DEEP NEURAL NETWORKS FOR MODELING NONLINEAR DYNAMICS
A comparison of Shannon's cross entropy and mean squared error ... Squared Error (MSE) vs Cross Entropy. (CE) ... 5 repetitions of 10-fold cross validation. 1. 1.
D EEP N EURAL N ETWORKS FOR M ODELING N ONLINEAR D YNAMICS
A comparison of Shannon‘s cross entropy and mean squared error N AJEEB K HAN AND I AN S TAVNESS D EPARTMENT OF C OMPUTER S CIENCE , U NIVERSITY OF S ASKATCHEWAN . I NTRODUCTION
R ESULTS
• Evaluation metric: root mean squared error (RMSE) • 5 repetitions of 10-fold cross validation
• Arm reaching movements can be modeled as a mapping [1]
Dataset partitioned into k folds 1 1
Joint Space
test
5 6
Torque trajectory
Hidden features α
Y-axis
Elbow Angle θ2
0
0.2
7 0 −0.2
−2
−1
0
1
2
3
Shoulder Angle θ1
−0.4 −0.8 −0.6 −0.4 −0.2
0
0.2
0.4
0.6
0.8
X-axis
test test test
Input features α
Hidden features β
Hidden features α
Hidden features β
Initial and final states
Reconstructed trajectory
Hidden layer α
Hidden layer β
6
#10
τˆ1
τˆ1
α1
α ˆ1
τ2
τˆ2
τˆ2
τˆ3
α2
α ˆ2
τ3
τˆ3
τˆ3
τ4
τˆ4
α3
α ˆ3
τ4
τˆ4
χ1
τˆ4
τ5
τˆ5
α4
α ˆ4
τ5
τˆ5
χ2
τˆ5
τ6
τˆ6
α5
α ˆ5
τ6
τˆ6
τˆ6
τ7
τˆ7
α6
α ˆ6
τ7
τˆ7
τˆ7
τ8
τˆ8
τ8
τˆ8
τˆ8
τ2
τˆ2
τ3
(a) Learning hidden features α from torque trajectory
(b) Learning hidden features β from hidden features α
100
200
300
α ˆ1
α2
α ˆ2
α3
α ˆ3
α4
α ˆ4
α5
α ˆ5
α6
α ˆ6
400
500
Cross Entropy Mean Squared Error
τ1
τˆ1
τ2
τˆ2
τ3
τˆ3
τ4
τˆ4
τ5
τˆ5
τ6
τˆ6
τ7
τˆ7
τ8
τˆ8
5 4 3 2 1
#10!3
6
200
300
400
(c) Pre-training a deep autoencoder
·10−3 4
τ1
τˆ1
τ2
τˆ2
τ3
τˆ3
τ4
τˆ4
τ5
τˆ5
τ6
τˆ6
τ7
τˆ7
τ8
τˆ8
CE MSE
(d) Deep network predicting torque trajectory from initail and final state
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