Chapter 3 Evolution Strategies
3.2 Neural Network Structure
3.3.5 Flow Chart
Figure 3.8 the flow chart of the evolution strategies
Chapter 4
Simulation Results
In this chapter, the objective system of the neural network is a first order LTI system, described as
( ) ( ) ( )
t yt uty& + = (4.1) for simplicity. According to the system, the training data are captured with given sampling times when the input function is a step function whose amplitude is equal to 1 and the system initial condition are idled. The fitness function is defined as the negative sum of the errors between the outputs of the first order LTI system and the NN system. The learning procedure indicated in the last chapter was implemented by a Matlab program.
In last chapter, it is mentioned that there are some settings which will affect the performance of the evolution strategies, such as the initial individuals and sampling times.
Here, the influence of these settings will be discussed, and then the learned neural network using the evolution strategies will be implemented as a controller. The influence of the sampling times will be discussed in Section 4.1, the influence of the initial individuals creation will be discussed in Section 4.2, and then Section 4.3 will show the results of the neural network trained as a controller.
4.1 Influence of The Sampling Time
The influence resolves itself into the following two points: one is the influence of the sampling time to learning result; the other is the abilities of two neural network structures to adapt different sampling times. The first point will be discussed in the following paragraph.
In the first order difference equation, it showed that the fitness increases as the sampling time decreases. It is concerned whether the fitness of the two neural network structures increase as the sampling time decrease like the first order difference equation. There remains a second question about whether the sampling time affects the success rate of the learning result or not. Here, the GS and SST are trained under the sampling time 0.01 and 0.001. Fig.
4.1.1 shows the learning result of the GS when the training data are under sampling time 0.01 at 1st, 3rd, and 5th time. Fig 4.1.2 shows the variation of negative fitness, sum of the error, during the learning process of the GS when the training data are under sampling time 0.01 at 1st, 3rd, and 5th time. Similarly, the objective of Fig. 4.2.1 and Fig 4.2.2 is the SST under the sampling time 0.01, the objective of Fig. 4.3.1 and Fig 4.3.2 is the GS under the sampling time 0.001, and the objective of Fig. 4.4.1 and Fig 4.4.2 is the SST under the sampling time 0.001. Table 4.1 presents the learning results whose initial individual are all given randomly.
We define the learning as success learning while the average of the error is smaller than 0.05, and show the result in the Table 4.2.1 and Table 4.2.2. Every case is learned by starting with three different initial random individuals.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
time(sec) response
real system output 1st NN system output (1) 3rd NN system output (2) 5th NN system output (3)
Figure 4.1.1 the learning result of the GS under sampling time 0.01
0 20 40 60 80 100 120
0 10 20 30 40 50 60 70 80 90
generation
error
the error
1st training (1) 3rd training (2) 5th training (3)
Figure 4.1.2 the change of the sum of the error of the GS under sampling time 0.01 (1)
(2) (3)
(2) (1) (3)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
time(sec) response
real system output 1st NN system output (1) 3rd NN system output (2) 5th NN system output (3)
Figure 4.2.1 the learning result of the SST under sampling time 0.01
0 20 40 60 80 100 120 140
0 5 10 15 20 25 30 35 40
generation
error
the error
1st training (1) 3rd training (2) 5th training (3)
Figure 4.2.2 the change of the sum of the error of the SST under sampling time 0.01 (1)
(2) (3)
(1)
(2) (3)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
time(sec) response
real system output 1st NN system output (1) 3rd NN system output (2) 5th NN system output (3)
Figure 4.3.1 the learning result of the GS under sampling time 0.001
0 20 40 60 80 100 120 140
0 100 200 300 400 500 600 700 800 900
generation
error
the error
1st training (1) 3rd training (2) 5th training (3)
Figure 4.3.2 the change of the sum of the error of the GS under sampling time 0.001 (1)
(2)
(3)
(3) (2) (1)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0
0.2 0.4 0.6 0.8 1 1.2 1.4
time(sec) response
real system output 1st NN system output (1) 3rd NN system output (2) 5th NN system output (3)
Figure 4.4.1 the learning result of the SST under sampling time 0.001
0 10 20 30 40 50 60 70 80
0 100 200 300 400 500 600 700 800
generation
error
the error
1st training (1) 3rd training (2) 5th training (3)
Figure 4.4.2 the change of the sum of the error of the SST under sampling time 0.001 (1)
