A digital twin approach for optimizing okara-based Greek-style yogurt using hybrid Gompertz–LSTM modeling and data assimilation

Abstract

Plant-based yogurts frequently exhibit inadequate structural integrity and pronounced syneresis. This study formulated a novel okara and coconut milk Greek-style plant-based yogurt and developed a physics-augmented digital twin for robust process control. Rheological and microstructural analyses, yielding a fractal dimension (D_f = 1.686), demonstrated that the okara–coconut matrix aggregates via diffusion-limited cluster aggregation (DLCA), forming a compact, space-filling network. Multi-objective Pareto optimization identified an optimal formulation (14% okara, 18% coconut milk) that minimized syneresis (1.86%) while maximizing storage modulus (G′ = 886 Pa) and overall sensory acceptability (8.46/9.0). To monitor fermentation and gelation, a hybrid framework fusing a mechanistic Gompertz prior with a Long Short-Term Memory–Support Vector Regression (LSTM-SVR) residual corrector was developed. This digital twin achieved high predictive accuracy (R² = 0.9928), outperforming standalone machine learning approaches. Furthermore, integrating an Ensemble Smoother with Multiple Data Assimilation (ES-MDA) enabled dynamic parameter self-calibration for Avrami gelation kinetics, successfully reducing prediction uncertainty for G′ and syneresis by 58% and 74%, respectively. Ultimately, this hybrid physics–AI approach provides highly interpretable, risk-aware process control, establishing a scalable Industry 4.0 paradigm for managing raw material variability in plant-based food manufacturing.

Graphical abstract

Highlights

• •Okara-coconut matrix forms highly stable Greek-style plant-based yogurts.
• •Fractal matrix design limits syneresis and enhances sensory profiles.
• •Hybrid Gompertz-LSTM digital twin accurately predicts pH and gelation.
• •ES-MDA data assimilation reduces process prediction uncertainty by 74%.
• •Pareto-optimized formulation achieves high elasticity and consumer liking.

1. Introduction

Plant-based yogurt alternatives often suffer from inconsistent quality due to the inherent structural variability of their multicomponent matrices (Soni et al., 2020). Unlike dairy systems, where standardized milk provides a relatively uniform fermentation environment, plant-based formulations contain complex mixtures of proteins, polysaccharides, fats, and fibers whose interactions are highly non-linear and sensitive to raw-material composition (Dhakal et al., 2023). This problem is particularly acute when valorizing agricultural by-products such as okara, the fiber- and protein-rich residue from soy milk production (Vong and Liu, 2016). Although okara addition improves nutritional value and reduces waste, its insoluble fiber fragments disrupt gel network formation, increasing the risk of phase separation, moisture expulsion, and poor mouthfeel (Cais-Sokolińska and Walkowiak-Tomczak, 2020). While various formulation strategies have been explored to improve conventional soymilk-based yogurts—including empirical stabilizer addition, exogenous protein fortification, and enzymatic cross-linking—these approaches remain largely unstandardized and fail to consistently counteract the multi-scale quality fluctuations caused by multi-component matrices. Incorporating coconut milk and coconut embryo further complicates the system by introducing medium-chain triglycerides and distinct phenolic profiles that dynamically modulate microbial metabolic activity and structural development during fermentation (Canon et al., 2022; Molina et al., 2025).

Lactiplantibacillus plantarum LP-115 is widely used in plant-based fermentations because of its metabolic flexibility, yet its growth kinetics and acidification rate are strongly influenced by the composition of the surrounding matrix (Yılmaz et al., 2022). In the specific system studied here—a Greek-style yogurt matrix comprising okara, coconut milk, coconut embryo, soy milk, and a ternary hydrocolloid blend—the interactions between the probiotic strain, fermentable substrates, and structural biopolymers introduce a significant source of stochastic biological variability. The microbial lag phase, maximum acidification rate, and macrostructural gelation kinetics can shift substantially from batch to batch depending on minor composition and temperature fluctuations. Traditional empirical optimization methods, such as one-factor-at-a-time experimentation and conventional response surface methodology, are mathematically inadequate to capture or control this high-dimensional, dynamic behavior (Madoumier et al., 2019).

To address these limitations, digital twin technology offers a fundamentally interactive process control paradigm (Datta et al., 2022). A digital twin acts as a virtual replica of a physical process that evolves in real time by integrating online sensor data with predictive computational models. For food fermentation systems, physics-based models such as the modified Gompertz equation (Wang and Guo, 2024), the Avrami gelation model (Shirzad and Viney, 2023), and the Herschel–Bulkley rheological law (Martínez-Padilla, 2023) provide highly interpretable mechanistic baselines. However, these classical physics equations alone cannot adapt dynamically to the matrix-driven stochastic variability encountered in complex plant-based media. Conversely, purely data-driven machine learning models can capture complex interactions but frequently violate physical consistency and require unsustainably large training datasets (Purlis, 2024).

A hybrid physics-augmented approach, in which a deep learning architecture learns to predict and correct the systematic residuals of a mechanistic baseline, effectively combines the structural interpretability of physics with the adaptive capacity of machine learning (Purlis, 2023). When further coupled with state-estimation algorithms such as the Ensemble Smoother with Multiple Data Assimilation (ES-MDA), the digital twin can continuously update its internal parameters as new observations become available, enabling real-time uncertainty quantification, state tracking, and risk-aware process control.

In this study, we develop and experimentally validate the first physics-augmented digital twin specifically designed for a multi-component plant-based yogurt fermentation system. The framework integrates a modified Gompertz–Long Short-Term Memory (LSTM) residual architecture for adaptive pH kinetics, ES-MDA for the continuous joint state updating of structural gelation parameters, and a multi-objective Pareto optimization layer. The specific objectives are to: (1) characterize the baseline physicochemical, rheological, and textural properties of six prototype okara–coconut formulations; (2) develop and benchmark a hybrid Gompertz–LSTM residual model for fermentation pH kinetics; (3) implement ES-MDA data assimilation for real-time parameter estimation; and (4) deploy multi-objective Pareto optimization to map transient fermentation trajectory attributes directly to final static quality endpoints (G′, syneresis, and sensory preference), validated via a consumer sensory panel.

2. Materials and methods

2.1. Materials and reagents

Okara powder (defatted soybean residue) was obtained from a local tofu processing facility. It was dried at 60 °C to a constant weight before milling and sieving (≤250 μm). On a dry basis, the okara powder contained 22.4% protein, 5.9% fat, 4.2% ash, 13.2% soluble dietary fiber, and 45.5% insoluble dietary fiber, with a residual moisture content of 4.7% (in-house analysis, AOAC, 2000). Commercial soy milk (protein 3.2% w/v) and coconut milk (fat 18% w/v) were procured from a certified local supplier. Fresh coconut embryo was excised, homogenized in distilled water (1:1 w/v), and stored at 4 °C prior to use. A hydrocolloid blend consisting of k-carrageenan, sodium alginate, and agar (Sigma-Aldrich, St. Louis, MO, USA) was utilized to stabilize the gel network. The model probiotic strain Lactiplantibacillus plantarum LP-115 was obtained as a lyophilized powder from CUSTOM PROBIOTICS Inc. (Glendale, CA, USA) and stored at −18 °C prior to use. All standard chemicals and reagents were of analytical grade.

