Soft computing models for predicting polymer viscosity across wide thermal and ionic-strength conditions

Abstract

Characterizing polymer properties under varying thermal and ionic-strength conditions is crucial across diverse fields, from materials science to environmental engineering. While experimental techniques are effective, they are often time-consuming and costly, underscoring the need for rapid, reliable computational alternatives. This study evaluates the performance of various soft computing techniques, ranging from traditional machine learning methods (MLP, SVR, RF, GBM, LGBM, and XGB) to advanced deep learning approaches (CNNs), for predicting polymer viscosity under diverse temperature and salinity conditions. The models are trained on a curated database of 4,366 experimental measurements spanning ten commercially used polymers and wide ranges of molecular weight, polymer concentration, temperature, salinity, and shear rate. Using hold-out validation, 10-fold cross-validation, and a strict unseen-polymer test, all models achieve high accuracy (test R² ≥ 0.98), with LGBM and CNN providing the best overall performance. Compared with traditional models (MLP and SVR), these methods reduce test-set error metrics by approximately 25–30%, and the CNN attains R² = 0.997 on the unseen-polymer dataset. Permutation-importance analysis identifies polymer concentration as the dominant predictor, followed by shear rate and molecular weight, in agreement with established rheological behavior. The results demonstrate that ensemble and deep learning models can serve as fast, reliable surrogates for polymer viscosity across high-temperature and high-salinity conditions, reducing experimental effort and supporting data-driven optimization of polymer selection and operating conditions.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-42546-4.

Introduction

The polymer industry has grown tremendously over the past century, with the development of synthetic polymers¹. These materials now underpin applications across materials science and polymer processing, chemical engineering, biotechnology, pharmaceuticals, food science, environmental engineering, and petroleum engineering². The technique of polymer flooding is a well-researched and commonly used method for enhanced oil recovery (EOR) that involves the recovery of residual or unswept oil from oil reservoirs³. Practically, polymers improve the microscopic sweep and displacement efficiency of the aqueous phase by increasing its viscosity and mobility ratio, subsequently⁴. Accurate characterization of apparent viscosity is therefore crucial, particularly in porous media where viscosity typically decreases with increasing flow velocity but can rise rapidly beyond a critical velocity due to shear-thickening. A commonly used polymer in this context is partially hydrolyzed polyacrylamide (HPAM) because of its high water solubility and cost-effectiveness^4,5. HPAM is a water-soluble polyelectrolyte bearing negative charges along its chain⁶, available in a range of molecular weights and degrees of hydrolysis⁷.

Chemical-enhanced oil recovery and polymer flooding are integral components of EOR strategies. Polymer flooding, in particular, has emerged as a widely adopted method for increasing oil recovery rates⁸. The fundamental principle underlying polymer flooding involves the introduction of water-soluble polymers into the reservoir, effectively increasing the viscosity of the injected water⁴. This increase in viscosity improves mobility control and reduces water’s relative permeability compared to oil, thereby enhancing the displacement of hydrocarbons⁹. The types of polymers used in polymer flooding can vary significantly, including both synthetic and biopolymer categories. HPAM and xanthan are the most commonly utilized polymers in polymer flooding operations¹⁰. However, the rheological behavior of polymer solutions is influenced by various factors, including temperature (T), shear rate (SR), and salinity, which are crucial to understanding their performance in the context of EOR^11–13.

Accurate measurement of polymer viscosity is foundational for characterizing such complex fluids. Conventional steady-shear measurements provide essential data but cannot fully capture fluid–structure interactions^14,15. Dynamic oscillatory tests complement these by probing viscoelastic response under sinusoidal loading and thus offering insight into both viscosity and microstructural behavior¹⁶. However, laboratory measurements are time-consuming and resource-intensive. Preparing homogeneous polymer solutions can require several days per sample. Moreover, viscosity depends on many interacting parameters—molecular weight, degree of hydrolysis, polymer concentration, cation content, shear rate, and temperature—leading to strongly nonlinear relationships^17–19. While experiments remain the gold standard, exhaustive testing becomes impractical as applications diversify and the design space expands^14,20. This complexity also hampers the derivation of general theoretical correlations that simultaneously account for shear rate, ionic strength/hardness, concentration, salinity, and temperature²⁰.

Data-driven modeling has therefore emerged as a promising complement to experimentation for predicting polymer viscosity under varying conditions^21–29. Methods spanning multilayer perceptron (MLP), gradient boosting (GBM, XGB, LGBM), voting regressors (VR), random forests (RF), support vector regression (SVR), and convolutional neural networks (CNN) have proved effective in modeling complex, multivariate systems across scientific domains^30–32. These soft-computing and machine-learning approaches accommodate nonlinear, interdependent relationships among inputs—polymer molecular weight, concentration, temperature, shear rate, salinity/ionic strength, and degree of hydrolysis—thereby improving prediction reliability and efficiency^33–38. In this work, we employ MLP, SVR, RF, GBM, LGBM, XGB, VR, and CNN to predict polymer viscosity under high-temperature and high-salinity (ionic-strength) conditions relevant not only to subsurface operations but also to environmental and process applications.

