Prediction and Interpretation of the Volumetric Mass Transfer Coefficient in Bioreactors Using a No-Code Platform for Autonomous Machine Learning Model Selection

Abstract

The volumetric mass transfer coefficient ($k_{L}a$) governs the design, operation, and scale-up of aerobic bioprocesses, yet its dependence on reactor geometry, impeller design, operating conditions, and fluid properties limits prediction by empirical correlations. Machine learning (ML) improves accuracy but faces two barriers in bioprocess practice: selecting the best model among many candidates requires expertise, and small, highly multicollinear data make models chosen based on test error alone prone to overfitting. Using a browser-based, no-code platform, we trained 14 regression algorithms under an identical pipeline on a published $k_{L}a$ dataset, and introduced a composite objective, the generalization-penalized error (GPE), which is the test RMSE plus the absolute train–test RMSE gap. Minimizing GPE rather than test RMSE expanded the top statistically equivalent group to include not only boosting ensembles but also simpler, interpretable models, indicating that black-box models hold no clear advantage once train–test consistency is assessed. Sensitivity analysis showed that tree models produce discontinuous responses, whereas algebraic learning via elastic net (ALVEN) yields smooth surfaces. Shapley additive explanations (SHAP) and an ontology graph, interpreted by a retrieval-augmented language-model agent, identified rotational speed and gas flow rate as dominant, reproducing the established mass transfer mechanism. The framework offers a reproducible, interpretable, expertise-light route to bioprocess model selection.

1. Introduction

Volumetric mass transfer coefficient ($k_{L}a$) is the single most critical parameter governing the design, operation, and scale-up of aerobic bioprocesses [1]. In microbial and animal-cell cultures, dissolved oxygen is rapidly depleted, owing to its low solubility in the aqueous phase, so the rate of oxygen supply from the gas phase to the liquid phase directly limits the specific growth rate and metabolic activity of the cells [2]. The oxygen transfer rate is expressed as the product of $k_{L}a$ and the concentration-difference driving force. Whereas the driving force behaves relatively independently of the mass transfer coefficient, $k_{L}a$ is a complex function that simultaneously depends on reactor geometry, impeller design, operating conditions, and the rheological properties of the broth [3]. Accurate prediction of $k_{L}a$ is therefore essential, both for defining operating windows that avoid oxygen transfer limitation and for performing reliable scale-up from laboratory to commercial scale [4,5].

Traditionally, $k_{L}a$ has been estimated using empirical correlations based on power input per unit volume and superficial gas velocity [6,7]. Although such dimensional-analysis-based correlations offer clear physical interpretability, their predictive accuracy deteriorates once the reactor geometry or fluid conditions move outside the narrow range for which the correlation was derived; multidimensional effects, such as impeller type and multistage configuration, are difficult to incorporate within a single correlation [8]. Against this background, data-driven machine learning (ML) approaches, which can simultaneously accommodate numerous structural, impeller, and operational variables as inputs, have recently been reported to significantly improve $k_{L}a$ prediction accuracy relative to empirical correlations [9,10]. Although demonstrated on a $k_{L}a$ dataset, the platform and the GPE-based selection workflow are dataset-agnostic and applicable to other multi-input, single-output regression problems in bioprocess and chemical engineering [11,12,13].

However, two structural barriers hinder the practical use of ML models in chemical and biological engineering. First, selecting the model best suited to a given dataset among many candidate algorithms, ranging from linear regression families to tree-based ensembles and artificial neural networks, requires considerable expertise, so model selection acts as a collaboration bottleneck between domain experts and data scientists [14]. Second, because bioprocess data are typically small and highly multicollinear [15,16], with few samples and strong inter-variable correlation, selecting a model on the basis of test error alone risks overfitting the training data [17]. This is especially pronounced for small datasets, and the common practice of reporting the model with the lowest single metric, irrespective of generalization performance, can undermine reproducibility.

To mitigate both barriers simultaneously, this study uses a browser-based, no-code ML modeling platform to predict and interpret $k_{L}a$ in gas–liquid stirred bioreactors. This platform (1) trains and compares a range of regression algorithms in parallel under an identical preprocessing and validation pipeline [14,18]; (2) introduces a composite metric that penalizes the train–test error gap together with the test error, thereby identifying models with superior generalization; and (3) integrates variable-importance analysis based on Shapley additive explanations (SHAP), sensitivity analysis, an ontology graph, and a domain-knowledge-based retrieval-augmented, large-language-model interpretation agent to extract engineering insight from the derived model [19,20,21,22,23,24].

The contributions of this study are threefold: (1) to quantify the uncertainty of model selection by identifying statistically equivalent groups of models in a no-code environment accessible to non-experts; (2) to propose a model-recommendation strategy for small, highly multicollinear data by comparing two objective functions, simple test-error minimization and generalization-penalized minimization; and (3) to quantitatively contrast the variable-importance results of the data-driven model with the established physical understanding of the oxygen transfer mechanism.