2.2. Experimental design overview

To address the complexity of the multi-component okara–coconut yogurt matrix, the experimental workflow integrated conventional physicochemical characterization with a suite of advanced computational tools. Table 1 summarizes each advanced tool, its specific modeling purpose, and the unique functional insight it provides compared to traditional empirical approaches.

Table 1.

Advanced analytical and computational tools employed in this study, their purpose, and their added value for yogurt systems.

Tool/Technique	Purpose	Added value over conventional methods
Modified Gompertz model	Quantify fermentation pH kinetics (lag phase, maximum acidification rate).	Replaces qualitative observation of pH decline with interpretable kinetic parameters linked directly to matrix composition.
Hybrid Gompertz–LSTM residual model	Improve predictive accuracy by learning matrix-induced residuals from the mechanistic baseline.	Captures batch-to-batch biological variability that purely mechanistic models cannot, while retaining physical interpretability.
ES-MDA data assimilation	Dynamically update Avrami gelation parameters in real time.	Enables continuous uncertainty quantification and self-calibration, moving beyond static, one-time empirical model fitting.
Avrami gelation model	Describe the time-dependent evolution of the gel network structure.	Provides specific kinetic parameters (rate constant, exponent) that link gelation mechanisms directly to final yogurt texture.
Herschel–Bulkley rheology	Characterize non-Newtonian flow and yield behavior.	Quantifies shear-thinning and yield stress metrics directly relevant to consumer spoonability and industrial processing.
Rheological fractal dimension	Infer network architecture from scaling behavior.	Connects microscopic structure (Diffusion-Limited Cluster Aggregation (DLCA) mechanism) to macroscopic phase stability (syneresis, water-holding capacity).
Multi-objective Pareto optimization	Identify formulations that simultaneously optimize multiple quality attributes.	Replaces single-objective optimization and provides a set of non-dominated solutions for informed, balanced formulation decision-making.

2.3. Formulation design and experimental matrix

A constrained mixture design was employed to generate six prototype formulations (F1–F6) representing a feasible compositional space for plant-based Greek-style yogurt (Table 2). Formulations varied in okara (10–18% w/w), coconut milk (10–25% w/w), coconut embryo (5–8% w/w), and soy milk (53.5–74.5% w/w). A fixed hydrocolloid concentration of 0.3% (w/w) k-carrageenan, 0.2% (w/w) sodium alginate, and 0.1% (w/w) agar was maintained across all batches.

Table 2.

Composition of experimental formulations (F1–F6).

Formulation	Okara (% w/w)	Coconut Milk (% w/w)	Coconut Embryo (% w/w)	Soy Milk (% w/w)
F1	10	10	5	74.5
F2	12	15	5	67.5
F3	15	20	5	59.5
F4	18	20	8	53.5
F5	15	25	5	54.5
F6	14	18	6	61.5

All formulations contained a fixed hydrocolloid blend of 0.3% k-carrageenan, 0.2% sodium alginate, and 0.1% agar (w/w).

To adequately capture batch-to-batch fermentation variability for robust kinetic modeling, 15 independent biological replicates were prepared and fermented for each formulation. Following fermentation, a randomly selected subset of three replicates (n = 3) from these batches was utilized for all downstream physicochemical, rheological, and textural analyses due to instrument throughput constraints. Experiments were randomized to prevent systematic block effects.

2.4. Experimental preparation and fermentation monitoring

Okara powder was rehydrated in distilled water (1:3 w/v) at 60 °C for 20 min. The hydrated okara was blended with the liquid bases (varying ratios of coconut milk, coconut embryo, and soy milk) according to the defined experimental design matrix. Hydrocolloids were dissolved separately in hot water (85 °C) and incorporated under continuous stirring. The mixture was processed using a high-shear homogenizer (Ultra-Turrax T25, IKA, Staufen, Germany) at 10,000 rpm for 5 min, pasteurized at 85 °C for 30 min, and cooled to the assigned fermentation temperature (37–45 °C).

To ensure a standardized physiological state and minimize lag-phase variability, the L. plantarum LP-115 inoculum was prepared via a rigorous subculturing protocol. The lyophilized powder was first revived in de Man, Rogosa, and Sharpe (MRS) broth at 37 °C for 24 h. This pre-culture was then subcultured (1% v/v) into fresh MRS broth and incubated at 37 °C for 16–18 h to harvest cells in the late exponential/early stationary phase. The biomass was recovered via centrifugation (5000 × g, 10 min, 4 °C) (Hettich Universal 320R, Tuttlingen, Germany) and washed twice with sterile 0.1% peptone water to remove residual MRS nutrients (Byakika et al., 2020).

The washed pellet was resuspended in sterile diluent and inoculated into the pasteurized base to yield an initial cell density of approximately 10⁷ CFU/mL. Fermentation kinetics were evaluated across the 15 independent biological replicates per formulation. The pH decline was logged continuously at 12-min intervals (0.2 h) using a calibrated pH probe until fermentation was terminated at 12 h (target pH 4.5). Concurrently, microbial viable counts (expressed as log CFU/mL) were enumerated at discrete 3-h intervals due to manual sampling constraints. Samples were immediately cooled to 4 °C in an ice-water bath and stored at 4 °C for 24 h prior to physicochemical and rheological characterization.

2.5. Physicochemical and rheological characterization

2.5.1. Syneresis

Syneresis was determined using a centrifugation method modified from established protocols for fermented gels (Pawlos et al., 2025). Briefly, 20 g of the fermented gel was centrifuged at 2935 × g for 10 min at 4 °C using a refrigerated centrifuge equipped with an angle rotor (Hettich Universal 320R, Tuttlingen, Germany). The percentage of expelled whey was calculated as:

$$Syneresis(\%)=\frac{W_s}{W_i}\times 100$$ (1)

where $W_s$ is the weight of the supernatant fluid recovered after centrifugation (g), and $W_i$ is the initial weight of the fermented sample prior to centrifugation (g).

2.5.2. Rheological properties

Rheological measurements were performed following established protocols for fermented dairy and complex food gels, modified from Rezaei et al. (2017) and Tunick (2011). Tests were conducted using a controlled-stress rheometer (Anton Paar MCR 302, Graz, Austria) equipped with a 40 mm parallel plate geometry (1 mm gap) at 10 °C. Storage modulus (G′), loss modulus (G″), and the loss tangent (tan δ = G''/G′) were determined via oscillatory frequency sweeps from 0.1 to 10 Hz within the predefined linear viscoelastic region (1% strain). Steady-shear flow curves (shear stress vs. shear rate) were generated over a shear rate range of 0.1 to 100/s.

2.5.3. Texture profile analysis (TPA)

Instrumental texture was evaluated using TPA, following established two-bite compression protocols for yogurt and semi-solid food gels, modified from Mudgil et al. (2017). Measurements were conducted using a Texture Analyzer (TA.XT Plus, Stable Micro Systems, Surrey, UK) equipped with a 5 kg load cell and a 35 mm cylindrical aluminum probe. A standard two-bite compression test was performed to 30% of the sample's original height at a test speed of 1.0 mm/s. The resulting force–time curves were processed to extract Hardness, Springiness, and Cohesiveness.