Accurate viscosity prediction is essential for the design and optimization of polymer-enabled processes. Reliable models support decisions on polymer selection, target concentration, and operating windows (e.g., temperature and shear), improving mobility control and process stability while reducing material use and cost. Beyond subsurface engineering, such predictive capability aids dosage optimization in flocculation and sludge conditioning, consistency control in food and pharma formulations, and flow assurance in microfluidic and additive-manufacturing platforms^39,40.

Prior studies often focused on limited datasets, narrow operating ranges, or single algorithm families, which constrain generalization across polymers and conditions. Unlike those efforts, the present study integrates ensemble and deep-learning models over a comprehensive database of 4,366 experimental measurements spanning ten polymers and wide ranges of temperature, salinity/ionic strength, concentration, and shear. We benchmark traditional, ensemble, and deep architectures under consistent protocols (hold-out, 10-fold cross-validation, and an unseen-polymer test) to provide a generalizable and interpretable framework suitable for cross-sector use.

Therefore, the overarching goal of this work is to develop a comprehensive approach to predict polymer viscosity across a broad range of operating conditions using the methods described above. Earlier investigations (Table 1) employed limited input data and modeling strategies. In contrast, we combine multiple learning paradigms and leverage an expansive database curated from experiments and literature, enabling a more inclusive analysis grounded in prior work. This multi-faceted methodology is intended to strengthen predictive fidelity, broaden applicability beyond oil-specific scenarios, and support environmentally relevant and sustainable polymer applications in modern process engineering. This study presents an innovative approach to understanding the behavior of polymer solutions in EOR applications, focusing on a broad range of operating conditions, including high-temperature and high-salinity environments. The research methodology encompasses comprehensive data collection, model development, and evaluation, offering several key contributions:

Table 1.

Summary of data-driven studies on predicting polymer’s viscosity across diverse conditions.

Reference	Target	Inputs	Databank	Algorithm
Duchesne et al. (2010)41	Predict slag viscosity	Temperature and components	Vargas 2001 and Persson 2007	ANN
Kang et al. (2016)33	Estimate the viscosity of pH-sensitive polymers	Shear rate, Cpolymer, CNaCl, CCa2+, and temperature	Lee et al. 2009	ANN
Alqahtani (2019)14	Polymer rheology prediction	Polymer concentration, brine composition, hardness, and temperature	UT’s Center, laboratory experiment, and Lee et al. 2009	RF and ANN
Shaik et al. (2019)34	Predict polymer viscosity	Temperature, shear rate, and salinity	Generated experimental data	XGB
Pérez et al. (2022)18	Predict models for the coefficients of the Carreau–Yasuda equation	Carreau-Yasuda parameters	Lee et al. 2009; Stavland et al. 2010; Moradi 2011; Jouenne 2020, and values obtained by the Brookfield viscometer	Genetic programming (GP)
Qiu et al. (2022)42	Predict epoxy resin viscosity	Temperature, epoxy resin type, diluent type, Molar ratios	Public science databases	Decision Trees (DT), Support Vector Machines (SVM), RF, MLP, and TensorFlow Sequential (TFS)
Abdullah et al. (2023)20	Predict the Carreau parameters	Polymer concentration, brine composition, and temperature	Lee et al. 2009; Rashidi et al. 2009; Wu 2016; Salih et al. 2016; Alqahtani 2019; Jouenne 2020; Al-Hamairi and AlAmeri 2020	Linear regression, ridge regression, polynomial regression, SVR, DT, RF, and ANN
Kikhavani et al. (2023)43	Predict polymer viscosity	Temperature, shear rate, and inorganic/organic additives	Measured experimental data	Radial basis functions (RBF), Gaussian process regression (GPR), and multilayer perceptron (MLP)
Shakeel et al. (2023)28	Predict polymer viscosity	Temperature, shear rate, and Polymer concentration	Measured experimental data	ANN, multilinear regression (MLR), SVM, regression DT
Wang et al. (2024)44	Predict various polymer properties	Polymer SMILES sequences, 3D structural information	Public polymer databases	Polymer Graph Convolutional Neural Network (PGCNN)
Kazemi-Khasragh et al. (2024)45	Predict multiple properties of linear polymers	Polymer structural features	Small datasets (13 to 18 samples)	Transfer learning with neural networks
Wang et al. (2024)46	Predict the viscosities of polymers in different states	Molecular descriptors, PC-SAFT parameters, MACCS fingerprinting	Measured experimental data	XGB, CNN, and MLP
Kondratieva and Chepurnenko (2024)47	Predict rheological parameters of polymers	Stress relaxation curves	Synthetic data from theoretical relaxation curves	K-nearest neighbor (KNN), support vector machine (SVM)
Jain et al. (2025)48	Predict polymer melt viscosity	Polymer melt characteristics	Simulated and experimental data	Physics-enforced neural network (PENN)
This work	Predict polymer viscosity	Mw, Cp, salinity, T, SR (10 polymers)	4,366 experimental points, high-T/high-salinity	MLP, SVR, RF, GBM, LGBM, XGB, VR, CNN; unseen-polymer validation