2. Methods

2.1. Dataset for Machine Learning Modeling

The dataset used in this study was obtained from the gas–liquid stirred-bioreactor oxygen transfer database reported by Chen et al. [10]. The data comprise $k_{L}a$ values measured or simulated under Newtonian fluid conditions together with the multidimensional input variables. The inputs (X features) are divided into three categories according to their physical character. The structural variables include reactor volume ($V$), diameter ($D$), and liquid height ($H$), which define the overall scale and shape of the reactor. The impeller variables include impeller diameter ($d_0$), disk diameter ($d$), blade length ($L$), blade width (W), blade inclination angle ($\theta$), number of impellers ($m$), number of blades ($b$), clearance ratio ($z/H$), and bottom impeller position ($h_0/H$). The operational variables include impeller rotational speed ($N$) and gas flow rate ($Q_g$), and viscosity ($\mu$) is included as a fluid property. All input features were used as model inputs. The output (Y feature) is the single value of the volumetric mass transfer coefficient, $k_{L}a$. After checking for missing values and unit consistency, the raw data were used without deterministic variable transformation, so that the subsequent ML models would learn the nonlinear relationships among variables on their own. The dataset comprises a total of 174 data points.

2.2. Algorithms for Autonomous ML Model Selection

All data analyses were performed using a prototype version of in-house PickMo (Prototype version; https://junwookim.com/software/; 18 June 2026) software, a no-code regression-modeling environment in which the entire workflow runs client-side within a web browser without server installation or coding. It allows for data uploading, preprocessing, parallel training of multiple models, automated hyperparameter search, model evaluation, variable-importance and sensitivity interpretation, and result storage within a single interface. The data-processing structure and module composition of PickMo are illustrated in Figure 1.

Figure 1. Algorithms for autonomous ML model selection of PickMo.

The data were split once into a 70% training set and a 30% test set using a fixed random seed (adjustable by the user within the platform), and all candidate models were trained under the same split and preprocessing pipeline to ensure a fair, reproducible comparison. Model evaluation was performed in parallel against two objective functions; the first is the conventional criterion of minimizing the root mean squared error of the test set (test RMSE), and the second is a composite metric defined in this study as the generalization-penalized error (GPE), given by

$$\mathrm{G}\mathrm{P}\mathrm{E}=\mathrm{R}\mathrm{M}\mathrm{S}{\mathrm{E}}_{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}}+\left|\mathrm{R}\mathrm{M}\mathrm{S}{\mathrm{E}}_{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}}-\mathrm{R}\mathrm{M}\mathrm{S}{\mathrm{E}}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}}\right|$$

(1)

Here, the first term on the right-hand side measures the absolute predictive accuracy of the model, while the second term measures the gap between training and test performance. The second term penalizes both models that overfit the training data, such that test performance falls well below training performance, and models whose test performance abnormally exceeds training performance owing to a fortunate split. Adopting GPE as the objective, therefore, favors not simply the model with the lowest test error, but the model whose train–test behavior is consistent and whose generalization to unseen data is reliable. This reflects a design intent to mitigate the risk of model overestimation arising from a single test split in bioprocess data with few samples and high multicollinearity. The platform does not diagnose overfitting in absolute terms. Instead, by weighting the train–test RMSE gap, GPE enables a relative comparison that penalizes models whose train–test inconsistency is comparatively large.

By applying the two objectives in parallel and comparing their results, this study quantitatively analyzes the influence of metric choice on model recommendation. Repeated k-fold cross-validation was performed within the training partition, with k = 5 and three repeats (15 validation folds in total). Across these 15 folds, models were ranked by the mean value of each objective function (test RMSE or GPE). Accordingly, the train RMSE and test RMSE values (±standard deviation) are reported as the across-fold mean ± standard deviation over these 15 cross-validation folds, not values from a single split.

Starting from the best-ranked model, consecutively ranked models were grouped together, as long as they were not significantly different from the top-ranked model, by a two-sided paired t-test on their per-fold differences (p > 0.05). This top group therefore contains only models that are statistically indistinguishable from the best-performing model. This grouping was introduced because the best-performing model can shift depending on the chosen dataset and objective function. The paired-test grouping makes it possible to assess whether such ranking changes are statistically meaningful rather than artifacts of a particular split, thereby avoiding over-reliance on a single nominally optimal model.