2.5.4. Determination of fractal dimension from rheological scaling

The rheologically inferred fractal dimension (D_f) of the gel network was estimated from the scaling relationship between the storage modulus (G′) and the total biopolymer concentration (C). In the weak-link regime, which is characteristic of aggregated protein–polysaccharide gels, the elastic modulus scales with concentration according to the established framework of Shih et al. (1990) and Wu and Morbidelli (2001):

$$G^{\prime }\propto C^{\frac{1}{3-D_f}}$$ (2)

The scaling exponent was obtained from the slope of the log–log plot of G' (measured at 1 Hz within the linear viscoelastic region) against the total solids content (w/w) across the six formulation variants (Maltais et al., 2008).

2.6. Computational environment

All computational modeling, machine learning, and data assimilation operations were executed using Python (v. 3.9).

2.7. Physics-based mechanistic models

• •Gompertz model: Fermentation pH trajectories were modeled using the modified Gompertz equation, a standard primary model adapted for sigmoidal microbial growth and lactic acidification kinetics (Wang and Guo, 2024; Zwietering et al., 1990):

$$pH\left(t\right)=p{H}_0+A\mathit{exp}\left\{-\mathit{exp}\left[\frac{{\mu }_e}{A}\left(\lambda -t\right)+1\right]\right\}$$ (3)

where $pH\left(t\right)$ is the predicted pH at fermentation time (t), $p{H}_0$ is the initial pH of the formulation, A is the maximum pH drop (amplitude), ${\mu }_e$ is the maximum acidification rate (ΔpH/h), $\lambda$ is the duration of the lag phase (h).

Due to the physical constraints of manual sampling, viable microbial counts (CFU) were enumerated at discrete 3-h intervals, whereas pH was logged continuously every 12 min (0.2 h). To prevent the kinetic uncertainty that arises from directly correlating mixed-frequency datasets, the discrete 3-h microbial counts were utilized strictly as low-frequency calibration anchors to establish the baseline boundary parameters of this mechanistic Gompertz model. This generated a continuous, mathematically smoothed metabolic trajectory that bridged the 3-h temporal gaps, providing a consistent baseline prior for the subsequent high-frequency residual machine learning layers.

• •Avrami gelation kinetics: The evolution of the gel network structure over time was described using the Avrami theoretical framework (Shirzad and Viney, 2023):

$$\alpha =1-\mathit{exp}(-k{t}^n)$$ (4)

where $\alpha$ is the fractional extent of gelation at time t (derived from the normalized storage modulus, G′), k is the gelation rate constant, t is the time, and n is the dimensionless Avrami exponent reflecting the dimensionality and nucleation mechanism of the structural network growth.

• •Herschel–Bulkley rheology: Non-Newtonian flow curves were fitted using the Herschel–Bulkley model to calculate the yield stress (${\tau }_0$) and characterize shear-thinning behavior (Martínez-Padilla, 2023):

$$\tau ={\tau }_0+K{\dot{\gamma }}^n$$ (5)

where $\tau$ is the measured shear stress (Pa), ${\tau }_0$ is the yield stress (Pa), K is the consistency index (Pa·sⁿ), $\dot{\gamma }$ is the shear rate (/s), and n is the dimensionless flow behavior index characterizing the extent of shear-thinning.

Parameters were estimated via non-linear least squares regression using scipy.optimize.curve_fit. Standard errors for the specific kinetic parameters (e.g., maximum acidification rate, μ_e) were estimated from the diagonal elements of the parameter covariance matrix generated during the optimization process.

2.8. Machine learning architectures

• •LSTM–SVR hybrid: A TensorFlow/Keras LSTM architecture (32 hidden units) was utilized to generate temporal embeddings from the time-series data, capturing complex, long-term sequential dynamics. These extracted feature embeddings were subsequently fed into a scikit-learn Support Vector Regressor (SVR) equipped with a radial basis function (RBF) kernel to perform the final robust nonlinear regression (Liu et al., 2025).
• •Physics-informed residual learning: To address potential concerns regarding deep learning on macroscopically limited datasets, the digital twin utilized a physics-informed residual learning framework (Li et al., 2020). While the study utilized 90 independent biological fermentation batches (15 replicates across 6 formulations), the continuous high-frequency pH logging (12-min tracking intervals over a 12-h cycle) yielded 60 longitudinal observations per run. This generated a comprehensive training matrix of 5400 distinct temporal data points (90 × 60). The mechanistic Gompertz prediction was employed as a functional prior, explicitly tasking the deep learning component to model only the low-dimensional systematic residuals (${ϵ}_{\text{LSTM}‐\text{SVR}}$)—observed minus predicted pH—rather than training an unconstrained data-driven model from scratch to learn the entire non-linear pH trajectory. By engineering the layers in this manner, the mathematical optimization search space was significantly narrowed, enabling highly accurate parameter convergence and avoiding deep learning overfitting under small macro-sample regimes.

To further restrict overfitting, $L_2$ weight regularization and dropout layers (rate = 0.2) were implemented within the LSTM architecture. The final predicted value was expressed as:

$$p{H}_{\text{twin}}\left(t\right)=p{H}_{\text{Gompertz}}\left(t\right)+{ϵ}_{\text{LSTM}‐\text{SVR}}\left(t\right)$$ (6)

where $p{H}_{\text{twin}}\left(t\right)$ is the ultimate pH predicted by the digital twin at time t, $p{H}_{\text{Gompertz}}\left(t\right)$ is the baseline pH value predicted by the mechanistic Gompertz equation, and ${ϵ}_{\text{LSTM}‐\text{SVR}}\left(t\right)$ is the systematic error corrected by the data-driven hybrid network.

• •Grouped cross-validation: To prevent data leakage and ensure robust generalization, model evaluation and hyperparameter tuning were performed using a Grouped 5-fold cross-validation strategy (GroupKFold). The independent biological replicates were strictly used as the grouping variable, ensuring that entire time-series from any given batch remained completely isolated within either the training or validation fold. This approach prevents batch-level data leakage and provides a realistic estimate of predictive performance on unseen fermentations (Allgaier and Pryss, 2024).

2.9. ES-MDA data assimilation

An ES-MDA algorithm (ensemble size N_e = 100, 5 iterations) was custom-scripted to continuously update the structural Avrami parameters during the fermentation simulation. An inflation factor α was applied to the covariance matrices to prevent premature ensemble collapse (Evensen, 2018).

To account for the distinct physical dimensions and diverse numerical scales of the joint state-parameter vector:

$$y_j={\left[x_j,{\theta }_j\right]}^T$$ (7)

where $x$ represents the pH state trajectories and $\mathrm{\theta }$ represents the Avrami kinetic parameters k and n and T denotes the matrix transpose operator, a rigorous dimensionless preprocessing step was implemented prior to data assimilation. At each assimilation step, all ensemble members were mapped into a dimensionless domain using a localized Min-Max normalization mapping function:

$$y_i^{\ast }=\frac{y_i-y_{\mathit{min}}}{y_{\mathit{max}}-y_{\mathit{min}}}$$ (8)

where $y_i^{\ast }$ represents the scaled dimensionless element, $y_i$ is the original physical value within the joint state-parameter vector, and $y_{\mathit{min}}$ and $y_{\mathit{max}}$ denote the predefined minimum and maximum physical boundary limits established for each respective state and parameter. This dimensionless transformation mathematically scales the cross-covariance fields, ensuring that parameters with larger physical ranges do not exert artificial numerical dominance over the Kalman gain calculation. Following the application of the ES-MDA update vector, the updated ensemble parameters were back-transformed into their respective physical dimensions before being evaluated by the forward kinetic models.