1. Extensive Rheological Database: We compile one of the most extensive rheological databases for polymer solutions, relevant to high-temperature, high-salinity operations, comprising 4,366 viscosity measurements across ten commercially used polymers. These data cover a wide range of polymer concentrations, temperatures, salinity, and shear rates, including extreme conditions up to 17,200 ppm salinity, 90 °C, and 250,000 ppm. Importantly, this study incorporates recently common and experienced polymers that have not been previously studied, expanding the understanding of polymer behavior in challenging EOR conditions.

2. Benchmarking Machine Learning Models: The study systematically evaluates eight implicit intelligence algorithms—ranging from machine learning to deep learning—using standardized protocols such as hold-out validation, 10-fold cross-validation, and testing on unseen polymers. This rigorous process enables a clear comparison of model accuracy, generalization ability, and computational efficiency, highlighting the strengths and limitations of each approach.

3. Model Development and Insights: Each algorithm is carefully explained, along with its technical benefits and efficiency. The study finds that the LightGBM (LGBM) algorithm is a fast and highly accurate surrogate for large-scale viscosity prediction, while Convolutional Neural Networks (CNN) provide the best generalization to completely unseen polymer chemistries. The results are further analyzed using statistical criteria, and the models are compared across all input parameters, offering valuable insights into the underlying rheological behavior of polymer solutions.

4. Identification of Key Factors: Through permutation-based variable importance and stratified error analyses, we quantitatively confirm and contextualize the dominant roles of polymer concentration, shear rate, and molecular weight in viscosity control. This analysis underscores the importance of these factors in optimizing EOR processes.

Ultimately, by utilizing soft-computing approaches in rheology, this research provides critical insights into the behavior of polymer solutions across a broad spectrum of temperature and salinity conditions, contributing to the optimization of EOR processes and distinguishing it from previous single-algorithm or narrowly scoped investigations.

For a comprehensive understanding of our modeling procedures and the validation protocols applied to the models we developed, please refer to the illustrative representation in Fig. 1.

Fig. 1.

The framework presents the essential steps for predicting polymer viscosity using soft-computing and machine-learning methods.

Data description

Recent research has consistently demonstrated that machine learning algorithms have superior predictive capabilities when applied to the intricate characterization of complex subsurface processes ^{30,31,49–52}. The reliability and robustness of predictions generated by these models are notably contingent upon the scope and quality of the data utilized to train them. Hence, in pursuit of scientific rigor, we have compiled one of the most comprehensive experimental datasets currently accessible to train various models to forecast polymer viscosity in porous media. This extensive dataset meticulously captures the variations in viscosity concerning independent variables such as molecular weight, salinity, hardness, temperature, shear rate, and polymer concentration, curated from relevant literature sources ^20,23,24,27. A detailed summary of the dataset used in this study is presented in Table 2.

Table 2.

A brief description of the database utilized in this study.

#	Polymer	Mw (MDa)	Polymer Conc. (ppm)	Salinity (g/L)	Temperature (oC)	Shear rate (1/s)	Hydrolysis degree	Description	Supplier
1	FP 3130	2	300-10000	29.2	25	0.1–94.5	20–30%	HPAM	SNF Floerger
2	FP 3230	5	1500–5000	44	25–55	0.2–100	20–30%	HPAM	SNF Floerger
3	FP 3330	8	100–5000	28.4–44	25–75	0.01–300	25–30%	HPAM	SNF Floerger
4	FP 3430	12	500-10000	13	25	0.1–100	25–30%	HPAM	SNF Floerger
5	FP 3530	16	500–5000	28.4–44	25–75	0.001-300	25–30%	HPAM	SNF Floerger
6	FP 3630	20	1-17200	0.4–100	25–75	0.0003-300	25–30%	HPAM	SNF Floerger
7	5110	6	500–3000	25	65	0.1–200	20–30%	HPAM	SNF Floerger
8	AN-125	8	1000–10,000	6-100	25	0.01–200	20–30%	Poly (AM-co-AMPS)	SNF Floerger
9	Sav 10	10	1000–10,000	6-250	25–90	0.01–200	20–30%	HPAM	SNF Floerger
10	ZLPAM 31,175	20	100-10000	35	25	0.001-135	20–30%	ZLPAM	ZL Chemicals

The dataset contains 4366 experimental data points measured in different conditions. The scatter matrix in Fig. 2 further elucidates the distributions of the individual input and target variables and their interdependencies. This matrix enhances data clarity by combining histogram plots, scatter plots, and kernel density estimation (KDE) plots along the central diagonal, the upper portion, and the lower portion, respectively. Although the degree of hydrolysis is known to strongly influence HPAM rheology, in the present curated dataset, all commercial polymers fall within a relatively narrow window (20–30% hydrolysis; Table 2). As a result, this variable does not provide sufficient variation to be treated as an independent predictor. For this reason, the degree of hydrolysis is reported in Table 2 for completeness, but was not included as a separate input feature in the statistical analyses (Fig. 2) or in the machine-learning models. Consequently, the present study focuses on the effects of molecular weight, polymer concentration, temperature, salinity, and shear rate, while acknowledging hydrolysis degree as an essential design variable beyond the scope of this dataset.