For the interpretation of the derived models, PickMo provides SHAP-based global and local variable-importance analysis, one- and two-dimensional sensitivity analysis, point simulation at arbitrary operating points, and an ontology graph that visualizes the inter-variable influence structure [19,20,21,22,23]. It also incorporates a domain-knowledge-based retrieval-augmented, conversational interpretation agent built on a large language model (Claude Opus 4.7, Anthropic, CA, USA), which summarizes the variable-importance and sensitivity results from an engineering perspective in natural language and responds to follow-up queries from the user [24]. In the current prototype (1 June 2026), the AI agent feature is disabled due to API key constraints, while all other functionalities remain fully operational.

While the underlying algorithms are identical to those available in conventional coding environments, the platform lowers the barrier for domain experts without programming expertise by providing an accessible interface and an identical, reproducible pipeline across all candidate models, as well as continual updates to the embedded model library. Its purpose is therefore not to replace hand-written code, but to standardize and broaden access to model selection so that domain knowledge, rather than coding proficiency, governs the workflow.

2.3. Machine Learning Models

PickMo implements fourteen regression algorithms spanning linear and regularized regression, partial-least-squares families, nonlinear sparse regression, kernel methods, artificial neural networks, and tree-based ensembles, and in this study we compared them all under identical conditions. The linear and regularized regressions included multiple linear regression (MLR), ridge regression (Ridge) [25], Lasso regression (Lasso) [26], and elastic net (EN), which combines L1 and L2 penalties [27]. For latent-variable methods, partial least squares (PLS) [28], widely used for multicollinear data, and sparse PLS (SPLS) [29], which performs dimension reduction and variable selection simultaneously, were used. For nonlinear sparse regression, algebraic learning via elastic net (ALVEN) was included, which generates features over a predefined family of nonlinear transformations and learns an interpretable nonlinear model via elastic net [30]. Unlike black-box models, ALVEN explicitly retains the transformed variable terms, pursuing interpretability and generalization simultaneously, and has been reported to be particularly suitable for small, highly multicollinear process data. For kernel methods, support vector regression (SVR) was used [31], and for nonlinear function approximation, an artificial neural network (ANN) was used [32]. For tree-based methods, random forest (RF) [33], gradient boosting machine (GBM) [34], XGBoost [35], LightGBM [36], and CatBoost [37] were included. The hyperparameters of each algorithm were optimized separately for the two objective functions defined above (test RMSE and GPE) using the automated search function of PickMo. For the search algorithm, we used random sampling of candidate combinations from predefined hyperparameter distributions [38]. The fourteen models represent the algorithms currently implemented in the platform, chosen to span the principal regression families. Because of the platform’s modular, browser-based architecture, the library can be extended in future releases.

3. Results and Discussion

3.1. Data Characterization

Prior to modeling, the linear correlation structure and nonlinearity of the data were diagnosed. The Pearson correlation coefficients among the input variables and between inputs and output are summarized in Figure 2 as a correlation heatmap and the top ten absolute-correlation pairs. The analysis confirmed strong multicollinearity among many structural variables, with particularly high mutual correlations among $D$, $d_0$, and $V$. This arises because, in conventional reactor design, volume, diameter, and impeller dimensions increase together according to geometric similarity laws, suggesting that these variables are effectively different expressions of a single scale factor. The high multicollinearity justifies the use of regularized or latent-variable-based sparse regression and implies that collinearity correction is needed when interpreting the individual importance of each variable.

Figure 2. Correlation heatmap and top 10 absolute correlations between features. The diagonal is left blank as it represents the self-correlation of each variable.

To evaluate nonlinear relationships not captured by linear correlation alone, the top ten variables ranked by nonlinearity index with respect to $k_{L}a$ are presented in Figure 3. The nonlinearity index was computed based on the concept of maximal correlation. Specifically, six univariate transformations (identity, square, cube, logarithm, exponential, and square root) were applied to each input variable. The maximum of the absolute Pearson correlation with Y was taken, and the difference obtained by subtracting the absolute value of the original linear correlation was defined as the degree of nonlinearity for that variable. The impeller variables $L$, $z/H$, and $W$ showed strong nonlinear relationships with $k_{L}a$. The presence of such nonlinearity suggests that purely linear models may be insufficient to capture these relationships, supporting the introduction of ALVEN, which explicitly generates nonlinear terms, and tree-based ensemble models.

Figure 3. Top 10 input variables ranked by nonlinearity. For each variable, three quantities are shown: the absolute linear correlation with $k_{L}a$ (|Pearson|, orange); the maximal absolute Pearson correlation obtained after applying six univariate transformations—identity, square, cube, logarithm, exponential, and square root (Max|corr|, blue); and the nonlinearity index, defined as their difference (Difference, gray).

