Algorithm 1 IT3FBLS modeling algorithm
Input: Training data {X, Y}; Initial model parameters; Regularization parameter; Learning rate; Maximum iteration number
Output: Model prediction output
(1). Initialization	Compute the outputs of the fuzzy subsystems P and enhancement nodes H // ~
	Concatenate P and H to construct matrix Q, and calculate the initial model prediction output //
(2). Calculate information entropy and regularity coefficient	For each column of Q , estimate the PDF using KDE and calculate the corresponding information entropy // ~
	Apply the exponential mapping function to transform the entropy into regularization weights and construct the diagonal matrix //
(3). Update	Update the output layer weights using the closed-form solution derived from the entropy-weighted adaptive regularization approach //
(4). Gradient descent for internal parameters:	Compute the gradient of loss function with respect to , and update these parameters using gradient descent //
(5). Gradient-based secondary optimization	After updating , recalculate the outputs of the enhancement nodes and the corresponding information entropy
	Update the output layer weights again using the updated
	Repeat Steps 2 through 5 until the maximum number of iterations is reached.
(6). Output results	After IT3FBLS training is completed, the model prediction output is calculated as the final output of the model

Algorithm 1 The pseudocode of the proposed IT3FBLS model is as follows:

Datasets	Methods	RMSE			MAE			R2
Datasets	Methods	Mean	Var	Best	Mean	Var	Best	Mean	Var	Best
Housing	BLS	1.8988E+01	1.7532E+01	4.9911E+00	5.0214E+00	1.7552E+00	3.3301E+00	-6.5651E+00	1.2706E+01	7.1119E-01
	FNN	6.1635E+00	9.4208E-01	5.4973E+00	4.0787E+00	6.4398E-01	3.6233E+00	5.5443E-01	1.3463E-01	6.4964E-01
	FBLS	5.9531E+00	1.1426E+00	4.7884E+00	3.9587E+00	3.9816E-01	3.3284E+00	5.7475E-01	1.6724E-01	7.8698E-01
	IT2FNN	4.1984E+00	4.1219E-01	3.5039E+00	3.0080E+00	1.9918E-01	2.6240E+00	7.9378E-01	4.1478E-02	8.5766E-01
	IT2FBLS	6.3649E+00	7.3477E-01	4.9374E+00	4.5117E+00	5.0624E-01	3.5953E+00	5.2438E-01	1.0990E-01	7.1737E-01
	IT3FBLS-1	6.2516E+00	7.3677E-01	5.5300E+00	4.6600E+00	4.8271E-01	4.1938E+00	5.4123E-01	1.1426E-01	6.4546E-01
	IT3FBLS-2	6.1518E+00	7.0482E-01	4.9612E+00	4.1655E+00	5.0065E-01	3.8651E+00	5.8664E-01	1.0812E-01	6.8520E-01
	IT3FBLS	5.8991E+00	3.6990E-01	4.2865E+00	3.8528E+00	2.2710E-01	3.2322E+00	6.0470E-01	4.7700E-02	7.9050E-01
CCS	BLS	3.5018E+01	3.8531E+01	1.0920E+01	9.9500E+00	4.3984E+00	6.3229E+00	-7.8145E+00	2.3569E+01	6.0136E-01
	FNN	1.2610E+01	6.2004E-01	1.1921E+01	8.8290E+00	9.0309E-01	8.0701E+00	4.6742E-01	5.3011E-02	5.2493E-01
	FBLS	9.9873E+00	1.0259E+00	8.1331E+00	7.5818E+00	8.2169E-01	6.0590E+00	6.6320E-01	6.8675E-02	7.4887E-01
	IT2FNN	8.3817E+00	3.6140E-01	7.4174E+00	6.3209E+00	2.7262E-01	5.5798E+00	5.2650E-01	1.9941E-02	7.1607E-01
	IT2FBLS	1.2468E+01	1.3018E+00	9.3524E+00	9.6935E+00	1.0329E+00	7.3395E+00	4.7491E-01	1.0449E-01	7.0760E-01
	IT3FBLS-1	1.3538E+01	2.8278E-01	1.3235E+01	1.1004E+01	1.9383E-01	1.0824E+01	3.8710E-01	2.5980E-02	4.1439E-01
	IT3FBLS-2	1.3860E+01	7.2619E-01	1.3084E+01	1.1275E+01	5.5857E-01	1.0557E+01	3.5637E-01	6.7368E-02	4.2767E-01
	IT3FBLS	2.4234E+00	1.5360E-01	2.1406E+00	1.8176E+00	1.2040E-01	1.6300E+00	7.6473E-01	5.9300E-02	7.3200E-01
Abalone	BLS	8.2271E+00	1.1380E+01	2.6158E+00	2.6303E+00	6.5151E-01	1.9711E+00	-1.1315E+01	4.3493E+01	5.5819E-01
	FNN	2.9319E+00	3.5560E-02	2.9068E+00	2.2705E+00	5.0842E-02	2.2346E+00	4.4492E-01	1.3464E-02	4.5444E-01
	FBLS	2.7202E+00	2.7033E-01	2.3118E+00	1.9126E+00	1.3661E-01	1.7431E+00	5.1777E-01	9.9518E-02	6.5492E-01
	IT2FNN	2.4935E+00	1.7485E-01	2.3083E+00	1.8871E+00	1.2127E-01	1.7381E+00	5.9677E-01	5.8821E-02	6.5598E-01
	IT2FBLS	2.3091E+00	1.3151E-01	2.1305E+00	1.7435E+00	1.1133E-01	1.5805E+00	6.5468E-01	3.9673E-02	7.0694E-01
	IT3FBLS-1	2.6090E+00	1.2095E-01	2.4669E+00	1.9028E+00	1.1005E-01	1.7313E+00	5.5973E-01	4.0807E-02	6.0707E-01
	IT3FBLS-2	2.3936E+00	6.6976E-02	2.3062E+00	1.7696E+00	4.6055E-02	1.7170E+00	6.2984E-01	2.0760E-02	6.5660E-01
	IT3FBLS	2.3975E+00	1.0250E-01	2.2039E+00	1.6459E+00	1.0730E-01	1.4590E+00	5.3750E-01	3.9600E-02	6.0990E-01