(3) (2)
(1)
(2) (3)
Table 4.1 the results of learning with different sampling times
∆T = 0.01 ∆T = 0.001
generation length learning time generation length learning time
GS SST GS SST GS SST GS SST
1st 78 89 01:34:55 00:08:13 62 47 01:21:54 00:33:07 2nd 36 88 00:50:07 00:09:01 485 27 15:27:51 00:16:57 3rd 102 66 01:50:32 00:09:27 86 76 01:12:17 00:55:54 4th 81 38 01:32:53 00:12:34 94 53 01:43:28 00:39:49 5th 54 133 01:34:51 00:07:21 127 34 04:42:18 00:28:33 average 70 83 01:28:40 00:09:19 171 47 04:53:34 00:24:52
Table 4.2.1 the error and the accurate rate of learning with sampling time 0.01
∆T = 0.01
Sum of error average of error (10-2) Success/Fail
GS SST GS SST GS SST
1st 0.0740 0.1784 0.01480 0.03568 S S
2nd 0.2939 0.1538 0.05878 0.03076 S S
3rd 1.5743 0.2525 0.31486 0.05050 S S
4th 1.8689 0.4242 0.37378 0.08484 S S
5th 0.1782 0.1291 0.03564 0.02582 S S
average 0.7979 0.2276 0.15957 0.04552 Success rate(%) 100 100
Table 4.2.2 the error and the accurate rate of learning with sampling time 0.001
∆T = 0.001
Sum of error average of error (10-2) Success/Fail
GS SST GS SST GS SST
1st 61.3104 85.9001 1.22621 1.71800 S S
2nd 144.4808 118.1299 2.88962 2.36260 S S
3rd 374.9141 4.1978 7.49828 0.08396 F S
4th 131.3523 32.7474 2.62705 0.65495 S S
5th 570.8313 15.3980 11.41663 0.30550 F S
average 256.5779 51.2746 5.13156 1.02500 Success rate(%) 60 100
The Fig 4.1.1 and Fig. 4.2.1 show that the learning results are almost the same with the system response when the sampling time is 0.01. Fig. 4.3.1 and Fig. 4.4.1 show that the learning results have some error from the first order LTI system response when the sampling time is 0.001, although the direction of the trend is the same. It means the fitness of the proposed neural network structures increases as the sampling time decrease. Besides, Table 4.1, Table 4.2.1, and Table 4.2.2 tell that the error has no relationship with generation length, and the learning time is independent with the value of the generation length. The Table 4.1 also shows that the learning time increases as the sampling time decrease in the same neural network structure, and the average learning time of the SST is shorter than the GS. Table 4.2.1 and Table 4.2.2 indicate that the larger sampling time produce smaller fitness. It is similar with the first order difference equation. Besides, compared to the success rate, it is clear that the success rate of the GS decreases as the sampling time decreases.
Since the simulation results show errors are small, i.e. fitness are large, take the best individual with ∆T=0.01 to test the dynamic of the NN system. Here, two significant conditions for a system are variable for validation: the one is the initial condition y(0) of the system, and the other is the input function. Let the initial condition y(0) be 0.5, 2, and -2, and the input function be 2p(t), -2 p(t), and sine function where p(t) is a unit step function. The testing results of the initial conditions are shown in Fig. 4.5.1 to Fig. 4.5.3, and the testing results of the input functions are shown in Fig. 4.6.1 to Fig. 4.6.3.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
time(sec) system response
input
real system output (1) output of GS (2) output of SST (3)
Figure 4.5.1 testing result when initial condition y(0) is 0.5 (1)
(2) (3)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
time(sec) system response
input
real system output (1) output of GS (2) output of SST (3)
Figure 4.5.2 testing result when initial condition y(0) is 2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-2 -1.5 -1 -0.5 0 0.5 1
time(sec) system response
input
real system output (1) output of GS (2) output of SST (3)
Figure 4.5.3 testing result when initial condition y(0) is -2 (1)
(2) (3)
(1) (2)
(3)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time(sec) system response
input
real system output (1) output of GS (2) output of SST (3)
Figure 4.6.1 testing result when input function u = 2p(t)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
time(sec) system response
input
real system output (1) output of GS (2) output of SST (3)
Figure 4.6.2 testing result when input function u = -2p(t) (1)
(2) (3)
(1) (2)
(3)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
time(sec) system response
input
real system output (1) output of GS (2) output of SST (3)
Figure 4.6.3 testing result when input function u = sin(2t)
Fig 4.5.1 to Fig 4.6.3 indicate the neural networks using evolution strategies don't just learn as a specific curve, but the input-output relationship of the objective system. In another words, the learned neural network system behaves similar with the first order LTI system no matter what the initial condition or the input function is. Thus, these results lead to the conclusion that that the neural network using evolution strategies could learn well as a first order LTI system.