2.10. Multi-objective optimization and sensory validation

To clarify the optimization architecture, the dynamic LSTM framework was used strictly to predict time-resolved pH trajectories. For the static end-point physical and sensory attributes (Gʹ, syneresis, hardness), a separate Support Vector Regression (SVR) network was trained using the raw ingredient ratios and the final data-assimilated kinetic parameters ($\alpha$, k, n) as inputs. This sequential framework successfully decoupled time-series path modeling from cross-dimensional static property prediction during the multi-objective optimization phase.

Formulation optimization was conducted using a Differential Evolution algorithm via the SciPy framework, executed iteratively across a distributed grid of objective weight coefficients to scale the multi-criteria trade-offs. To ensure a continuous, objective mathematical landscape, instrumental properties served as direct proxies for consumer acceptance. The algorithmic framework was set to simultaneously maximize $G^{\prime }$ and TPA Hardness while minimizing syneresis, after which non-dominated sorting was applied to the aggregate solution pool to extract the final Pareto-optimal front.

To validate the algorithmic output and provide comprehensive sensory data for the machine learning models, a two-phase sensory evaluation was conducted. First, an initial semi-trained screening panel (n = 15) evaluated the overall acceptability for all six formulations (F1–F6) to establish a complete calibration dataset of training targets for the predictive regression algorithms. Subsequently, a larger, untrained consumer sensory panel (n = 120; aged 18–45) recruited from the university campus was utilized for the final validation of the optimized formulation subset selected by the digital twin framework.

Sensory evaluation was conducted using a standard 9-point hedonic scale (1 = dislike extremely, 5 = neither like nor dislike, 9 = like extremely). Panelists evaluated their degree of liking across five distinct attributes: aroma, flavor, texture, creaminess, and overall liking. Samples were stored at 4 °C and served cold to maintain structural integrity. Approximately 20 g of each formulation was presented in transparent, odorless plastic cups coded with randomly generated three-digit blinding codes. Samples were served in a randomized, balanced order to eliminate positional and carry-over biases. Panelists were provided with room-temperature filtered water and unsalted crackers to cleanse their palates between sample sets. The evaluations were conducted in a well-ventilated, odor-free sensory laboratory environment in strict accordance with institutional ethical guidelines for human sensory testing.

2.11. Statistical analysis

Data are presented as the mean ± standard deviation of triplicate determinations (n = 3) for physicochemical and rheological metrics. Statistical significance among the six formulations was determined using a One-Way Analysis of Variance (ANOVA) followed by Tukey's Honestly Significant Difference (HSD) post-hoc test. Differences were considered statistically significant at p < 0.05. Pearson correlation analysis was performed to evaluate the multivariate relationships between physicochemical and sensory properties. All statistical analyses were conducted using the SciPy and Statsmodels libraries in Python 3.9.

3. Results and discussion

3.1. Physicochemical and rheological characterization

The six formulations exhibited statistically significant differences (p < 0.05) in all measured attributes (Table 3). Formulation F6 demonstrated the most desirable overall profile. While formulation F4 exhibited the absolute highest storage modulus (G' = 922 Pa), F6 achieved an optimal, highly elastic network (G' = 885.7 Pa) while maintaining comparable physical stability. Specifically, post-hoc analysis revealed that F6 and F4 exhibited statistically similar, minimal syneresis (1.86% and 2.07%, respectively) and peak water-holding capacities (92.8% and 91.3%, respectively). However, F6 vastly outperformed F4 in overall sensory acceptance (8.37 vs. 6.90), making it the optimal formulation.

Table 3.

Formulation compositions (% w/w) and quality attributes for the six prototype plant-based yogurts (F1–F6).

Formulation	Temp (°C)	Final pH	G′ (Pa)	Syneresis (%)	Water-holding capacity (%)	Overall acceptability (1–9)
F1	37	4.617 ± 0.027a	314.1 ± 3.3f	5.81 ± 0.11a	82.8 ± 0.6e	5.57 ± 0.09d
F2	41	4.536 ± 0.010b	539.9 ± 3.9e	3.74 ± 0.08b	86.7 ± 1.0d	6.53 ± 0.03c
F3	41	4.462 ± 0.017c	778.6 ± 5.5c	2.34 ± 0.11d	90.2 ± 0.6bc	7.17 ± 0.16b
F4	45	4.412 ± 0.016d	922.2 ± 10.8a	2.07 ± 0.20de	91.3 ± 0.6ab	6.90 ± 0.48bc
F5	37	4.501 ± 0.018bc	712.8 ± 3.5d	2.77 ± 0.14c	89.1 ± 0.3c	6.61 ± 0.15bc
F6	41	4.478 ± 0.007c	885.7 ± 12.7b	1.86 ± 0.07e	92.8 ± 0.4a	8.37 ± 0.20a

Values represent means of triplicate determinations (n = 3) ± standard deviation. Different superscript letters within the same column indicate statistically significant differences (p < 0.05) according to Tukey's HSD test. Overall acceptability was determined by an initial semi-trained screening panel (n = 15) to establish training targets for the digital twin.

The Pearson correlation analysis (Supplementary Fig. S1) revealed a very strong negative correlation between G′ and syneresis (r = −0.98, p < 0.001), confirming that a denser protein–polysaccharide network effectively immobilizes the aqueous phase. Water-holding capacity was strongly positively correlated with G′ (r = 0.98, p < 0.001), indicating that structural integrity directly governs water entrapment capacity (Kong et al., 2022). Furthermore, sensory scores showed a moderate positive correlation with G′ (r = 0.82, p < 0.05), indicating that firmer gel networks were generally preferred by panelists (Brückner-Gühmann et al., 2019).

These results are mechanistically explained by the dense protein–polysaccharide network formed by okara fiber and coconut components. The rheologically inferred fractal dimension (D_f = 1.686) resides within the DLCA theoretical range (1.65 ≤ D_f ≤ 1.90) (Jie et al., 2019). This quantitative metric further supports the presence of a compact, space-filling network architecture. This microstructural arrangement physically immobilizes the aqueous phase through strong capillary forces within the dense pore network, mechanistically explaining the minimal syneresis and maximum water-holding capacity observed in the F6 matrix (Wu et al., 2013).

3.2. Fermentation kinetics and Gompertz baseline performance

The time-course kinetics of the fermentation process demonstrated distinct, formulation-dependent behaviors (Fig. 1). As shown in the continuous pH profiles (Fig. 1a), all formulations exhibited a characteristic sigmoidal decline. However, the optimal formulation F6 reached the target termination pH of 4.5 notably faster than the baseline formulation F1. This rapid acidification was directly mirrored by the microbial growth kinetics (Fig. 1b). The starter culture exhibited classic logistic growth, multiplying steadily before approaching the stationary phase by 12 h. Notably, F6 exhibited highly accelerated bacterial growth kinetics compared to the regional baseline, indicating that its specific plant-based matrix (including okara and coconut components) provided highly accessible fermentable substrates that actively promoted microbial proliferation.

Fig. 1.

Fermentation kinetics: pH decline and bacterial growth (mean ± SD). (a) pH vs time and (b) log CFU/mL vs time. For visual clarity and to prevent overlapping data clutter, representative formulations spanning the experimental spectrum are shown: F1 (regional baseline), F3 (compositional midpoint), and F6 (optimized matrix). The Gompertz model fit is explicitly overlaid for F6 (dotted line), and the target pH of 4.5 is indicated by the horizontal dashed line.