Table 3:Performance comparison results of the testing set data on the benchmark dataset

Datasets	Methods	RMSE			MAE			R2
Datasets	Methods	Mean	Var	Best	Mean	Var	Best	Mean	Var	Best
Mackey – Glass	BLS	2.9764E-01	3.9052E-01	3.3935E-02	2.2310E-02	2.7710E-02	3.7310E-03	2.5341E+01	9.2931E-01	9.8257E-01
	FNN	2.3121E-02	6.6905E-03	1.2152E-02	1.8167E-02	4.8734E-03	9.7597E-03	9.9130E-01	4.7095E-03	9.9776E-01
	FBLS	2.4116E-03	1.5833E-04	2.2007E-03	1.9114E-03	1.1725E-04	1.7442E-03	9.9861E-01	1.2254E-05	9.9913E-01
	IT2FNN	4.3339E-02	8.2000E-03	3.0138E-02	3.2803E-02	5.5523E-03	2.2987E-02	9.7060E-01	1.1591E-02	9.8625E-01
	IT2FBLS	2.3594E-02	1.0772E-02	1.1484E-02	1.8609E-02	8.4352E-03	9.9888E-03	9.8991E-01	9.7716E-03	9.9800E-01
	IT3FBLS-1	8.7532E-02	4.6528E-03	7.9216E-02	7.0162E-02	4.2087E-03	6.2543E-02	8.8373E-01	1.2219E-02	9.0502E-01
	IT3FBLS-2	1.4820E-02	7.8595E-04	1.3528E-02	1.1452E-02	6.0698E-04	1.0386E-02	9.9667E-01	3.5497E-04	9.9723E-01
	IT3FBLS	1.3573E-02	3.8224E-04	1.2991E-02	1.0227E-02	2.7178E-04	9.8179E-03	9.9721E-01	1.5743E-04	9.9845E-01
Rossler system	BLS	5.1297E-01	3.2984E-03	4.1499E-01	6.4590E-02	2.0603E-04	3.6554E-04	9.4233E+01	1.7868E-04	9.7453E+00
	FNN	1.4850E-01	2.8354E-01	1.8088E-01	5.9146E-02	8.9973E-02	5.1575E-03	9.9580E-01	1.2641E-02	9.9129E-01
	FBLS	1.7019E-03	3.1604E-03	1.5816E-06	7.4489E-04	2.3595E-03	1.0278E-06	9.9911E-01	1.5678E-04	9.9929E-01
	IT2FNN	1.1028E+00	2.1280E-01	5.1417E-01	7.4501E-01	1.2732E-01	3.7827E-01	9.4394E-01	2.0560E-02	9.8823E-01
	IT2FBLS	5.7999E-01	6.0014E-01	1.6707E-01	3.4652E-01	3.9376E-01	8.3696E-02	9.6979E-01	7.5461E-02	9.9876E-01
	IT3FBLS-1	1.7675E+00	2.2979E-01	1.5419E+00	1.4358E+00	2.5266E-01	1.1590E+00	8.5878E-01	3.7874E-02	8.9415E-01
	IT3FBLS-2	2.0261E-01	2.2748E-02	1.7888E-01	7.7464E-02	2.2375E-03	6.3005E-02	9.9815E-01	4.3197E-04	9.9858E-01
	IT3FBLS	2.0795E-01	5.9879E-02	1.2618E-01	5. 3515E-02	1.0391E-02	4.4466E-02	9.9793E-01	1.2413E-03	9.9929E-01

Table 4:Performance comparison reslts of the testing set in the uncertain function dataset