Since the influence of the sampling time to the fitness does exist, the abilities of two structures to adapt different sampling times must be recalled here. To verify the abilities, take the best learning result under the sampling time 0.01, to test the system under the smaller sampling time 0.005 whose result is shown as Fig. 4.7, and to test under the much smaller sampling time 0.001 whose result is shown as Fig. 4.8.
(1) (2)
(3)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0
0.2 0.4 0.6 0.8 1 1.2 1.4
time(sec) system response
input
real system output (1) output of GS (2) output of SST (3)
Figure 4.7 the testing result with ∆T=0.005 of the learned neural network with ∆T =0.01
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0 0.2 0.4 0.6 0.8 1 1.2 1.4
time(sec) system response
input
real system output output of GS output of SST
Figure 4.8 the testing result with ∆T=0.001 of the learned neural network with ∆T=0.01 (1)
(3) (2)
The Fig 4.7 tells that the GS fails, but the SST successes. Fig. 4.7 and Fig. 4.8 indicate that the GS can only be used under a fixed sampling time, but the SST can be used under larger range near the sampling time of the training data. Therefore, the SST is trained with different sampling times, 0.01, 0.001, and 0.0001 using evolution strategies, and then the results will show in Fig. 4.9 and Fig 4.10.1 to Fig. 4.10.3. It demonstrates that the SST can adapt larger rage of the sampling time when it is trained under larger range. It concludes that the SST is better than the GS concerning about the sampling time for learning the first order LTI system.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0 0.2 0.4 0.6 0.8 1 1.2 1.4
time(sec) system response
input
real system output (1)
output of SST: sampling time 0.01 (2) output of SST: sampling time 0.001 (3) output of SST: sampling time 0.0001 (4)
Figure 4.9 the learning result of the SST with sampling time 0.01, 0.001, 0.0001 (1)
(2) (3)
(4)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0
0.2 0.4 0.6 0.8 1 1.2 1.4
time(sec) system response
input
real system output
output of SST: sampling time 0.00001 output of SST: sampling time 0.00005
Figure 4.10.1 the testing result under the sampling time smaller than trained sampling time
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
time(sec) system response
input
real system output (1)
output of SST: sampling time 0.0005 (2) output of SST: sampling time 0.005 (3)
Figure 4.10.2 the testing result under the sampling time between trained sampling time (1)
(2) (3)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
time(sec) system response
input
real system output
output of SST: sampling time 0.05 output of SST: sampling time 0.5
Figure 4.10.3 the testing result under the sampling time larger than trained sampling time
4.2 Influence of The Initial Weights Setting
In the last section, it is said that the success rate decreases and the learning time increases as the sampling time decreases. How to let the error reach the global minimum is an important issue for neural network investigators because it is easy to consider the local minimum as the global minimum. Thus, the local minimum may the main reason for failure learning. However, it is worthy noticing that the first order difference equation provides a set of adequate parameters to approach the first order LTI system. In the following, one of the initial individual is given depending on the parameters of the first order difference equation and compare with the random initial individual. The general structure using the initial individual given by the first order difference equation is called GSi. Since the SST has used the parameters of the first order difference equation in the network, SST is not discussed in this
section. The learning objective is the first order LTI system with sampling time 0.001, and one of the initial individual is given by the first order difference equation with sampling time 0.01.
The learning result is shown in Figure 4.11.1 to Fig. 4.11.2, and Table 4.3.
