Design of modified structure multi-layer perceptron networks based on decision trees for the prediction of flow parameters in 90° open-channel bends

Figures & data

Figure 1. Experimental model geometry.

Figure 2. Experimental model scheme.

Table 1. Different experimental hydraulic properties.

Test no.	Normal depth Y (cm)	Discharge rate Q (l/s)	Velocity (m/s)	Froude number	Reynolds number
1	4.5	5	0.273	0.42	12460
2	6	7.8	0.321	0.42	18460
3	9	13.6	0.374	0.40	28940
4	12	19.1	0.394	0.36	36860
5	15	25.3	0.419	0.34	44705

Figure 3. DT-MLP procedure.

Figure 4. (a) Three-dimensional view of the cross sections, (b) example of the 13 points in a cross section and (c) example of the point coordinates.

Figure 5. DT-MLP velocity classification results.

Figure 6. Comparison of the velocity values predicted by the MLP and DT-MLP models with the experimental values for the training dataset.

Figure 7. DT-MLP water surface.

Figure 8. Scatter plot of experimental values with the MLP and DT-MLP models for the test dataset and with separate results relating to each discharge rate for one run.

Figure 9. Transverse distribution of depth-averaged velocity for the MLP and DT-MLP models compared with the experimental values at different discharge rates.

Table 2. MAE error values of the velocity transverse profile for the MLP and DT-MLP models at different discharge rates and cross sections.

	Q = 5		Q = 7.8		Q = 13.6		Q = 19.1		Q = 25.3
	MLP	DT-MLP	MLP	DT-MLP	MLP	DT-MLP	MLP	DT-MLP	MLP	DT-MLP
40 cm before	1.014	0.56	1.39	1.28	0.88	0.7	1.25	0.99	2.58	0.53
0°	1.89	1.058	1.38	0.95	1.2	1.04	1.52	2.11	1.72	1.24
45°	1.9	1.09	1.36	1.25	1.33	0.97	1.23	1.05	1.86	1.18
90°	2.72	2.03	2.61	2	2.2	1.32	2.84	1.88	3.3	1.51
40 cm after	1.43	0.64	1.49	1.37	1.015	0.99	1.1	1.046	1.24	1.23
Averaged	1.8	1	1.446	1.37	1.325	1	1.6	1.41	2.14	1.14

Figure 10. The transverse profile of water surface predicted by the MLP and DT-MLP models compared with the experimental results at different discharge rates at cross sections of (a) 0°, (b) 45°, (c) 90° and (d) 40 cm after the bend.

Table 3. MAE error values of the water surface transverse profile for the MLP and DT-MLP models at different discharge rates and cross sections.

	Q = 5		Q = 7.8		Q = 13.6		Q = 19.1		Q = 25.3
	MLP	DT-MLP	MLP	DT-MLP	MLP	DT-MLP	MLP	DT-MLP	MLP	DT-MLP
0°	0.035	0.021	0.038	0.023	0.024	0.026	0.044	0.018	0.09	0.06
45°	0.025	0.035	0.041	0.032	0.046	0.046	0.031	0.025	0.057	0.05
90°	0.042	0.016	0.055	0.045	0.07	0.05	0.084	0.03	0.11	0.055
40 after	0.025	0.011	0.024	0.034	0.02	0.014	0.06	0.048	0.04	0.042
Averaged	0.032	0.02	0.04	0.033	0.04	0.035	0.055	0.03	0.074	0.051

Table 4. Evaluation of the MLR, MLP and DT-MLP models in terms of velocity and water depth prediction with the testing dataset.

	Velocity prediction				Water depth prediction
	RMSE	MAE	R^2	δ(%)	RMSE	MAE	R^2	δ(%)
MLP	2.40	1.76	0.91	4.60	0.054	0.040	0.99	0.035
DT-MLP	2.20	1.52	0.92	3.96	0.058	0.042	0.99	0.036
MLR	5.46	4.28	0.50	77.91	0.295	0.235	0.85	0.945

Figure 11. The MAE error bar graph for the MLP and DT-MLP models with the test dataset at different discharge rates for predicting (a) water surface depth and (b) velocity.