GEKKO - 神经网络 - 求解器不工作

Question

I am trying to create a Neural Network to predict the behavior of variable "miu".

Since I only have 6 data points, I tried to use spline to find more points that follow the behavior of the system to afterwards use all those points in the neural network.

I am trying to use 2 inputs, which are time and cell concentration. And the expected output would be the miu value, which is given as the derivative dy/dx where y is the cell concentration and x the time.

I implemented the following code:

from gekko import brain
import numpy as np
import matplotlib.pyplot as plt  
from numpy import diff
from scipy.interpolate import CubicSpline
xm = np.array([ 0.0 , 23.0 , 47.0 , 49.0 ,\
                71.5 , 95.0 , 119.0 , 143.0 ])

def spline(cell):    
    m = GEKKO()
    m.options.IMODE=2
    c = [m.FV(value=0) for i in range(4)]
    x = m.Param(value=xm)
    cell = np.array(cell)
    y = m.CV(value=cell)
    y.FSTATUS = 1
    # polynomial model
    m.Equation(y==c[0]+c[1]*x+c[2]*x**2+c[3]*x**3)
    c[0].STATUS=1
    m.solve(disp=False)
    c[1].STATUS=1
    m.solve(disp=False)
    c[2].STATUS=1
    c[3].STATUS=1
    m.solve(disp=False)
    pbr = [c[3].value[0],c[2].value[0],\
           c[1].value[0],c[0].value[0]]
    print(pbr)
    xp = np.linspace(0,144,100)
    plot1 = plt.figure(1)
    if cell[0] == cell_br2[0]:
        plt.plot(xm,cell_br2, 'ko', label ='BR2')
        plt.plot(xp,np.polyval(pbr,xp),'g:',linewidth=2)
    elif cell[0]  == cell_br1[0] :
        plt.plot(xm,cell_br1, 'mo', label ='BR1')
        plt.plot(xp,np.polyval(pbr,xp),'r:',linewidth=2)

    plt.xlabel('time(hr)')
    plt.ylabel('cells')
    plt.legend()
    dx = diff(xp)
    dy1 = diff(np.polyval(pbr,xp))
    deriv1 = dy1/dx
    time =np.linspace(0,144,99)
    plot1 = plt.figure(2)
    if cell[0] == cell_br2[0]:
        plt.plot(time,deriv1,'b:',linewidth=2, label ='BR2')
    elif cell[0] == cell_br1[0]:
        plt.plot(time,deriv1,'m:',linewidth=2, label ='BR1')
    plt.xlabel('time(hr)')
    plt.ylabel('miu(1/h)')
    plt.legend()
    plt.show()
    return(deriv1)
    
m = GEKKO()
cell_br1 = (0.63*10**6 , 1.10*10**6, 2.06*10**6, 2.08*10**6,\
            3.73*10**6, 3.89*10**6, 3.47*10**6,2.312*10**6)
cell_br2=  (0.58*10**6 , 0.96*10**6, 2.07*10**6, 1.79*10**6,\
            3.57*10**6, 3.34*10**6, 2.62*10**6, 1.75*10**6)

b = brain.Brain()
b.input_layer(2)
b.layer(linear=5)
b.layer(tanh=5)
b.layer(linear=5)
b.output_layer(1)

x_s = np.linspace(0,144,99)
xg = np.array([ 0.0 , 23.0 , 47.0 , 49.0 , 71.5 ,\
                95.0 , 119.0 , 144.0 ])
cells_spline = CubicSpline(xm, cell_br1) 
y_cells = cells_spline(x_s)
miu_1 = spline(cell_br1)
miu_2 = spline(cell_br2)
x = (x_s, y_cells)#, y_glucose) #Inputs (3)
y = (miu_1)    #Output (2)

b.learn(x,y) # train
xp = np.linspace(0,144,99)
yp = b.think(x) # validate
yyp = np.array(yp)
miu = np.reshape(yyp, (99,))

plot1 = plt.figure(3)
plt.plot(xp,miu,'r-', label = 'Predicted ')
plt.plot(x_s,miu_1,'bo', label = 'Experimental points')
plt.xlabel('Time [hr]')
plt.ylabel('miu [1/h]')
plt.legend()
plt.show()

Although solver finds a solution, it is constant, which indicated that the solver is not working. My output is the following :

Can someone please help? I can't find what is failing. Thanks

Answer 1

Here are a couple issues with your current approach:

• The training uses two inputs while the validation uses only one input
• The data is not scaled. It generally helps if you scale the data to -1 to 1. I included a simple scalar but there are better ways to do this that also zero-center the data.

from gekko import brain
from gekko import GEKKO
import numpy as np
import matplotlib.pyplot as plt  
from numpy import diff
from scipy.interpolate import CubicSpline
xm = np.array([ 0.0 , 23.0 , 47.0 , 49.0 ,\
                71.5 , 95.0 , 119.0 , 143.0 ])

def spline(cell):    
    m = GEKKO()
    m.options.IMODE=2
    c = [m.FV(value=0) for i in range(4)]
    x = m.Param(value=xm)
    cell = np.array(cell)
    y = m.CV(value=cell)
    y.FSTATUS = 1
    # polynomial model
    m.Equation(y==c[0]+c[1]*x+c[2]*x**2+c[3]*x**3)
    c[0].STATUS=1
    m.solve(disp=False)
    c[1].STATUS=1
    m.solve(disp=False)
    c[2].STATUS=1
    c[3].STATUS=1
    m.solve(disp=False)
    pbr = [c[3].value[0],c[2].value[0],\
           c[1].value[0],c[0].value[0]]
    print(pbr)
    xp = np.linspace(0,144,100)
    plot1 = plt.figure(1)
    if cell[0] == cell_br2[0]:
        plt.plot(xm,cell_br2, 'ko', label ='BR2')
        plt.plot(xp,np.polyval(pbr,xp),'g:',linewidth=2)
    elif cell[0]  == cell_br1[0] :
        plt.plot(xm,cell_br1, 'mo', label ='BR1')
        plt.plot(xp,np.polyval(pbr,xp),'r:',linewidth=2)

    plt.xlabel('time(hr)')
    plt.ylabel('cells')
    plt.legend()
    dx = diff(xp)
    dy1 = diff(np.polyval(pbr,xp))
    deriv1 = dy1/dx
    time =np.linspace(0,144,99)
    plot1 = plt.figure(2)
    if cell[0] == cell_br2[0]:
        plt.plot(time,deriv1,'b:',linewidth=2, label ='BR2')
    elif cell[0] == cell_br1[0]:
        plt.plot(time,deriv1,'m:',linewidth=2, label ='BR1')
    plt.xlabel('time(hr)')
    plt.ylabel('miu(1/h)')
    plt.legend()
    #plt.show()
    return(deriv1)
    
cell_br1 = np.array([0.63*10**6 , 1.10*10**6, 2.06*10**6, 2.08*10**6,\
            3.73*10**6, 3.89*10**6, 3.47*10**6,2.312*10**6])
cell_br2=  np.array([0.58*10**6 , 0.96*10**6, 2.07*10**6, 1.79*10**6,\
            3.57*10**6, 3.34*10**6, 2.62*10**6, 1.75*10**6])

b = brain.Brain(remote=True)
b.input_layer(1)
b.layer(linear=1)
b.layer(tanh=4)
b.layer(linear=1)
b.output_layer(1)

x_s = np.linspace(0,144,99)
xg = np.array([ 0.0 , 23.0 , 47.0 , 49.0 , 71.5 ,\
                95.0 , 119.0 , 144.0 ])
cells_spline = CubicSpline(xm, cell_br1) 
y_cells = cells_spline(x_s)
miu_1 = spline(cell_br1)
miu_2 = spline(cell_br2)
scale = [1.0e6,1.0e4]
x = (y_cells/scale[0]) #, y_glucose) #Inputs (3)
y = (miu_1/scale[1])    #Output (2)

b.learn(x,y) # train
yp = b.think(x) # validate

xp = np.linspace(0,144,99)
yyp = np.array(yp)
miu = np.reshape(yyp, (99,))

plot1 = plt.figure(3)
plt.plot(xp,miu*scale[1],'r-', label = 'Predicted ')
plt.plot(x_s,miu_1,'bo', label = 'Experimental points')
plt.xlabel('Time [hr]')
plt.ylabel('miu [1/h]')
plt.legend()
plt.show()

Recommendations:

• Adjust the number of nodes and types of layers.
• Use a package such as Keras or PyTorch for this type of problem. Here is a tutorial on Keras . Gekko is especially good at problems that need extra things such as constraints, non-standard activation functions, and hybrid machine learning where the model is a combination of physics-based and empirical elements.
• Gekko uses gradient-based solvers that may get stuck at local minima.