A hybrid method based on particle swarm optimization and machine learning algorithm for predicting droplet diameter in a microfluidic T-junction

doi:10.1007/s10483-026-3334-9

Abstract

Abstract:

Droplet-based microfluidics is a transformative technology with applications across diverse scientific and industrial domains. However, predicting the droplet size generated by individual microchannels before experiments or simulations remains a significant challenge. In this study, we focus on a double T-junction microfluidic geometry and employ a hybrid modeling approach that combines machine learning with metaheuristic optimization to address this issue. Specifically, particle swarm optimization (PSO) is used to optimize the hyperparameters of a decision tree (DT) model, and its performance is compared with that of a DT optimized through grid search (GS). The hybrid models are developed to estimate the droplet diameter based on four parameters: the main width, side width, thickness, and flow rate ratio. The dataset of more than 300 cases, generated by a three-dimensional numerical model of the double T-junction, is used for training and testing. Multiple evaluation metrics confirm the predictive accuracy of the models. The results demonstrate that the proposed DT-PSO model achieves higher accuracy, with a coefficient of determination of 0.902 on the test data, while simultaneously reducing prediction time. This methodology holds the potential to minimize design iterations and accelerate the integration of microfluidic technology into the biological sciences.

Key words: droplet-based microfluidics, decision tree (DT), particle swarm optimization (PSO), double T-junction, grid search (GS)

2010 MSC Number:

O361.3

F. ESLAMI, R. KAMALI. A hybrid method based on particle swarm optimization and machine learning algorithm for predicting droplet diameter in a microfluidic T-junction. Applied Mathematics and Mechanics (English Edition), 2026, 47(1): 203-214.

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Figures/Tables 15

Fig. 1

Table 1

The physical parameters of the two fluids"

Parameter	Fluid 1 (water)	Fluid 2 (oil)
Dynamic viscosity/ $(Pa ⋅ s)$	$1.95 × 10 − 3$	$6.7 × 10 − 3$
Density/ $(kg ⋅ m − 3)$	$1 × 103$	$1 × 103$
Interfacial tension/ $(N ⋅ m − 1)$	$5 × 10 − 3$	$5 × 10 − 3$
Contact angle/ $(∘)$	135	135

Table 1

Table 2

Fig. 2

Fig. 3

Fig. 4

Table 3

Table 4

Parameters of PSO"

Parameter	Value
Swarm size	50
Iteration	100
$r 1$	0.4
$r 2$	0.5
$c 1$	2
$c 2$	2
$w$	0.8

Table 4

Table 5

Train and test performance using various metrics"

Model	MAE	MAPE	$R 2$	RMSE
DT-GS (train set)	5.034	7.171	0.835	7.442
DT-GS (test set)	4.984	6.674	0.772	7.863
DT-PSO (train set)	4.029	6.173	0.837	7.561
DT-PSO (test set)	3.012	5.883	0.902	6.923

Table 5

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

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Parameter	Range
Main width/μm	60–120
Side width/μm	60–120
Thickness/μm	30–60
Flow rate ratio	0.25–0.33

Hyperparameter	Minimum	Maximum	Number
Max_depth	3	10	6
Min_samples_split	3	8	6
Min_samples_leaf	2	8	6