Fig. 2.

A statistical representation of the dataset employed for model development is presented. The matrix under consideration offers a comprehensive view of the data, focusing on the main diagonal to reveal the distribution patterns of key predictors, encompassing molecular weight (M_w), salinity, temperature (T), polymer concentration (C_p), shear rate (SR), and the target value, which is the polymer viscosity denoted as µ_p. To explore the interrelationships between these variables, the matrix employs non-diagonal sections featuring Kernel Density Estimation (KDE) plots in the lower part (depicted in red) and scatter plots in the upper part (highlighted in green).

Model development

We applied machine-learning methods to model polymer viscosity by using two distinct methodologies: Ensemble Learning and Deep Learning. These approaches were meticulously applied to develop intelligent models that establish a robust relationship between the viscosity of polymers in porous media and a range of independent variables, including molecular weight (M_w), salinity, temperature (T), polymer concentration (C_p), and shear rate (SR). For the deep learning facet of our study, we used a convolutional neural network (CNN) ⁵³ as a well-known smart technique. In parallel, the Ensemble Learning dimension employed Random Forest (RF) ^36,54, Gradient Boosting Machine (GBM) ³⁶, Light Gradient Boosting Machine (LightGBM) ⁵⁵, and Extreme Gradient Boosting (XGBoost) ^56,57, along with the inclusion of Multilayer Perceptron (MLP) ^58,59 and Support Vector Regression (SVR) ⁶⁰ as a traditional machine learning model, to enhance the reliability of our predictions, we further integrated the Voting Regressor (VR) algorithm, which combines predictions from multiple ensemble models. A detailed description of the machine learning algorithms assessed in this study is presented in supplementary material.

The utilized models were chosen based on their strengths in handling nonlinear relationships, high-dimensional data, and complex interactions among polymer properties (Table 3 presents a short comparison of assessed models).

Table 3.

Strengths and weaknesses of various AI models utilized in this study.

Model	Strengths	Weaknesses
MLP (Neural Network)	Captures nonlinear relationships, suitable for moderate datasets	May overfit if not tuned properly
SVR (Support Vector Regression)	Effective for high-dimensional data, it prevents overfitting	Computationally expensive for large datasets
RF (Random Forest)	Robust to noise, interpretable, and handles missing data well	Can be slow with extensive datasets
GBM (Gradient Boosting Machine)	High accuracy, reduces model bias over iterations	Computationally intensive, prone to overfitting
LGBM (LightGBM)	Faster than GBM, efficient with large datasets	Requires careful parameter tuning
XGB (Extreme Gradient Boosting)	Very accurate, suitable for structured data	Needs extensive tuning, can overfit
VR (Voting Regressor)	Combines multiple models for improved stability	May not outperform the best individual model
CNN (Convolutional Neural Network)	Learns deep patterns, excellent for unseen data	Requires large datasets and computational power

MLP and SVR are efficient and straightforward but may underperform on highly nonlinear datasets. Ensemble learning models, such as RF, GBM, LGBM, and XGB, improve predictive performance by combining multiple weak learners, making them more resilient to overfitting and capable of capturing intricate parameter interactions. These models have demonstrated strong results in previous studies, particularly in handling structured data with diverse input variables. Among these, LGBM was the most computationally efficient and highly accurate, making it an attractive choice for large-scale viscosity prediction. The VR was also included to explore whether combining multiple models could enhance prediction stability, but its performance did not significantly exceed that of the most substantial individual models. Finally, the deep learning model, a CNN, was selected for its ability to identify hidden patterns in high-dimensional data and to generalize well, particularly in capturing complex dependencies among polymer molecular structure, shear rate, temperature, and viscosity.

Data preprocessing

The rheological dataset was assembled from four primary sources, each comprising different measurements. All salinity values initially reported in ppm were converted to g/L (1 g/L ≈ 1000 ppm) for consistency, and viscosity values were converted to cP. Data entries with missing or ambiguous temperature, shear rate, or polymer concentration were discarded, and obvious typographical errors were corrected by cross-checking with the original tables/figures. We screened for non-physical outliers (e.g., negative viscosities, extreme values inconsistent with the original publications) and removed many points. The final curated dataset contains 4,366 measurements.