3.2. Hyperparameter Optimization

Hyperparameters were optimized to minimize the cross-validation RMSE or GPE on the training partition. The held-out test set was used only for final evaluation. Across folds and under both objective functions, the hyperparameter search converged consistently and showed stable behavior. The hyperparameters of the fourteen models optimized for the two objective functions are summarized in Table 1. Even for the same algorithm, the selected hyperparameters differed depending on the objective function. In particular, for the regularized-regression family, a tendency was observed for stronger regularization to be selected when GPE was the objective, demonstrating that the train–test gap penalty term of GPE acts to suppress model complexity and thereby structurally controls overfitting. This behavior is consistent with the bias–variance trade-off, in which the training error decreases but the validation error increases beyond a certain point as model complexity grows [39]. The gap term of GPE effectively incorporates train–test consistency into the optimization target, guiding the model to remain near the optimum of the trade-off. For tree-based ensembles as well, tree depth or the number of estimators tended to be selected more conservatively under the GPE criterion. This is consistent with the fact that the number of estimators and tree depth are the principal determinants of complexity, such that constraining them regularizes the model, reducing variance and improving generalization [34].

Table 1. Summary of hyperparameter optimization results.

ML Model	Hyperparameter	Symbol *	Minimum Test RMSE	Minimum GPE
PLS	Number of latent components	$n$	13	3
SPLS	Number of latent components	$n$	16	2
	Sparsity strength	$\lambda$	0.17	0.42
Ridge	L2 regularization strength	$\lambda$	0.1	0.1
Lasso	L1 regularization strength	$\lambda$	0.05	0.05
EN	L1/L2 mixing ratio	$\alpha$	0.8	0.5
	Regularization strength	$\lambda$	0.0175	0.0248
ALVEN	Nonlinear degree	$d$	3	1
	EN L1/L2 mixing ratio	$\alpha$	0.77	0.95
	regularization strength	$\lambda$	0.59	0.36
RF	Number of trees	$n$	66	44
	Maximum depth	$d$	10	9
SVR	Regularization strength	$C$	2.44	0.15
	Tolerance margin	$ϵ$	0.1	0.27
	Kernel	-	rbf	rbf
	Kernel width parameter	$\gamma$	0.463	0.424
ANN	Epochs	$ep$	285	208
	Learning rate	$lr$	0.0123	0.0087
	Hidden layer structure	$h$	22, 18	37, 18
GBM	Boosting rounds	$n$	176	138
	Learning rate	$\eta$	0.174	0.189
	Tree depth	$d$	5	3
XGBoost	Boosting rounds	$n$	86	80
	Learning rate	$\eta$	0.172	0.1
	Tree depth	$d$	4	4
CatBoost	Boosting rounds	$n$	151	85
	Learning rate	$\eta$	0.276	0.07
	Tree depth	$d$	4	2
LightGBM	Boosting rounds	$n$	131	87
	Learning rate	$\eta$	0.13	0.023
	Tree depth	$d$	3	3

* Hyperparameter symbols follow those used in PickMo and apply only within this table, disregarding overlap with other variables in this paper.

3.3. Model Recommendation

The recommendation rankings of the fourteen models for each objective function are presented in Table 2 and Table 3. For minimizing test RMSE, the top statistically equivalent group (p > 0.05) was classified as CatBoost, GBM, XGBoost, and LightGBM. That is, the four boosting-family models in the top group were not statistically distinguishable from one another, consistent with the original data-source study, in which CatBoost and XGBoost were reported as the optimal models [10]. For the present dataset, the purely linear models may be insufficient, as empirically confirmed by their comparatively low test accuracy (Table 2 and Table 3). This does not preclude linear models from being adequate on other datasets.

Table 2. Model ranking when minimizing test RMSE is the objective function. ± indicates standard deviation.

Rank	Model	Train RMSE (1/s)	Test RMSE (1/s)	GPE (1/s)	Test R2	Test MAPE (%)
1	CatBoost	0.0011 ± 0.0001	0.0079 ± 0.0029	0.0146 ± 0.0058	0.8096 ± 0.127	41.54 ± 19.70
2	GBM	0.00078 ± 0.0001	0.0082 ± 0.0029	0.0156 ± 0.0058	0.7936 ± 0.123	45.74 ± 20.51
3	XGBoost	0.0023 ± 0.0002	0.0082 ± 0.0027	0.0140 ± 0.0055	0.7975 ± 0.108	39.55 ± 14.58
4	LightGBM	0.0024 ± 0.0002	0.0083 ± 0.0029	0.0142 ± 0.0057	0.7880 ± 0.137	46.13 ± 18.14
5	ALVEN	0.0050 ± 0.0003	0.0107 ± 0.0036	0.0165 ± 0.0073	0.6254 ± 0.297	48.94 ± 11.51
6	RF	0.0074 ± 0.0004	0.0116 ± 0.0024	0.0157 ± 0.005	0.6184 ± 0.109	65.92 ± 22.40
7	ANN	0.0078 ± 0.0020	0.0118 ± 0.0031	0.0160 ± 0.0053	0.5869 ± 0.190	64.57 ± 17.80
8	SVR	0.0093 ± 0.0008	0.0133 ± 0.0028	0.0175 ± 0.0056	0.4892 ± 0.189	65.29 ± 22.74
9	PLS	0.0132 ± 0.0006	0.0148 ± 0.0022	0.0171 ± 0.0042	0.3806 ± 0.109	90.64 ± 15.37
10	SPLS	0.0132 ± 0.0006	0.0148 ± 0.0021	0.0170 ± 0.0041	0.3797 ± 0.105	90.13 ± 15.30
11	MLR	0.0132 ± 0.0006	0.0148 ± 0.0021	0.0170 ± 0.0041	0.3797 ± 0.105	90.13 ± 15.30
12	EN	0.0133 ± 0.0005	0.0148 ± 0.0020	0.0170 ± 0.0039	0.3813 ± 0.089	86.45 ± 16.46
13	Lasso	0.0139 ± 0.0005	0.0153 ± 0.0019	0.0174 ± 0.0035	0.3419 ± 0.080	91.32 ± 18.64
14	Ridge	0.0143 ± 0.0005	0.0153 ± 0.0019	0.0175 ± 0.0032	0.3373 ± 0.067	93.98 ± 19.73