Datasets	Methods	RMSE			MAE			R2
Datasets	Methods	Mean	Var	Best	Mean	Var	Best	Mean	Var	Best
MSWI	BLS	4.2869E+01	1.9537E+01	1.8329E+01	1.5747E+01	3.7545E+00	1.1383E+01	5.0442E+00	5.3205E+00	7.7202E-02
	FNN	1.8251E+01	1.2451E-01	1.8123E+01	1.4488E+01	1.4933E-01	1.4272E+01	8.4970E-02	1.2505E-02	9.7762E-02
	FBLS	1.3577E+01	5.2206E-01	1.2591E+01	1.0832E+01	4.4640E-01	1.0033E+01	4.9293E-01	3.8997E-02	5.6452E-01
	IT2FNN	1.4292E+01	3.6553E-01	1.3677E+01	1.1703E+01	3.7454E-01	1.0961E+01	4.3853E-01	2.8698E-02	4.8619E-01
	IT2FBLS	1.6738E+01	4.4930E-01	1.6093E+01	1.3856E+01	2.8086E-01	1.3419E+01	2.2992E-01	4.1974E-02	2.8854E-01
	IT3FBLS-1	1.0180E+01	1.6538E-01	9.9034E+00	7.7655E+00	8.6324E-02	7.5849E+00	7.1528E-01	9.2605E-03	7.3059E-01
	IT3FBLS-2	9.9189E+00	1.5445E-01	9.7150E+00	7.5013E+00	1.3604E-01	7.2075E+00	7.2968E-01	8.4652E-03	7.4074E-01
	IT3FBLS	9.1893E+00	1.7770E-01	8.6750E+00	7.2749E+00	1.8865E-01	7.0219E+00	7.4554E-01	2.0913E-02	7.6339E-01
TE process	BLS	5.9468E+01	1.2131E+02	4.7453E+00	9.2506E+00	1.1582E+01	2.5239E+00	-1.1923E+04	4.4483E+04	-1.4329E+01
	FNN	1.0766E+00	6.7992E-02	9.9546E-01	8.5259E-01	5.8838E-02	7.9559E-01	2.0840E-01	1.0168E-01	3.2542E-01
	FBLS	1.1011E+00	4.5375E-02	1.0186E+00	8.7610E-01	2.7352E-02	8.2791E-01	1.7336E-01	6.7758E-02	2.9376E-01
	IT2FNN	1.0602E+00	6.8572E-02	9.7473E-01	8.3604E-01	5.4041E-02	7.6247E-01	2.3175E-01	1.0123E-01	3.5323E-01
	IT2FBLS	1.1221E+00	6.5875E-02	9.8712E-01	9.1175E-01	5.2059E-02	8.1825E-01	1.4005E-01	9.9557E-02	3.3667E-01
	IT3FBLS-1	1.1244E+00	4.1835E-02	1.0725E+00	9.2102E-01	3.6988E-02	8.7056E-01	1.3816E-01	6.4728E-02	2.1698E-01
	IT3FBLS-2	1.0650E+00	4.2266E-02	1.0090E+00	8.7492E-01	3.8107E-02	8.1693E-01	1.8214E-01	6.2344E-02	3.0688E-01
	IT3FBLS	1.1102E+00	8.1063E-02	9.4379E-01	8.8026E-01	5.8782E-02	7.4087E-01	1.5670E-01	1.2226E-01	3.9363E-01

Table 6:Performance comparison results of the testing set in complex industrial process data sets

Datasets	Hyperparameters of IT3FBLS
Datasets	J	K	M	L
Range	[1,100]	[1,50]	[1,1000]	[1,2050]	[0,1]	[0,1]
CCS	66	7	460	140	0.7	0.07
Abalone	8	4	835	176	0.3	0.2
Housing	48	11	271	490	0.45	0.4
Mackey–Glass	8	40	20	800	0.3	0.3
Rossler attractor	6	4	10	800	0.3	0.3
MSWI process	42	6	125	1610	0.22	0.16
TE chemical process	4	40	10	600	0.3	0.2

	Source of uncertainty	Sampling interval (min)	Input variable	Output variable
MSWI process	Fuel composition fluctuations, air supply, combustion dynamics, strong nonlinear combustion, turbulent heat transfer, operating condition fluctuations	1	Primary air volume, secondary air volume, average feeder speed, average drying grate speed, ammonia injection volume	Furnace temperature
TE process	Reaction kinetics, heat exchange, material transport, multi-step chemical reactions, thermodynamics, fluid transport, transport failures and abnormal conditions	3	D feed rate (stream 2), E feed rate (stream 3), A feed rate (stream 1), total feed rate (stream 4), compressor recirculation valve, discharge valve (stream 9), separator tank liquid flow (stream 10), stripper liquid product flow (stream 11), stripper water flow valve, reactor cooling water flow, condenser cooling water flow	Reactor temperature

Tables at a glance

Figures at a glance

Datasets	Details of datasets	No. of Samples			Input variables
Datasets	Details of datasets	Train	Validation	Test	Input variables
1	Concrete compressive strength (CCS)	515	257	258	8
	Abalone	2088	1044	1045	8
	Housing	253	126	127	13
2	Mackey–Glass	400	200	200	4
2	Rossler attractor	500	250	250	3
3	MSWI process	428	214	215	5
3	TE chemical process	250	125	125	11