The databank did not initially follow a normal distribution (see the histograms in the diagonal plots of Fig. 2). To address this, we applied a log transformation to mitigate the data’s skewness and approximate it as closely as possible to a bell-curve distribution. The log transform normalization reduces the influence of the extreme value, stabilizes the variance, and makes the data distribution suitable for training the machine learning models (refer to the histograms in the diagonal plots of Fig. S1 for the data normalized using the log transform technique). Furthermore, it should be noted that using a log transform for normalization facilitates denormalizing the predicted viscosities, as the log transform is not dependent on extreme values, unlike other normalization methods, such as the min-max scaler and standard scaler.

For the traditional machine learning models, the input data were used in their tabular numerical format without any spatial reshaping. However, to adapt the tabular rheological dataset for CNN modeling, the input data were reshaped into a 2D format suitable for one-dimensional convolutions. For this purpose, the features were transformed into tensors of shape (N, n, 1), where N is the number of samples and n is the number of features. This structure enables convolutional layers to learn local feature interactions and hierarchical patterns that traditional fully connected architectures may not capture effectively.

Model training

Following data preprocessing, the dataset was partitioned to facilitate model training. The Hold-out technique achieved this objective by randomly designating 80% of the dataset as the training set. In comparison, the remaining 20% was assigned to the test set to facilitate validation. Following this, the input parameters encompassing molecular weight (mDA), salinity (g/L), temperature (°C), polymer concentration (ppm), and shear rate (1/s) alongside the target value (µ_p, cP) were directly fed into the deep and ensemble learning algorithms for the development of viscosity models via supervised learning.

After a thorough series of examinations, the parameters for refining the intelligent algorithms were fine-tuned. In this study, tuning was performed by manually adjusting the most influential hyperparameters (e.g., number of layers and kernel sizes for the CNN models, and number of estimators and maximum depth for the ensemble learning techniques). The goal was to achieve the highest possible performance while keeping computational costs low. The objective was to strike an optimal balance between addressing high variance (overfitting) and mitigating bias (underfitting). The configuration of tuning parameters for the traditional, learning, and ensemble learning methodologies, and the architectural specifications for the deep learning models, are comprehensively detailed in Tables 4, 5 and 6, respectively.

Table 4.

The tuning parameters of the traditional learning models, MLP and SVR, developed in this study.

MLP	Parameter	Hidden layer size	Activation function	Solver	Iterations	Learning rate
	Value	(20, 15, 10)	Relu	lbfgs	3000	Adaptive
SVR	Parameter	Kernal	Kernel coefficient	Stopping criteria	Iteration	Shrinking heuristic
	Value	RBF	scale	1 × 10− 3	Hard limit	True

Table 5.

The tuning parameters of the ensemble learning models (RF, GBM, LGBM, XGB, and VR) developed in the current study.

	RF	GBM	XGB	LGBM
Number of Estimators	1500	1000	1000	1000
Maximum Depth	20	5	5	5
Learning Rate	-	0.04	0.3	0.35
Eta	-	-	0.3	-
Alpha	-	-	0.15	-
Lambda	-	-	1	-
Boosting Technique	-	-	GBDT	DART

Table 6.

The architecture of the convolutional neural network (CNN) developed in this study.

	Network					Compile		Fit
Layers	Output shape	Activation function	Filters	Kernel size	Padding	Optimizer	Loss function	Batch size	Epoch
Conv1D	(5, 32)	ReLU	32	3	Same	Adam	MSE	50	500
MaxPooling1D	(3, 32)	-		-	-
Conv1D	(3, 64)	ReLU	64	3	Same
MaxPooling1D	(2, 64)	-		-ePara>	-
Conv1D	(2, 128)	ReLU		3	Same
MaxPooling1D	(1, 128)	-	128	-	-
Flatten	128	-		-	-
Dense	128	ReLU		-	-
Dense	1	-		-	-

Model validation

The precision and applicability of the created models were scrutinized using a range of statistical methodologies and visualization tools. The following subsections outline the steps for the model assessment.

Hold-out validation

The databank was shuffled and randomly split into two subsets: the training set (80%) and the test set (20%). It is important to note that the model’s performance can vary depending on the chosen train/test split. Different splits may yield different outcomes, potentially leading to inaccurate overestimations or underestimations during model evaluation.

10-fold cross-validation

We employed a 10-fold cross-validation approach to assess the models’ performance and generalizability further. This technique involves dividing the dataset into ten equal parts or folds. In the iterative process, one of the ten folds is employed as the validation set, while the remaining nine folds serve as the training set. This tenfold repetition ensures that each fold functions as the validation set precisely once. Accordingly, we obtained a more robust evaluation of the models’ performance and confirmed their ability to generalize well to unseen data.

Validation of completely unseen data

We further assessed the generalizability of our models by predicting viscosities for an unseen dataset, specifically the FP 3230 polymer. This dataset was excluded from the training and testing phases to ensure novelty. For further details about this unseen dataset, please refer to Table 2.