Table 3. Model ranking when minimizing GPE is the objective function. ± indicates standard deviation.

Rank	ML Model	Train RMSE (1/s)	Test RMSE (1/s)	GPE (1/s)	Test R2	Test MAPE (%)
1	XGBoost	0.0036 ± 0.0002	0.0088 ± 0.0028	0.0139 ± 0.0056	0.7717 ± 0.117	40.35 ± 11.98
2	LightGBM	0.0082 ± 0.0004	0.0115 ± 0.0022	0.0148 ± 0.0046	0.6253 ± 0.096	84.16 ± 24.22
3	GBM	0.0024 ± 0.0002	0.0086 ± 0.0030	0.0149 ± 0.0060	0.7742 ± 0.135	44.64 ± 16.57
4	RF	0.0076 ± 0.0004	0.0118 ± 0.0025	0.0159 ± 0.0052	0.6056 ± 0.112	67.05 ± 21.83
5	CatBoost	0.0093 ± 0.0005	0.0125 ± 0.0025	0.0159 ± 0.0050	0.5566 ± 0.125	69.66 ± 16.88
6	ALVEN	0.0130 ± 0.0004	0.0143 ± 0.0018	0.0162 ± 0.0033	0.4203 ± 0.090	87.08 ± 17.39
7	ANN	0.0079 ± 0.0019	0.0123 ± 0.0033	0.0169 ± 0.0061	0.5560 ± 0.187	64.86 ± 22.57
8	MLR	0.0132 ± 0.0006	0.0148 ± 0.0021	0.0170 ± 0.0041	0.3797 ± 0.105	90.13 ± 15.30
9	EN	0.0134 ± 0.0006	0.0149 ± 0.0020	0.0171 ± 0.0038	0.3756 ± 0.084	87.02 ± 16.64
10	Lasso	0.0139 ± 0.0005	0.0153 ± 0.0019	0.0174 ± 0.0035	0.3419 ± 0.080	91.32 ± 18.64
11	Ridge	0.0143 ± 0.0005	0.0153 ± 0.0019	0.0175 ± 0.0032	0.3373 ± 0.067	93.98 ± 19.73
12	PLS	0.0149 ± 0.0005	0.0159 ± 0.0017	0.0179 ± 0.0028	0.2815 ± 0.082	92.65 ± 19.43
13	SPLS	0.0155 ± 0.0005	0.0161 ± 0.0018	0.0180 ± 0.0023	0.2638 ± 0.105	96.31 ± 21.31
14	SVR	0.0153 ± 0.0010	0.0163 ± 0.0029	0.0194 ± 0.0043	0.2494 ± 0.163	96.15 ± 22.59

For minimizing GPE, the top statistically equivalent group (p > 0.05) was reorganized into XGBoost, LightGBM, GBM, RF, CatBoost, ALVEN, ANN, MLR, and EN. Notably, under the generalization-penalized criterion, the size of the top equivalent group expanded substantially, so that not only the boosting family but also relatively simple models such as ALVEN, ANN, MLR, and EN were included in the statistically equivalent performance category. This means that, when train–test consistency is jointly assessed, complex black-box models hold no clear advantage over simpler models, providing grounds for reassessing the generalization reliability of simple models on small datasets. It is common for no clear accuracy difference to exist between complex black-box models and inherently interpretable models, and in such cases it may be preferable to adopt the interpretable model first [40].