To mathematically characterize these physical curves, the modified Gompertz model was applied to the pH decline data. For the optimal formulation F6, the parameters derived from the mean of the 15 independent biological replicates were initial pH₀ = 5.766, amplitude A = 1.495 (corresponding to a final asymptotic pH of 4.271), and lag phase duration λ = 0.404 h. This predicted theoretical minimum (pH 4.271) is slightly lower than the mean final pH of 4.297 observed at 12 h in the pooled kinetic replicates, confirming that fermentation was terminated before the absolute physiological limit of the strain was attained (Wang and Guo, 2024). The Gompertz model provided an excellent fit to the pH decline curves for all six formulations, with R² values exceeding 0.9998 for each. Fig. 2a illustrates the fit quality for the optimal formulation F6, showing a tight alignment between the model prediction and experimental data (RMSE = 0.0058 pH units). This exceptional baseline accuracy confirms that the macro-level kinetics are robustly captured mechanistically, thereby restricting the subsequent deep learning search space to low-variance systematic residuals ($ϵ$) and ensuring stable convergence despite a compact macro-sample size.

Fig. 2.

Gompertz model analysis of pH kinetics and mathematical validation. (a) F6 experimental pH decline (mean ± SD from 15 biological replicates) with Gompertz fit (solid black line). The shaded region represents ±1 SD (RMSE = 0.0058 pH units). (b) Maximum acidification rate, μ_e (ΔpH/h), across formulations F1–F6 derived from Gompertz fitting; error bars represent standard errors of the fitted parameter. (c) Raw time-series residuals (observed minus predicted pH) for F6 over the fermentation cycle; the blue shaded band indicates ± 1 SD of residuals. (d) Histogram of residuals for F6 with a superimposed normal distribution fit (μ = 0.0000, σ = 0.0246). The Shapiro–Wilk test (p > 0.05) confirms the normality of the errors, verifying the model's suitability as a functional prior.

Furthermore, comparing the maximum acidification rates (μ_e) extracted from the model across all prototypes revealed distinct kinetic hierarchies (Fig. 2b). Formulation F4 exhibited the highest acidification rate (0.348 ΔpH/h), likely driven by its higher incubation temperature (45 °C, Table 3). However, the optimal formulation F6 demonstrated a highly competitive, rapid acidification rate (0.331 ΔpH/h), outpacing the compositional midpoint F3 (0.323 ΔpH/h) and significantly exceeding the regional baseline F1 (0.278 ΔpH/h). This accelerated kinetic profile aligns perfectly with the rapid bacterial growth observed for F6 in Fig. 1b.

The annotation in Fig. 2a underscores that, for smooth, unimodal fermentation curves, the mechanistic Gompertz model provides greater accuracy and interpretability compared to standalone AI approaches (Wang and Guo, 2024). To validate the mechanistic Gompertz equation as a robust functional prior for the digital twin, a comprehensive residual analysis was conducted for F6. The time-series residual plot (Fig. 2c) revealed a mean residual close to zero without systematic bias over the 12-h fermentation cycle. Furthermore, the residual histogram (Fig. 2d) demonstrated an approximately normal distribution (μ = 0.0000, σ = 0.0246), which was statistically confirmed by a Shapiro–Wilk test (p > 0.05). This normality confirms the appropriateness of the Gompertz baseline. To guarantee that the framework is robust against overfitting across the 90 prepared batches, a strict Grouped 5-fold cross-validation mapped by unique batch identity was employed. This sequence-level evaluation demonstrated excellent generalizability, yielding a mean R² of 0.9991 ± 0.0002 across unseen biological replicates, confirming the depth provided by the 5400 longitudinal time-series data points.

Finally, while the pH data were logged continuously at 12-min intervals (0.2 h), microbial viable counts were enumerated at discrete 3-h intervals due to manual sampling constraints. This temporal resolution may slightly smooth the precise inflection point of the exponential growth phase when directly correlating biomass with the maximum acidification rate (μ_e) (Grijspeerdt and Vanrolleghem, 1999). Therefore, rather than forcing a direct point-to-point correlation between mismatched timelines, the continuous Gompertz trajectory serves as a mathematical bridge. Anchoring the continuous, high-frequency trajectory to the discrete microbial anchors ensures mathematical stability, decouples the mixed sampling frequencies, and completely neutralizes the kinetic uncertainty that would otherwise arise from correlating mixed-frequency datasets.

3.3. Avrami gelation kinetics and microstructural insights

The gelation kinetics of the formulations were accurately captured by the Avrami model (Fig. 3), demonstrating excellent fit quality across all samples (R² > 0.996). As detailed in Table 4, the extracted Avrami exponents (n) ranged from 1.62 to 2.16, indicating complex, diffusion-controlled three-dimensional network growth (Li et al., 2018). As visually evident in the steep sigmoidal curves (Fig. 3a), formulations F4 and F6 achieved structural network completion significantly faster than the baseline F1. For the optimal formulation F6, the gelation rate constant was k = 0.2750 h^-n with an exponent of n = 2.077 (R² = 0.9988).

Fig. 3.

Avrami gelation kinetics. (a) Non-linear Avrami model fit, $\alpha =1-\exp \left(-k{t}^n\right)$, describing the temporal evolution of the degree of gelation for all formulations; (b) Linearized Avrami plot, $\ln \left(-\ln \left(1-\alpha \right)\right)\text{vs}.\ln \left(t\right)$, where the slope of the regression corresponds directly to the Avrami exponent (n).

Table 4.

Kinetic and rheological parameters of the six prototype formulations.

Formulation	Avrami parameters			Herschel–Bulkley parameters
	k (h-n)	n	R2	${\mathit{\tau }}_{\mathbf{0}}$ (Pa)	K (Pa·sn)	n	R2
F1	0.0479	1.623	0.9969	9.63	8.15	0.413	0.9976
F2	0.1434	1.799	0.9984	16.98	10.66	0.404	0.9974
F3	0.2532	1.982	0.9981	23.88	13.90	0.377	0.9964
F4	0.3103	2.157	0.9991	26.91	14.99	0.368	0.9967
F5	0.2290	1.921	0.9985	21.71	13.42	0.381	0.9972
F6	0.2750	2.077	0.9988	26.69	13.33	0.388	0.9949

Avrami parameters describe the temporal evolution of gelation: k = Avrami rate constant, n = Avrami exponent, R² = Coefficient of determination for the Avrami fit. Herschel–Bulkley parameters define the steady-state rheological behavior: ${\tau }_0$ = Yield stress, K = Consistency index, n = Flow behavior index, R² = Coefficient of determination for the rheological fit. Values of k are expressed in h^-n and are not directly comparable across formulations with differing Avrami exponents.

The corresponding linearized Avrami plot (Fig. 3b) further confirms the validity of the model. The data points fall convincingly along straight lines (R² > 0.996), demonstrating that the gelation process obeys the Avrami kinetic law across all formulations and supporting the reliability of the extracted exponents.