Permutation importance assessment

During the final stage of our analysis, we used the Permutation Importance technique to assess the importance of individual predictive features within the models developed in this investigation. Permutation importance is a technique for evaluating feature importance in a predictive model. It entails the random rearrangement of values associated with a specific feature, followed by quantifying the resultant decline in the model’s performance. The greater the decrease, the more influential the feature is in shaping the model’s predictions.

Results and discussion

As mentioned, multiple traditional, ensemble, and deep learning models were trained to predict the viscosity of polymers under high-temperature, high-salinity conditions. These models were designed to predict viscosity as a function of various input parameters, including molecular weight, salinity, temperature, shear rate, and polymer concentration. The training data for these models were derived from an extensive dataset comprising experimental viscosity measurements for 10 polymers across a wide range of conditions, totaling 4366 data points.

To comprehensively evaluate the efficacy of these models, diverse statistical metrics, such as R-squared, Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and Average Absolute Relative Error (AARE), were computed for the training (80%), testing (20%), and unseen data subsets. The results of these evaluations are presented in Table 7 and further illustrated in Fig. 3. As shown Fig. 3, the developed models achieve high performance, with overall R-squared values exceeding 0.98 and RMSE values below 0.0006 cp. in both the training and testing phases. Table 7 summarizes the performance of all models, including R², RMSE, AARE, and training, testing, 10-fold cross-validation, and unseen-polymer (FP 3230) execution times. Compared with traditional ML models (MLP and SVR), the best ensemble and deep learning models (particularly LGBM and CNN) reduce test-set AARE by approximately 25–30% while maintaining R² ≥ 0.99. For the unseen polymer, the CNN achieves R² = 0.997 and the lowest AARE, highlighting its superior generalization.

Table 7.

The accuracy of the models was developed using the machine learning techniques employed.

Model	R-squared			MSE			RMSE			AARE
	Train	Test	Unseen	Train	Test	Unseen	Train	Test	Unseen	Train	Test	Unseen
MLP	0.9932	0.9927	0.9847	0.0098	0.0106	0.0057	9.6 × 10− 5	1.1 × 10− 4	0.0000	0.0662	0.0705	0.0555
SVR	0.9840	0.9865	0.8903	0.0236	0.0207	0.0413	5.6 × 10− 4	4.3 × 10− 4	0.0017	0.1016	0.0994	0.1460
RF	0.9997	0.9979	0.9849	0.0004	0.0030	0.0160	1.6 × 10− 7	9.0 × 10− 6	0.0003	0.0116	0.0320	0.1107
GBM	0.9997	0.9991	0.9809	0.0004	0.0013	0.0265	1.6 × 10− 7	1.7 × 10− 6	0.0007	0.0145	0.0242	0.1366
LGBM	0.9997	0.9993	0.9222	0.0004	0.0010	0.0417	1.6 × 10− 7	1.0 × 10− 6	0.0017	0.0125	0.0204	0.1656
XGB	0.9995	0.9987	0.9071	0.0007	0.0019	0.0595	4.9 × 10− 7	3.6 × 10− 6	0.0035	0.0194	0.0304	0.2113
VR	0.9995	0.9989	0.9778	0.0008	0.0016	0.0228	6.4 × 10− 7	2.6 × 10− 6	0.0005	0.0186	0.0266	0.1285
CNN	0.9960	0.9951	0.9970	0.0064	0.0079	0.0010	4.1 × 10− 5	6.2 × 10− 5	0.0000	0.0519	0.0581	0.0253

Fig. 3.

The model accuracy evaluation in this study is presented along the x-axis, employing two statistical metrics: (a) R-squared and (b) RMSE. The depiction uses blue and red to represent the performance scores of all models during training and testing, respectively. The results for the unseen data are shown in yellow.

Figure 4 (a) to (h), the cross-plot representations provide additional insight into the quality of the predictions, displays the deviation between predicted viscosity values and measured values for all models. These plots show that most data points cluster closely around the unit-slope line, indicating a low prediction error. However, analysis of the relative residuals (Fig. S2) shows that the ensemble learning models, particularly LGBM, exhibit minimal residuals across the viscosity range. In contrast, the MLP and SVR models display larger error spreads. This observation is further supported by the error distribution histogram (Fig. S3), which shows that most relative errors in the LGBM model are near zero, whereas the MLP and SVR models exhibit a higher frequency of large errors.

Fig. 4.

The evaluation of the predictive performance of the created models, which encompass (a) MLP, (b) SVR, (c) RF, (d) GBM, (e) LGBM, (f) XGB, (g) VR, and (h) CNN models for the estimation of polymer viscosity. The figure includes a solid line that corresponds to the 1:1 line.