In both paths, because there is no statistically significant performance difference among models within the same group, the rank-1 model cannot be clearly superior to the other models in its group. The final choice within a group should therefore be made on the basis of secondary criteria such as interpretability, computational cost, and continuity of the prediction surface, rather than predictive accuracy. In this study, for the subsequent interpretation analysis, the black-box-family XGBoost and the interpretable nonlinear model ALVEN were selected as representatives of the top group for each objective function and compared.

The predictive accuracy of the selected models was verified through the parity plots in Figure 4 and the residual distributions in Figure 5. Each figure jointly presents the results of CatBoost, XGBoost, and ALVEN under the two objective functions (minimum test RMSE and GPE). In the parity plots, the predicted and measured values cluster around the diagonal, confirming high goodness of fit, and the residual distributions are centered on zero without pronounced bias, indicating small systematic error. As shown in Figure 4f and Figure 5f, the ALVEN result under the minimum-GPE objective shows lower accuracy but smoother response compared with the tree-based methods.

Figure 4. Parity plots of CatBoost, XGBoost, and ALVEN models for $k_{L}a$ prediction.

Figure 5. Residual distributions of CatBoost, XGBoost, and ALVEN models for $k_{L}a$ prediction.

3.4. Sensitivity Test

To quantify the contribution of each variable to $k_{L}a$ prediction, SHAP-based global variable importances were computed, as shown in Figure 6. Although there is rank variability depending on the objective function, the ML model, and the hyperparameters, the same result was obtained: $Q_g$ and $N$ were the two most important key variables.

Figure 6. Mean absolute SHAP-based variable importance for $k_{L}a$ prediction.

To quantitatively examine the variable–output relationships learned by the models, a two-dimensional sensitivity analysis was performed over a ±20% range around the median of each of the two most influential operational variables, $Q_g$ and $N$, as shown in Figure 7. The figure jointly presents the XGBoost and ALVEN results under the two objective functions, and analysis showed that increases in both variables acted to increase $k_{L}a$, consistent with the physical behavior of oxygen transfer [2]; however, an essential difference was observed in the shape of the response surfaces between the two model families. XGBoost, a decision-tree-based model, exhibited a stepwise, discontinuous response at split boundaries, whereas ALVEN produced a smooth, continuous response surface. Such discontinuity degrades prediction reliability near split boundaries on small datasets and acts as a constraint in applications requiring output continuity, such as gradient-based optimization or model predictive control [41].

Figure 7. Two-dimensional sensitivity test for $k_{L}a$ prediction. All values of X features are median. Sensitivity of $Q_g$ (cm³/s) and $N$ (RPM) is ±20%. Adjacent dots displayed on each heatmap indicate equal predicted $k_{L}a$ values.

This difference has practical implications for model selection. In applications requiring a continuous, differentiable output response to the input variables, such as process optimization or model predictive control, it is preferable to adopt a continuous-response model such as ALVEN over tree-based models, even when the predictive accuracy is statistically equivalent. This is because the discontinuity of tree-based models can impede the convergence of gradient-based optimization algorithms or induce non-physical local behavior. Therefore, when data continuity is required, ALVEN can be justifiably recommended even if its test accuracy is somewhat lower than that of the boosting-family models.

3.5. Ontology-Based Interpretation

The ontology interpretation in Figure 8 was derived based on the correlation coefficients among the X features and their SHAP-based variable importance. The input metadata are summarized in Box 1, and the engineering analysis by the interpretation agent regarding the variable-importance results is summarized in Box 2. The primary factors governing $k_{L}a$ prediction were impeller rotational speed ($N$) and gas flow rate ($Q_g$), followed by viscosity ($\mu$), which is consistent with mass transfer theory, in which, in the relation $k_{L}a=k_L·a$, both the liquid-side mass transfer coefficient ($k_L$) and the gas–liquid interfacial area ($a$) depend on agitation intensity and gas flow rate [2]. An increase in rotational speed promotes turbulence intensity and bubble breakup, simultaneously increasing the interfacial area and the liquid-side transfer coefficient, while an increase in gas flow rate determines oxygen availability through gas holdup and bubble residence time [6]. The appearance of viscosity as the third factor is consistent with the understanding that lower viscosity reduces boundary-layer thickness and promotes bubble breakup, thereby enhancing mass transfer [2]. Impeller geometry variables such as bottom clearance ratio ($h_0/H$), disk diameter ($d$), and blade length ($L$) were classified as secondary influencing factors.

Figure 8. Ontology graph with XGBoost and minimum GPE. Orange and gray curves indicate Y importance and X-X correlation. Curve thickness indicates strength.

Box 1. Metadata input for AI agent.

Project Name: Volumetric Mass Transfer Coefficient (kLa) in Gas–Liquid Stirred Bioreactors—Newtonian Fluids

Data Source: Chen et al., Chinese Journal of Chemical Engineering 84 (2025) 211–226. Compiled dataset of 174 experimental and CFD-derived kLa measurements for Newtonian fluids across laboratory- (1 L) to pilot-scale (100 L) stirred bioreactors.

