$ (9)

$H=\left(\begin{array}{ccc}{h}_{1}\left({x}_{1}\right)& \dots & {h}_{L}\left({x}_{N}\right)\\ ⋮& \ddots & ⋮\\ {h}_{1}\left({x}_{1}\right)& \dots & {h}_{L}\left({x}_{N}\right)\end{array}\right)$ (10)

${\beta }^{*}=\underset{\beta \in {R}^{L}}{\mathrm{arg}\mathrm{min}}\frac{1}{2}{‖H\beta -Y‖}_{2}^{2}+\frac{\lambda }{2}{‖\beta ‖}_{2}^{2}$ (11)

$J\left(\beta \right)$ 的梯度为零，即：

$\frac{\partial J}{\partial \beta }={H}^{\text{T}}H\beta -{H}^{\text{T}}Y+\lambda \beta =0$ (12)

${\beta }^{*}={\left({H}^{\text{T}}H+\lambda I\right)}^{-1}{H}^{\text{T}}Y$ (13)

${\left({H}^{\text{T}}H+\lambda I\right)}^{-1}{H}^{\text{T}}={H}^{\text{T}}{\left(H{H}^{\text{T}}+\lambda I\right)}^{-1}$ (14)

${\beta }^{*}={H}^{\text{T}}{\left(H{H}^{\text{T}}+\lambda I\right)}^{-1}Y$ (15)

${t}_{k}\left[n+1\right]={t}_{k}\left[n\right]+\gamma \left({\beta }_{k}\left[n+1\right]-Ζ\left[n+1\right]\right)$ (16)

${\beta }_{k}\left[n+1\right]={\left({H}_{k}^{\text{T}}{H}_{k}+\gamma Ι\right)}^{-1}\left({H}_{k}^{\text{T}}{Y}_{k}-{t}_{k}\left[n\right]+\gamma Ζ\left[n\right]\right)$ (17)

$Ζ\left[n+1\right]=\frac{\gamma \stackrel{^}{\beta }+\stackrel{^}{t}}{\frac{\lambda }{L}+\gamma }$ (18)

$\stackrel{^}{\beta }=\frac{1}{L}{\sum }_{k=1}^{L}{\beta }_{k}\left[n+1\right]$ (19)

$\stackrel{^}{t}=\frac{1}{L}{\sum }_{k=1}^{L}{t}_{k}\left[n+1\right]$ (20)

${\left({H}_{k}^{\text{T}}{H}_{k}+\gamma Ι\right)}^{-1}={\gamma }^{-1}\left[Ι-{H}_{k}^{\text{T}}\left(\gamma Ι+{H}_{k}{H}_{k}^{\text{T}}\right){H}_{k}\right]$ (21)

${r}_{k}\left[n\right]={\beta }_{k}\left[n\right]-Ζ\left[n\right]$ (22)

$s\left[n\right]=-\gamma \left(Ζ\left[n\right]-Ζ\left[n-1\right]\right)$ (23)

${‖{r}_{k}\left[n\right]‖}_{2}<{ϵ}_{\text{primal}}$ (24)

${‖s\left[n\right]‖}_{2}<{ϵ}_{\text{dual}}$ (25)

${ϵ}_{\text{primal}}=\sqrt{L}{ϵ}_{\text{abs}}+{ϵ}_{\text{rel}}\mathrm{max}\left\{{‖{\beta }_{k}\left[n\right]‖}_{2},{‖Ζ\left[n\right]‖}_{2}\right\}$ (26)

${ϵ}_{\text{dual}}=\sqrt{L}{ϵ}_{\text{abs}}+{ϵ}_{\text{rel}}{‖{t}_{k}\left[n\right]‖}_{2}$ (27)

4. 实验仿真结果与分析

4.1. 实验对象描述

Figure 1. Signal acquisition of experimental ball mill

4.2. MATLAB分布式计算集群配置

4.3. 球磨机实验结果和分析

$\text{MAE}=\frac{1}{L}{\sum }_{\left({X}_{i},{y}_{i}\right)\in T}|f\left({X}_{i}\right)-{y}_{i}|$ (28)

$\text{MSE}=\frac{1}{L}{\sum }_{\left({X}_{i},{y}_{i}\right)\in T}{\left(f\left({X}_{i}\right)-{y}_{i}\right)}^{2}$ (29)

$\text{RMSE}=\frac{1}{L}\sqrt{{\sum }_{\left({X}_{i},{y}_{i}\right)\in T}{\left(f\left({X}_{i}\right)-{y}_{i}\right)}^{2}}$ (30)

(a) (b)(c)

Table 3. Comparison of forecasting results of filling rate with different modeling methods

5. 结论

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