The comparison of model performance indicates a marginal difference among the algorithms. The close agreement between training and test metrics suggests that the model is resistant to overfitting. Among them, the LGBM outperforms the alternatives, achieving an R-squared of 0.99, while the SVR shows higher errors, with an R-squared of 0.986 on the test set. Assessing the generality of the models on unseen data (the FP 3230 polymer, which was excluded entirely from the training and test procedures) reveals that SVR, XGB, and LGBM have R-squares of 0.890, 0.907, and 0.922, respectively. The CNN shows the greatest generality on unseen data, with an R-squared of 0.997. It should be mentioned that the VR algorithm that attempts to provide a more appropriate prediction by voting multiple individual models (MLP, RF, GBM, LGBM, and XGB) does not improve the overall performance, as the RF and GBM outperform the VR with R-squares of 0.984, 0.980, and 0.978, respectively.

We also conducted a comprehensive performance analysis of the models across various variables, including polymer type, concentration, salinity, and temperature, with the results portrayed in Fig. 5 (a) to (d), respectively. It should be noted that the previously reported values for the statistical parameters were direct outputs of the models, without denormalization (a log transformation was used as the normalization technique). In this analysis, we have denormalized the outputs to assess the models’ differences better. As depicted in Fig. 5 (a), there are no statistically significant differences between the performance of models for the prediction of the viscosity of SAV 10, AN-125, FP 3130, FP 3330, FP 3430, FP 3530, FP 3630, 5110, and ZLPAM 31,175 polymers. This trend is consistently observed in Fig. 5 (b) to (d), demonstrating that the robust performance of the models is not limited to specific ranges of polymer concentration, salinity, or temperature. It is noteworthy, however, that the SVR and neural network-based algorithms (MLP and CNN) exhibit higher fluctuations in accuracy across different conditions. For instance, Fig. 5 (b) demonstrates the poor performance of the SVR and MLP in polymer concentrations of 2001–4000 ppm, or the most errors of CNN occur at the temperature of 65 °C (see Fig. 5 (d)). A more granular evaluation of model performance across different polymer types, concentrations, salinity levels, and temperatures is provided in Table S1 (Supplementary material), complementing the visual summary in Fig. 5.

Fig. 5.

The performance assessment of all formulated algorithms was conducted under diverse input variables, including (a) polymer type, (b) polymer concentration (Cp), (c) salinity, and (d) temperature. The selection of input variable ranges for analysis ensured comparable data points. R-squared was used to quantify the extent to which the independent variables in the model explain the variance in the dependent variable. The reader is pointed to Table S1 in the supplementary material for more details regarding the results.

Although Table 7; Figs. 3, 4 and 5 effectively convey statistical information demonstrating the reasonable performance of the developed models in predicting polymer viscosity across varying conditions, they do not provide insights into the models’ robustness to data leakage, sensitivity to train-test splits, or performance on unseen data. Moreover, beyond the precision of these models, the computational resources demanded by each model constitute another critical factor influencing the selection of an appropriate predictor model. To compensate for this, we have carried out 10-fold cross-validation, in which the dataset is partitioned into 10 equal-sized subsets: the model is trained on 9 subsets and tested on the remaining 1, and this process is repeated 10 times. The calculated R-squares and execution times for each iteration are represented in Fig. 6.

Fig. 6.

A boxplot is employed to depict the (a) R-squared (R²) visually and (b) computational fit time values that are linked to the 10-fold cross-validation process conducted across multiple algorithms within the scope of this study.

As shown in Fig. 6 (a), the median R² values (represented by the solid line inside the boxes) exceed 0.97 for both training and testing stages across all models, except for SVR, which exhibited a median test R² of 0.89. The close agreement between training and testing results across all folds further indicates the models’ robustness against overfitting. Moreover, the consistently small interquartile ranges (IQRs) across all models reflect strong consensus on predictive accuracy across the ten learning iterations, further demonstrating the models’ resilience to potential data leakage during the train-test split. Fig. 6 (b) shows that the developed models also achieved low execution times, with the CNN being the most computationally intensive, requiring 130 s.

Apart from the 10-fold cross-validation, we evaluated the generalizability of our models by predicting viscosities for an unseen dataset, the FP 3230 polymer, which had been entirely excluded from both the training and testing phases. The FP 3230 dataset encompasses experimental viscosity measurements at shear rates (ranging from 0 to 100 s^-1), temperatures (25 to 55 °C), and polymer concentrations (1500 to 5000 ppm). The results of this evaluation are visually presented in Fig. 7. As shown in Fig. 7 (a) to (h), all models effectively captured and tracked the variations in viscosity concerning these influential parameters. Notably, CNN emerged as the most accurate model, as evidenced in Fig. 7 (a). For example, many alternative models tended to overestimate viscosity levels at a polymer concentration of 1500 ppm and a temperature of 55 °C. In contrast, the CNN exhibited a closer fit to the experimental data. This observation underscores the superior generalizability of the CNN compared to the other models.

Fig. 7.

A comparative analysis is conducted to contrast the predictions made by a suite of models, denoted as (a) MLP, (b) SVR, (c) RF, (d) GBM, (e) LGBM, (f) XGB, (g) VR, and (h) CNN, against the actual experimental measurements.