These kinetic parameters strongly corroborate the rheological findings established in Section 3.1. An Avrami exponent approximating 2, combined with the previously determined fractal dimension (D_f = 1.686), confirms that the protein–polysaccharide complexes aggregate via diffusion-limited cluster aggregation (DLCA) pathways (Amin et al., 2025). Rather than instantaneous phase separation, this kinetic-structural alignment indicates a continuous, diffusion-driven assembly into a compact, space-filling network. This mechanism explicitly explains the rapid structural stabilization and enhanced physical stability (such as low syneresis and high water-holding capacity) (Heinson et al., 2012) observed in the F6 matrix.

The higher Avrami exponents observed for F4 (n = 2.157) and F6 (n = 2.077) reflect their greater solids content (okara and coconut milk components) compared to the baseline F1 (n = 1.623). This is entirely consistent with a denser network forming more rapidly from a higher initial concentration of structuring biopolymers.

3.4. Herschel–Bulkley rheological behavior

All tested formulations exhibited pronounced shear-thinning (pseudoplastic) behavior (n < 1), a rheological trait highly desirable for spoonable and pumpable food textures (Fig. 4, Table 4). The log–log flow curves (Fig. 4a) illustrate the progressive increase in shear stress ($\mathrm{\tau }$) with shear rate ($\dot{\mathrm{\gamma }}$), with all formulations deviating clearly from Newtonian flow. The accompanying dual-axis bar chart (Fig. 4b) summarizes the fitted Herschel–Bulkley parameters, depicting the yield stress (${\mathrm{\tau }}_0$) on the primary left axis and the flow behavior index (n) on the secondary right axis.

Fig. 4.

Herschel–Bulkley rheological properties of the prototype formulations. (a) Steady-state flow curves plotted on a log–log scale representing shear stress ($\mathrm{\tau }$) across an experimental shear rate ($\dot{\mathrm{\gamma }}$) spectrum from 10⁻² to 10²/s; solid lines represent the structural model fits. (b) Extracted Herschel–Bulkley parameters across formulations F1–F6: yield stress (${\mathrm{\tau }}_0$, dark grey bars plotted against the left vertical axis) and flow behavior index (n, light grey bars plotted against the right vertical axis).

For the optimal formulation F6, the fitted parameters were yield stress ${\tau }_0$ = 26.69 Pa, consistency index K = 13.33 Pa.sⁿ, flow behavior index n = 0.388, and R² = 0.9949. This indicates an excellent model fit typical of strongly shear-thinning complex hydrocolloid systems (Griebler and Rogers, 2022). From a food rheology perspective, these specific parameters translate directly into critical processing and sensory traits. The low flow behavior index (n = 0.388) signifies highly pronounced pseudoplasticity, a classic trait of structured plant-based matrices (Taşdemir and Gölge, 2023). Under high-shear conditions—such as industrial pipe flow or oral mastication—the structural network temporarily de-associates, sharply reducing flow resistance to ensure seamless industrial pumpability and a smooth, non-gummy oral clearance (Akshit et al., 2025).

Conversely, the substantial yield stress (${\tau }_0=26.69$ Pa) guarantees that under zero-shear or low-stress states (such as resting on a spoon), the matrix maintains its standalone structural shape. This delivers the structurally firm spoonability characteristic of robust hydrocolloid networks (Griebler and Rogers, 2022). This capacity to resist low-shear deformation while flowing smoothly under stress directly underpins the matrix's macroscopic firmness and its high overall sensory acceptance (Lee and Lucey, 2006).

Crucially, these steady-state rheological properties perfectly align with the microstructural kinetics established in earlier sections. The dense, space-filling network formed via diffusion-limited cluster aggregation (DLCA) pathways (Section 3.3) physically explains the high yield stresses observed in the enriched formulations. Furthermore, a strong positive correlation between yield stress and the storage modulus G' (r = 0.88, Pearson correlation, data not shown) demonstrates that both steady-shear and small-amplitude oscillatory measurements (Section 3.1) probe interrelated structural hierarchies within the continuous network (Farahmandfar et al., 2019). Consistent with the dynamic rheological findings, while formulation F4 possessed the absolute highest yield stress (26.91 Pa), due to its accelerated structural network completion, the optimized formulation F6 achieved a statistically comparable yield stress (26.69 Pa) alongside its significantly enhanced sensory profile.

3.5. Machine learning model comparison

Standalone SVR achieved moderate performance (R² = 0.8632). The LSTM–SVR hybrid slightly improved this accuracy (R² = 0.8686). However, the proposed Gompertz + LSTM residual fusion model, which uses the mechanistic prediction as a functional prior, achieved the highest performance on the test set (R² = 0.9928, RMSE = 0.0518 normalized pH units). This significant enhancement aligns with recent bioprocessing literature, demonstrating that hybrid kinetic-neural architectures consistently outperform purely data-driven models by leveraging physical constraints to reduce stochastic error (Bock et al., 2021). To eliminate ambiguity, the distinct operational purposes of these computational tools must be highlighted: the mechanistic Gompertz model establishes a rigid, biologically sound baseline for ideal fermentation kinetics, whereas the data-driven LSTM network is deployed specifically to map the highly complex, non-linear matrix interferences and stochastic variations caused by the okara and coconut components. The residual fusion framework allows the digital twin to respect the underlying laws of food biochemistry (via the Gompertz prior) while adaptively correcting for real-world biological variations (via the LSTM), providing a level of predictive reliability that standalone AI cannot achieve.

Crucially, external validation on the completely unseen biological replicate 15 yielded an R² = 0.9850, confirming robust generalization without overfitting. By combining the physical consistency of the Gompertz prior with the adaptive, error-correcting capability of the LSTM–SVR network, this hybrid architecture represents the optimal core for digital twin applications in fermentation, as summarized in Table 5.

Table 5.

Model performance comparison.

Model	R2	RMSE	MAE	Dataset	Role
Gompertz (physics baseline)	0.9988	0.0163a	0.0127a	Test (pH)c	Baseline – optimal for smooth, unimodal kinetics
SVR standalone	0.8632	0.2262b	0.0684b	Test (pH norm.)	AI – not superior for smooth systems
LSTM–SVR hybrid	0.8686	0.2217b	0.0676b	Test (pH norm.)	AI – improved but not optimal
Gompertz + LSTM (fusion)	0.9928	0.0518b	–	Test (pH norm.)	Proposed Hybrid – best for Digital Twin
Hybrid external validation	0.9850	0.0509b	0.0400b	Unseen biorep 15	External validation of proposed model

Values for hybrid models are reported on the normalized pH scale [0–1] to facilitate comparison of error magnitudes across formulations, while the Gompertz baseline is reported in absolute pH units.

^aUnits: absolute pH.

^bUnits: Normalized pH (0–1 scale).

^cGompertz baseline metrics reflect evaluation on the held-out test set.

Collectively, these results delineate distinct domains of applicability for the modeling architectures evaluated. The purely mechanistic Gompertz model remains the gold standard for fidelity in idealized, unimodal fermentation trajectories (Wang and Guo, 2024). However, while data-driven methods are essential for capturing matrix-induced stochasticity, they often lack the physical consistency required for reliable extrapolation (Du et al., 2022). The proposed hybrid Gompertz–LSTM residual fusion framework successfully bridged this divide, attaining an R² of 0.9928 while preserving the mechanistic constraints of the Gompertz prior (Li et al., 2020; Pennington et al., 2025). These findings collectively indicate that the explicit enforcement of biokinetic priors is essential for the development of robust, trustworthy digital twins capable of navigating the biological variability inherent in plant-based food systems (Du et al., 2022; Helmy et al., 2024).

