Enhancing hydrogel predictive modeling: an augmented neural network approach for swelling dynamics in pH-responsive hydrogels

doi:10.1007/s10483-025-3290-9

Abstract

Abstract:

The pH-sensitive hydrogels play a crucial role in applications such as soft robotics, drug delivery, and biomedical sensors, as they require precise control of swelling behaviors and stress distributions. Traditional experimental methods struggle to capture stress distributions due to technical limitations, while numerical approaches are often computationally intensive. This study presents a hybrid framework combining analytical modeling and machine learning (ML) to overcome these challenges. An analytical model is used to simulate transient swelling behaviors and stress distributions, and is confirmed to be viable through the comparison of the obtained simulation results with the existing experimental swelling data. The predictions from this model are used to train neural networks, including a two-step augmented architecture. The initial neural network predicts hydration values, which are then fed into a second network to predict stress distributions, effectively capturing nonlinear interdependencies. This approach achieves mean absolute errors (MAEs) as low as 0.031, with average errors of 1.9% for the radial stress and 2.55% for the hoop stress. This framework significantly enhances the predictive accuracy and reduces the computational complexity, offering actionable insights for optimizing hydrogel-based systems.

Key words: transient swelling, pH-responsive hydrogel, neural network, data-driven model, hydration and stress dynamics

2010 MSC Number:

O313

M. A. FARAJI, M. ASKARI-SEDEH, A. ZOLFAGHARIAN, M. BAGHANI. Enhancing hydrogel predictive modeling: an augmented neural network approach for swelling dynamics in pH-responsive hydrogels. Applied Mathematics and Mechanics (English Edition), 2025, 46(9): 1787-1808.

TrendMD

Figures/Tables 18

Fig. 1

Table 1

Model parameters: chemical properties from experiments and mechanical properties fitted to data"

Property type	Hydrogel domain
Mechanical	$M = 13.27$ MPa $G = 0.227$ MPa
Chemical^[42]	$D H = 9.3 × 10 − 9$ m²/s $D HB = 8.79 × 10 − 10$ m²/s $c fix = 1.8$ mol $c salt = 0.2$ mol $K = 10 − 5$ mol $H 0 = 0.2$

Table 1

Fig. 2

Fig. 3

Fig. 4

Table 2

Evaluation metrics for predicting hydration with ML algorithms using an 80% training and 20% testing split"

ML algorithm	MAE	$R 2$	MAPE
Neural network	0.031	0.988	0.023

Table 2

Fig. 5

Table 3

Evaluation metrics for predicting hydration with ML algorithms in the cross-validation"

ML algorithm	MAE	$R 2$	MAPE
Neural network	0.061	0.953	0.06

Table 3

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Fig. 11

Fig. 12

Fig. 13

Fig. 14

Fig. 15

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