Table 4.3 learning results of GS with sampling time 0.001 GS with ∆T = 0.001
generation
length learning time sum of error average of error
(10-2) success/fail
1st 70 03:02:29 3.9376 0.07875 S
2nd 49 02:40:50 25.6901 0.51380 S
3rd 850 06:34:49 1.1476 0.02295 S
4th 235 14:45:59 22.7393 0.45479 S
5th 117 05:01:20 93.8409 1.87682 S
average 264 06:25:05 29.4711 0.58942 success rate (%) 100
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0
0.2 0.4 0.6 0.8 1 1.2 1.4
time(sec) response
real system output 1st NN system output (1) 3rd NN system output (2) 5th NN system output (3)
Figure 4.11.1 the learning result of the GS under sampling time 0.001
0 100 200 300 400 500 600 700 800 900
0 100 200 300 400 500 600 700 800
generation
error
the error
1st training (1) 3rd training (2) 5th training (3)
Figure 4.11.2 the change of the sum of the error of the SST under sampling time 0.001 (1)
(2) (3)
(1) (2) (3)
Compared with the Table 4.1 to Table 4.2.2, Table 4.3 indicates that the success rate of GSi is larger than GS. It is worthy noticing that the sum of error of 3rd learning is smaller than the first order difference equation, 2.0302. It means the GS could perform better than the first order difference equation under the sampling time 0.001. However, the average of the learning time is larger than the last section. It reminds that the learning time is independent of the success rate. Similar to Section 4.1, two significant conditions for a system are also variable for validation: the one is the initial condition of the system, and the other is the input function. The influences of these two conditions are shown in Fig. 4.12.1 to Fig. 4.13.3 using the 3rd learning result. Thus, as the figures indicate, no matter what the initial condition and the input functions are, the learned neural network behaves corresponding to the first order LTI system.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5
0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
time(sec) system response
input
real system output (1) output of GS (2)
Figure 4.12.1 testing result when initial condition y(0) is 0.5 (1) (2)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
time(sec) system response
input
real system output (1) output of GS (2)
Figure 4.12.2 testing result when initial condition y(0) is 2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-2 -1.5 -1 -0.5 0 0.5 1
time(sec) system response
input
real system output (1) output of GS (2)
Figure 4.12.3 testing result when initial condition y(0) is -2 (1) (2)
(1) (2)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time(sec) system response
input
real system output (1) output of GS (2)
Figure 4.13.1 testing result when input function u = 2p(t)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
time(sec) system response
input
real system output (1) output of GS (2)
Figure 4.13.2 testing result when input function u = -2p(t) (1) (2)
(1) (2)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
time(sec) system response
input
real system output (1) output of GS (2)
Figure 4.13.3 testing result when input function u = sin(2t)
4.3 Implement as a Controller
According to the results of the last two sections, it concludes that GS and SST using the evolution strategies have good performance on learning a first order LTI system. Further, implement it as a first order LTI controller to control the given plant to verify its ability. Here, a first order and a second order LTI system are the objective plant of the learned controller.
Section 4.3.1 use the first order LTI plant to test the learned neural network structures, and Section 4.3.2 use the second order LTI plant to test.
(1) (2)
4.3.1 System with First Order LTI Plant
Figure 4.14 the feedback system
In the beginning, a first order LTI system, described as
( ) ( ) ( )
t yt u ty& + = (4.2) where y is the system output and u is the input of the plant, is given as a plant of a feedback system, which is shown as Fig. 4.14. In the system, the neural network is trained as a first order LTI system and replaces the original controller, which is designed as
( )
t u( )
t e( )
tu& +6.5 =2.5 (4.3) where the e is the error between the desired output and the system output.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
time(sec) system response
input
real system output (1) output of GS (2) output of SST (3)
Figure 4.15 the learning result of a controller: u&
( )
t +6.5u( )
t =2.5e( )
t( ) ( ) ( )
t y t u ty& + =
( )
t u( )
t e( )
tu& +6.5 =2.5
∑
-+ e u y
yd
(1) (2) (3)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
time(sec) system response
input
real system output (1) output of GS (2) output of SST (3)
Figure 4.16 the testing result of the feedback system with a neural network controller
Figure 4.17 the feedback controller with disturbance
The neural networks, SST and GSi, are trained as (4.3) with sampling time 0.001, the input function is a unit step function. The Fig. 4.15 shows the training result of two structures when they use the same input function and initial condition. It shows that the output of neural network controller is very close to the original controller, thus take it to replace the original one. Fig. 4.16 shows the system response with neural network controller when the y(0) is 0.5 and the input function u is sin(2t). It was observed that the controller control the system well even if the initial condition or the input function change.