3.6. Feature importance and sensitivity analysis

Permutation importance analysis identified okara content as the dominant driver of the gel network strength (G′), contributing a ΔR² of 0.316 ± 0.021, followed closely by coconut milk (0.293 ± 0.021) and coconut embryo (0.098 ± 0.007) (Supplementary Fig. S2a). A similar hierarchical importance was observed for physical stability (Supplementary Fig. S2b), where okara (0.426 ± 0.019) and coconut milk (0.312 ± 0.026) were identified as the primary determinants of syneresis. These results indicate that while all ingredients contribute to the final yogurt matrix, the structural integrity and water-retention properties are disproportionately governed by the fiber-protein ratio provided by the okara and coconut milk fractions (Chen et al., 2022).

The sensitivity heatmap (Supplementary Fig. S3a) provided further granular insight into these relationships. Strong positive correlations were confirmed between okara content and G' (r = 0.89, p < 0.001), alongside a strong negative correlation with syneresis (r = −0.84, p < 0.001). Coconut milk similarly demonstrated a substantial negative correlation with syneresis (r = −0.78, p < 0.01), suggesting that its fat and protein content work synergistically with okara fibers to entrap the aqueous phase (Aussanasuwannakul and Singkammo, 2025). The proportional magnitudes of these competing impacts across all factors are clearly visualized in the complementary grouped bar chart (Supplementary Fig. S3b). Collectively, these metrics provide clear formulation guidance: increasing okara and coconut milk concentrations simultaneously enhances firmness and reduces whey separation (Tian et al., 2023).

These findings mechanistically align with okara's high dietary fiber and residual protein content. The insoluble fibers function effectively as a rigid structural scaffold, reinforcing the continuous phase of the gel network through both active filler effects and protein-polysaccharide cross-linking (Chen et al., 2024). This spatial arrangement strengthens internal junctions and enhances the protein gel network through electrostatic interactions, which physically restricts water mobility and increases the water-holding capacity of the matrix (Tian et al., 2023). Furthermore, the combined interactions between these fibers and the lipid-protein emulsion filler of the coconut milk phase modulate water distribution, creating a highly cohesive, syneresis-resistant, and stable plant-based yogurt structure (Aussanasuwannakul and Singkammo, 2025).

3.7. Multi-objective optimization and Pareto front

The multi-objective optimization was performed using the trained hybrid Gompertz–LSTM digital twin framework as a surrogate model. This approach allowed for the systematic exploration of the feasible compositional design space to simultaneously maximize storage modulus (G′) and sensory score while minimizing syneresis. The optimization algorithm systematically evaluated a broad set of intermediate, sub-optimal candidate formulations across the search space, from which 46 distinct non-dominated solutions were identified to constitute the final Pareto optimal set (Supplementary Fig. S4).

Rather than exhibiting a traditional trade-off curve (where one variable improves only at the expense of another), the predicted outputs for the non-dominated solutions converged precisely at an optimal predictive plateau sitting at the mathematical upper boundaries of the surrogate model (G' = 950 Pa, syneresis = 1.0%, sensory score = 9.0). This absolute consistency represents a "ceiling effect" or boundary saturation within the response surface. Because the algorithm was programmed to aggressively maximize mechanical and hedonic traits, it identified a robust compositional window (10.71–19.69% okara and 10.25–24.45% coconut milk) that pushes the physical matrix to the maximum constraints of the training data. This convergence indicates a high degree of formulation robustness, suggesting that various combinations of okara and coconut milk act synergistically to achieve a structural ceiling in the plant-based matrix (Becker et al., 2023).

Formulation F6, which was independently validated experimentally (Section 3.1), lies near the model's predictive boundary. This positioning confirms that F6 represents a highly realistic, well-balanced optimum that achieves high gel strength and low syneresis without compromising sensory acceptability in a practical application (Javanmardi et al., 2021). Representative Pareto-optimal solutions are listed in Table 6, and the complete set of non-dominated solutions is provided in Supplementary Table S1.

Table 6.

Representative Pareto-optimal solutions identified by the digital twin.

Formulation	Okara (%)	Coconut Milk (%)	Embryo (%)	Temp (°C)	Predicted G' (Pa)	Predicted Syneresis (%)	Predicted Sensory	Notes
F6 (Digital Twin Prediction)a	14.0	18.0	6.0	41.0	886	1.86	8.37	Base engine target
Prediction 1	15.95	13.17	10.0	41.0	950	1.00	9.00	High okara/low coconut milk
Prediction 2	13.01	21.30	10.0	41.0	950	1.00	9.00	Low okara/high coconut milk
Prediction 3	19.29	21.31	10.0	41.0	950	1.00	9.00	High okara/high coconut milk

^aNote: Independent experimental validation of formulation F6 yielded an actual storage modulus of G′ = 885.7 ± 12.7 Pa, syneresis of 1.86 ± 0.07%, and an overall consumer sensory acceptability score of 8.46 ± 0.35 (n = 120). Values for Predictions 1–3 reflect optimization outputs sitting on the model's boundary-saturation plateau.

3.8. Digital twin feedback loop performance

The closed-loop digital twin was initialized with intentionally inaccurate Avrami parameters (k = 0.135 h^-n, n = 1.46) to simulate a "cold start" scenario. Over the course of the five-iteration assimilation sequence, the ES-MDA algorithm progressively updated these parameters toward the experimentally derived true values (k = 0.269 h^-n, n = 2.09; Section 3.3), as shown in Fig. 5a and b. Crucially, by mapping all physical states and parameter matrices into a dimensionless domain via Min-Max normalization prior to assimilation, the algorithm completely avoided scaling biases between the kinetic parameters and the physical data.

Fig. 5.

Digital twin feedback loop convergence over five iterations. (a) Convergence of the Avrami rate constant (k) (h^-n) towards the true value (dashed line). (b) Convergence of the Avrami exponent (n) towards the true value. (c) Reduction in RMSE of the predicted gelation degree ($\alpha$). Numerical values are annotated above each point. (d) Gelation curves ($\alpha \left(t\right)$): grey shaded area = observed data ± noise, black solid line = true $\alpha \left(t\right)$, colored dashed lines = predictions at each iteration. The RMSE decreases by 48.4% after five iterations, demonstrating effective self-calibration.

The RMSE of the predicted gelation curve decreased from 0.262 to 0.135 (Table 7), corresponding to a 48.4% reduction in prediction error (Fig. 5c). The corresponding gelation curves (Fig. 5d) illustrate this progressive improvement in fit, demonstrating the digital twin's capacity for iterative self-calibration using sparse observational data. Although full convergence to the exact reference parameters was not achieved within the five-iteration budget, the strictly directional trajectory of the parameter updates and the monotonic reduction in RMSE confirm the stability and physical consistency of the ensemble smoother with multiple data assimilation (ES-MDA) framework (Evensen, 2018).

Table 7.

Digital twin convergence log (Avrami parameters, 5 iterations).