Process Description: Structural, impeller, fluid, and operational features are used to predict the volumetric oxygen mass transfer coefficient kLa, a key indicator for aerobic fermentation performance. The original paper showed that CatBoost outperforms classical empirical correlations on this dataset, with rotational speed (N) and gas flow rate (Qg) emerging as the top predictors. A separate non-Newtonian subset (with consistency coefficient K and power-law index n in place of μ) is also reported in the paper.

Related Links: Chen et al. 2025—original paper (https://doi.org/10.1016/j.cjche.2025.06.010)

Per-Variable Metadata

Symbol	Variable Name	Category	Unit	Description
kLa	Volumetric mass transfer coefficient	target	1/s	Volumetric oxygen mass transfer coefficient (target variable)
V	Reactor volume	reactor	L	Total bioreactor working volume
D	Reactor diameter	reactor	mm	Inner tank diameter
H	Liquid height	reactor	mm	Static liquid height
m	Number of impellers	impeller	count	Number of impellers mounted on the shaft
z/H	Clearance ratio	impeller	-	Vertical clearance between impellers, normalised by liquid height
b	Blade number	impeller	count	Number of blades per impeller
d0	Impeller diameter	impeller	mm	Overall impeller (swept) diameter
d	Disk diameter	impeller	mm	Diameter of the central disk (0 for non-disk impellers)
L	Blade length	impeller	mm	Length of a single blade
W	Blade width	impeller	mm	Width of a single blade
h0/H	Bottom impeller position	impeller	-	Distance from tank bottom to the lowest impeller, normalised by liquid height
theta	Blade inclination angle	impeller	°	Blade pitch angle (90° = straight Rushton-style; <90° = pitched-blade up; >90° = pitched-blade down)
μ	Viscosity	fluid	Pa·s	Dynamic viscosity of the liquid phase
Qg	Gas flow rate	operation	cm3/s	Sparged gas (air) volumetric flow rate
N	Rotational speed	operation	rpm	Impeller rotational speed

Box 2. Response of AI agent to the query “Interpret the variable importance prediction results from an engineering perspective”.

Variable Importance Analysis for Bioreactor kLa Prediction

Based on the XGBoost model (Score: 0.0139, best performing), here’s the engineering interpretation of variable importance for predicting volumetric oxygen mass transfer coefficient (kLa):

Primary Drivers (High Impact Variables)

Variable	SHAP Importance	Engineering Significance
N (Rotational speed)	0.0098	Primary driver of turbulence and bubble breakup. Higher impeller speed increases interfacial area and liquid-side mass transfer coefficients
Qg (Gas flow rate)	0.0084	Controls gas holdup and bubble residence time. Direct relationship with oxygen availability and contact time
μ (Viscosity)	0.0041	Affects bubble formation, coalescence, and liquid-side mass transfer resistance. Lower viscosity enhances mass transfer

Secondary Influencers (Moderate Impact)

Variable	SHAP Importance	Engineering Role
h0/H (Bottom clearance)	0.0025	Affects flow patterns and gas distribution efficiency
d (Disk diameter)	0.0019	Influences bubble dispersion mechanism for disk-style impellers
L (Blade length)	0.0018	Controls swept volume and shear generation

Engineering Process Insights

1. Mass Transfer Mechanism

• N and Qg dominate because kLa = kL × a, where both liquid-side coefficient (kL) and interfacial area (a) scale with agitation intensity and gas flow
• The 1D sensitivity confirms: N shows +0.0069 response for +10% increase, Qg shows +0.0029 response

2. Fluid Properties Impact

• Viscosity (μ) ranks third, showing strong negative sensitivity (−0.0080 for −10% change)
• This aligns with theory: lower viscosity reduces boundary layer thickness and enhances bubble breakup

3. Impeller Design Hierarchy

The importance ranking reveals design priorities:

• Geometric factors: d0, d, L (impeller sizing)
• Configuration factors: h0/H, theta (positioning/orientation)
• Multiple impeller effects: m, z/H show lower individual importance

4. Scale-Up Implications

• Reactor geometry (V, D, H) shows relatively low individual importance, suggesting good scale-up potential when maintaining similar N and Qg/V ratios
• High collinearity between V↔D (0.9166) and V↔d0 (0.9036) indicates these scale together in typical designs

Operational Recommendations

Primary Control Variables

• Impeller speed (N): Most sensitive parameter—optimize for maximum mass transfer without excessive power consumption
• Gas flow rate (Qg): Second priority—balance between oxygen supply and gas holdup efficiency

Design Considerations

• Impeller positioning (h0/H): Significant impact warrants careful optimization during reactor design
• Blade geometry (L, d): Secondary but measurable effects on mass transfer performance

Process Monitoring

Focus monitoring efforts on N, Qg, and μ (via temperature/composition control) as these drive 70%+ of kLa variability according to SHAP analysis.