( ) ( ) ( )
t y t u t y& + =( )
t u( )
t e( )
t u& +6.5 =2.5∑
-+ e u y
yd
+ 0.1sin(100t)
+
∑
(1) (2) (3)
In reality, the feedback system may contain some unexpected disturbances, so we add a sin function with high frequency 100 rad/sec and small amplitude 0.1 as the unexpected disturbance to the feedback system, shown as Fig. 4.17. Fig 4.18 shows the testing result of the feedback system with disturbance. It tells that the neural network controller performs similar to the original one, and can reject the influence of the disturbance.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
time(sec) system response
input
real system output (1) output of GS (2) output of SST (3)
Figure 4.18 the testing result of the feedback system with unexpected disturbance
4.3.2 System with Second Order LTI Plant
Figure 4.19 the feedback system
∑
-+ e u y
yd u&
( ) ( ) ( )
t +u t =et y&&( )
t +12.5y&( )
t +42.5y( )
t =13u( )
t (1) (2) (3)Similar to the last section, a second order LTI plant, described as
( )
t y( )
t y( )
t u( )
ty&& +12.5& +42.5 =13 (4.4) in the feedback system, which is shown as Fig. 4.19. In the feedback system, the neural network is trained as a first order LTI system and replaces the original controller, which is designed as
( ) ( ) ( )
t u t e tu& + = . (4.5)
Since the original controller is designed the same with (4.1), let the best learning result of the above simulations be the neural network controller here.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
time(sec) system response
input
real system output (1) output of GS (2) output of SST (3)
Figure 4.20 the testing result of the feedback system with a neural network controller Fig. 4.20 shows the system response with neural network controller when y(0) is 0.5, and the input function u is sin(2t). It was observed that the controller control the second order system stable. Then, we also add an unexpected disturbance corresponding to the last section
(1) (2) (3)
to the feedback system. The system is shown as Fig. 4.21. Fig 4.22 shows the testing result of the feedback system with disturbance, and it shows that the neural network controller can reject the influence of the disturbance. Fig. 4.23(a) and Fig. 4.23(b) show the testing results of the plant input and plant output respectively while the unexpected disturbance is given as a random function. According to the above two simulations by different plants, it can be said that the neural network can learn well as a first order LTI controller by the evolution strategies.
As long as the original controller can do, the neural network can also do.
Figure 4.21 the feedback controller with disturbance
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
time(sec) system response
input
real system output (1) output of GS (2) output of SST (3)
Figure 4.22 the testing result of the feedback system with unexpected disturbance
∑
-+ e u y
yd u&
( ) ( ) ( )
t +u t =et y&&( )
t +12.5y&( )
t +42.5y( )
t =13u( )
t 0.1sin(100t)∑
(1) (2) (3)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.6
-0.4 -0.2 0 0.2 0.4 0.6 0.8
time(sec) input of plant
real controller GS controller SST controller
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
time(sec) system response
input real system output output of GS output of SST
(a) (b)
Figure 4.23 the testing result of the feedback system with unexpected random disturbance
Chapter 5 Conclusion
In this thesis, we provide two neural network structures, GS and SST, to learn a first order LTI system. These two structures are both simple structures using only one output delay.
The GS can be used under a fixed sampling time, so it should be trained under sampling time small enough to increase its workable range. The SST contains the parameters of the first order difference equation in the network, so the parameters help SST adapt larger range of sampling time around that of the training data. Besides, we provide evolution strategies to help the neural network learn. It increases the searching space and tries to avoid local minima.
For the purpose, the initial individual can be given by a set of known parameters from the first order difference equation. In the evolution strategies, the objective of the learning process is to find the individuals which lead to larger fitness. In the learning process, the change of the individuals of last step is important for creating the individuals of the next step. In the reproduction process, we use two ways to generate children such that the chance of finding a better individual is increased.
Originally, it is expected that the cost of the computation time is low because the neural
Originally, it is expected that the cost of the computation time is low because the neural