We employed the permutation variable importance technique to assess the impact of input parameters on model performance, specifically evaluating the relationship between each input variable and the target variable (viscosity). This method involves randomly permuting the variable sequence and measuring the resulting variation in model performance. Figure 8 presents the outcomes across various models, consistently ranking polymer concentration as the most significant predictor, followed by shear rate, molecular weight, temperature, and salinity.

Fig. 8.

The evaluation of the importance of the permutation-based variable is performed across various developed models using the dataset. In this context, the association between the mentioned variable and the target outcome, viscosity, is nullified through the random rearrangement of the variable values. Subsequently, the model’s RMSE is recalculated to assess the influence of the specific variable on predictive accuracy. This process is iterated five times, and the resulting RMSE values are computed. Variable significance is determined by the difference between the initial RMSE value and the averaged value obtained after five rounds of permutation.

The findings align with classical polymer rheology, where increasing polymer concentration increases viscosity by increasing the overlap and entanglement density of polymer chains, leading to a transition from the dilute to the semi-dilute or entangled regime. Shear rate influences viscosity due to the shear-thinning or shear-thickening behavior of HPAM and similar polymers, which is a result of chain orientation and stretching in flow. Molecular weight controls chain length and hydrodynamic volume, affecting zero-shear viscosity and the onset of non-linear viscoelasticity.

Temperature and salinity have a lesser impact on viscosity in our models, consistent with the narrower ranges explored for these variables. Their effects are mostly indirect, mediated by coil contraction, ion screening, and possible thermo-mechanical degradation, rather than the more direct impact of concentration and shear rate on network formation. Notably, this pattern holds across all models, except for MLP and SVR, which rank salinity as more influential than temperature. Overall, the data-driven importance ranking is physically plausible and in line with prior rheological studies ^61,62.

Conclusions

This study has demonstrated the capability of soft-computing techniques, including traditional machine learning (MLP and SVR), ensemble learning (RF, GBM, LGBM, XGB), and deep learning (CNN), to accurately predict polymer viscosity under diverse rheological conditions influenced by salinity/ionic strength, temperature, concentration, shear rate, and molecular weight. Polymer viscosity depends on nonlinear interactions among concentration, molecular weight, temperature, salinity, and shear rate, which complicate empirical modeling.

By accommodating these complexities, data-driven models enable precise characterization of polymer behavior across a broad range of application-relevant environments. The models developed here provide accurate viscosity predictions across broad rheological conditions, reducing experimental effort in formulation and process optimization, as well as in other sustainable process-engineering workflows. In these settings, reliable viscosity prediction helps optimize polymer selection, dosage, and operating windows (temperature and shear), thereby improving process stability, product quality, resource efficiency, and cost effectiveness while reducing reliance on extensive laboratory testing. The trained models were evaluated through several statistical techniques, with the following results.

• Both the hold-out and 10-fold cross-validation reveal high performance for all models. The SVR model’s minimum test R-squared of 0.986 constituted the lowest among the candidates, underscoring the importance of methods that effectively capture nonlinear, multivariate dependencies in polymer rheology.

• Among the models, LGBM demonstrated superior accuracy, attaining an R-squared value of 0.99 with a runtime below 1 s, making it suitable for large-scale viscosity prediction.

• Generalization analysis on the completely unseen dataset (FP 3230) showed the strongest performance for CNN, with an R-squared of 0.997.

• Permutation-importance analysis identified polymer concentration as the most influential factor, followed by shear rate and molecular weight, reinforcing their fundamental roles in viscosity modeling.

A natural extension of this work is to couple our rheological database with molecular representations of polymer structure, enabling unified models that interpolate across broader families of chemistries, as recently explored in structure-aware polymer ML studies ⁴². Such models could bridge formulation-level and molecular-level design for viscosity control.

Overall, these findings confirm that soft-computing, data-driven viscosity prediction can substantially reduce experimental burden and accelerate development cycles for polymer-enabled technologies in environmental and process engineering contexts. Future work can expand the databank to additional polymer families and operating regimes; incorporate physics-informed learning, uncertainty quantification, and explainability; and explore transferability to online monitoring and digital-twin implementations for sustainable, resource-efficient manufacturing.

Supplementary Information

Below is the link to the electronic supplementary material.

Author contributions

F.N. conceived the study, curated the data, provided resources, prepared visualizations, developed the methodology, and wrote the original draft. M.M. prepared visualizations, implemented the software, conducted the investigation, and reviewed and edited the manuscript. M.S. validated the results, supervised the research, and reviewed and edited the manuscript.

Data availability

The datasets generated and/or analysed during the current study are available in the [GitHub repository](https:/github.com/farhoodnavaie/Dataset-of-Soft-Computing-Models-for-Predicting-Polymer-Viscosity-.git) .

Declarations

Competing interests

The authors declare no competing interests.

Declaration of Generative AI and AI-Assisted Technologies in the Writing Proces

During the preparation of this work, the authors used ChatGPT v4 in order to improve the language and readability. After using this tool/service, the authors reviewed and edited the content as needed and take full responsibility for the content of the publication.