This analysis confirms the original Chen et al. findings that operational parameters (N, Qg) dominate kLa prediction, with viscosity effects being the primary fluid property concern for Newtonian systems.

The relatively low individual importance of structural variables such as reactor volume ($V$), diameter ($D$), and height ($H$) should be interpreted together with the high multicollinearity among structural variables confirmed in Section 3.1. When features are strongly correlated, SHAP attributions become less reliable, since the shared contribution among correlated features is difficult to allocate unambiguously to any single variable, and the underlying approximation may evaluate the model on unrealistic feature combinations [42].

That is, for a group of structural variables whose mutual correlation exceeds 0.9, such as $V$ and $D$, their combined contribution to oxygen transfer is not consolidated into a single variable but is attributed across several. Thus, the absence of physical influence cannot be concluded from the magnitude of individual importance alone. Moreover, under a similarity condition in which these variables increase together, oxygen transfer performance can be relatively well preserved when $N/V$ and $Q_g/V$ are maintained, which is consistent with the standard scale-up principle of stirred bioreactors that takes the maintenance of constant power input per unit volume and gas flow rate per unit volume as key criteria [2]. Therefore, this result does not imply that the structural variables themselves are irrelevant to oxygen transfer; rather, it indicates that the individual influence of the absolute scale variables appears relatively small as long as the intensive variables obtained by normalizing the operational variables by volume are preserved. This demonstrates that this data-driven model independently reproduces the established physical understanding of the oxygen transfer mechanism. The conclusion of this analysis, that operational variables dominate mass transfer and that viscosity is the principal fluid-property factor, is also consistent with the qualitative findings of the original data-source study [10]. Furthermore, future studies could further enhance the robustness of scale-up predictions by incorporating key hydrodynamic parameters extracted from computational fluid dynamics (CFD) simulations, such as mixing time, maximum shear stress, turbulent dissipation rate, wall shear stress, and turbulent kinetic energy, as derived features in the machine learning model. Additionally, securing metadata on these localized flow indicators by compartmentalizing the reactor into analogous zones would provide an even stronger predictive framework [4].

However, the interpretation-agent analysis in Box 2 went only as far as directly interpreting the low SHAP importance of the structural variables as a positive signal of scale-up potential, and did not capture the deeper implication that this low importance may partly result from the shared contribution being distributed among correlated variables owing to the high multicollinearity among structural variables. It means that the variable-importance attribution itself can be distorted by the correlation structure. This is because the current interpretation agent reasons on the basis of the model’s quantitative outputs (SHAP values, sensitivity, correlation coefficients) and general process-engineering knowledge, suggesting a limitation in automatically identifying the influence of inter-variable dependence on statistical attribution. This limitation is expected to be improved by providing richer, domain-expertise-based structured metadata in the future. For example, if causal and dependency relationships among variables, the physical units of each variable and the intensive variables that should be preserved, and the constraints during scale-up, are specified as metadata and included in the agent’s retrieval-augmented context, the agent could be guided beyond a mere listing of importance rankings to recognize the influence of multicollinearity on attribution and to present a correspondingly corrected interpretation. This represents a direction for further refining an interpretation framework that combines quantitative model outputs with a domain ontology, and we plan to pursue such metadata-enhanced refinement of the interpretation agent in future work.

4. Conclusions

This study used the no-code machine learning platform PickMo to predict and interpret $k_{L}a$ in gas–liquid stirred bioreactors and to statistically quantify the uncertainty of model selection. When comparing fourteen regression models under identical conditions, the two objective functions, minimizing test RMSE and GPE, yielded different model recommendations. In particular, under the GPE criterion, black-box and simple models formed a statistically equivalent performance group, showing that the superiority of complex models is not clear within the limits of the data. The statistically-equivalent-group analysis based on the paired t-test recommends avoiding the practice of reporting a single optimal model and instead selecting a model within the group according to interpretability, continuity, and computational cost. The sensitivity analysis contrasted the discontinuous response of tree-based models with the continuous response of ALVEN, showing that continuous-response models are advantageous for optimization and control applications. Finally, the SHAP and ontology-based interpretation confirmed that the data-driven model identifies $N$ and $Q_g$ as the key factors and, at the same time, that the low importance of structural variables is interpreted in terms of high multicollinearity arising from geometric similarity, thereby quantitatively reproducing the established physical understanding of the oxygen transfer mechanism. This study presents a model selection strategy that jointly considers generalization performance and interpretability in a modeling environment accessible to non-experts, and it can be practically applied to the data-driven design and scale-up of bioprocesses.