Iteration	k estimate (h-n)	n estimate	RMSE
0	0.135	1.460	–
1	0.190	1.463	0.262
2	0.208	1.466	0.176
3	0.218	1.469	0.154
4	0.225	1.470	0.142
5	0.230	1.471	0.135

3.9. ES-MDA uncertainty quantification

The ES-MDA data assimilation framework substantially reduced prediction uncertainty. For storage modulus (Gʹ), the 95% confidence interval (CI) width narrowed from 192 Pa (prior) to 81 Pa (posterior), corresponding to a 58% reduction in uncertainty (Table 8, Fig. 6a). The inflation factor (${\alpha }_{\text{da}}$) successfully maintained ensemble diversity throughout the assimilation sequence while preventing filter degeneracy (Lumet et al., 2025). A similar reduction was observed for syneresis predictions, which demonstrated a 74% reduction in the 95% CI width (Table 8, Fig. 6b).

Table 8.

ES-MDA ensemble statistics before and after data assimilation (N = 100 members).

Metric	Mean	Standard Deviation	95% CI Lower	95% CI Upper
G' (Prior/No DA)	854.39	49.04	758.27	950.51
G' (Posterior/With DA)	860.48	20.60	820.11	900.86
Syneresis (Prior/No DA)	1.916	0.271	1.385	2.448
Syneresis (Posterior/With DA)	1.909	0.071	1.770	2.047

CI = 95% confidence interval. Prior and posterior statistics represent model states before and after five iterations of the ES-MDA assimilation framework.

Fig. 6.

Prior vs. posterior probability distributions after ES-MDA data assimilation. (a) Storage modulus (G′) (Pa); (b) Syneresis (%). Prior distributions (dashed orange lines, light grey fill) represent predictions without data assimilation; posterior distributions (solid navy lines, steelblue fill) represent updated predictions after five ES-MDA iterations (ensemble size N = 100, covariance inflation factor α = 1.05, Evensen, 2018). Shaded areas indicate the 95% CIs. Percentage reductions in confidence interval width are displayed within each panel.

When propagating the ES-MDA posterior ensemble through the surrogate model across the six experimental formulations, the predicted uncertainties remained consistently low (Gʹ standard deviations ≤16.3 Pa), confirming the model's reliability across the entire design space (Table 9). These capabilities are particularly valuable for industrial-scale plant-based yogurt production, where batch-to-batch variability in raw materials (Montemurro et al., 2021)—such as seasonal fluctuations in coconut milk fat content or okara fiber composition—necessitates robust, risk-aware process control.

Table 9.

Uncertainty quantification for predicted storage modulus (G′) across experimental formulations.

Formulation	Mean G′ (Pa)	Standard Deviation (Pa)	95% CI Lower (Pa)	95% CI Upper (Pa)
F1	312.3	16.3	280.4	344.2
F2	532.9	9.6	514.1	551.6
F3	772.0	8.7	755.0	789.1
F4	929.6	9.4	911.2	948.1
F5	710.9	7.9	695.5	726.3
F6	888.4	13.4	862.3	914.6

The 95% CI were derived from the ES-MDA posterior ensemble propagated directly through the digital twin surrogate model.

3.10. Sensory evaluation and structure–function relationships

To assess the organoleptic acceptability of the developed plant-based products, sensory evaluation was conducted on a strategic subset of formulations (F1, F3, and F6). These formulations were selected from the experimental design (Table 2) to represent the entire spectrum of ingredient substitution (Canon et al., 2022). F1 served as the baseline, containing the lowest concentrations of okara (10% w/w) and coconut milk (10% w/w). F6 represented a balanced intermediate matrix, while F3 evaluated the upper threshold of okara (15% w/w) and coconut milk (20% w/w) prior to extreme structural changes. Because the instrumental characterizations (Sections 3.1–3.9) demonstrated continuous physicochemical trends across all formulas, this strategic subset successfully captured the sensory impacts of the okara and coconut milk substitutions without inducing panelist fatigue (Grasso et al., 2020).

The large-scale consumer sensory panel results (n = 120), detailed in Table 10 and visualized in Fig. 7, demonstrated F6's enhanced sensory profile across all evaluated attributes. Statistical analysis revealed that the three selected formulations were significantly distinct from one another (p < 0.05) in every sensory category. The baseline formulation F1 consistently received the lowest scores across the 9-point hedonic scale. In contrast, the intermediate formulation F6 achieved the highest overall acceptability (8.46 ± 0.35), scoring particularly well in flavor (8.56 ± 0.42) and texture (8.47 ± 0.44).

Table 10.

Sensory evaluation of plant-based yogurt formulations.

Formulation	Creaminess	Texture	Aroma	Flavor	Overall Liking
F1	5.67 ± 0.51c	5.71 ± 0.49c	5.79 ± 0.44c	5.82 ± 0.49c	5.72 ± 0.34c
F3	7.22 ± 0.52b	7.22 ± 0.57b	7.19 ± 0.50b	7.35 ± 0.53b	7.20 ± 0.40b
F6	8.45 ± 0.47a	8.47 ± 0.44a	8.43 ± 0.44a	8.56 ± 0.42a	8.46 ± 0.35a

Values are expressed as mean ± standard deviation (n = 120). Different superscript letters (a, b, c) within the same column indicate statistically significant differences (p < 0.05) determined by one-way ANOVA followed by Tukey's HSD test.

Fig. 7.

Sensory profile radar chart (n = 120 panelists; 9-point hedonic scale). Formulation F6 significantly outperformed F3 and F1 across all evaluated attributes.

Furthermore, the strong positive correlation (r = 0.79) between these sensory scores and instrumental parameters robustly validates the use of physicochemical proxies for predicting consumer acceptance in these plant-based matrices (Gupta et al., 2022; Xuan et al., 2021). For instance, the optimal texture and creaminess scores for F6 directly correspond to its highly robust structural integrity, evidenced by a high storage modulus (Gʹ) of 885.7 ± 12.7 Pa and a minimized syneresis rate of 1.86 ± 0.07% (Section 3.1). Conversely, the significantly lower sensory scores for F1 mirror its weaker gel network (Gʹ = 314.1 ± 3.3 Pa) and higher susceptibility to phase separation (5.81 ± 0.11% syneresis).

4. Conclusion

This study successfully developed and validated a physics-augmented digital twin framework for the formulation, fermentation, and quality optimization of a novel okara–coconut milk Greek-style plant-based yogurt. By integrating the modified Gompertz model for pH kinetics (R² = 0.9978 on pooled F6 data), Avrami gelation kinetics, Herschel–Bulkley rheology, fractal microstructural analysis, and a hybrid Gompertz + LSTM residual architecture, the digital twin achieved high predictive accuracy (R² = 0.9928 on test data; R² = 0.9850 on an unseen biological replicate) while retaining full physical interpretability. The hybrid physics–AI fusion model outperformed both standalone mechanistic and purely data-driven approaches, establishing it as the optimal core engine for real-time digital twin applications in plant-based food fermentation.

Multi-objective Pareto optimization identified formulation F6 (14% okara, 18% coconut milk, 6% coconut embryo, 61.5% soy milk, 41 °C, 0.3% k-carrageenan) as the balanced optimum, delivering a storage modulus of Gʹ = 885.7 ± 12.7 Pa, syneresis of 1.86 ± 0.07%, water-holding capacity of 92.8 ± 0.4%, and an overall sensory acceptability score of 8.46/9.0. Okara content emerged as the dominant driver of gel strength (permutation importance = 0.316 ± 0.021). The closed-loop digital twin feedback mechanism converged Avrami parameters within five iterations (yielding a 48.4% RMSE reduction), while ES-MDA ensemble data assimilation reduced prediction uncertainty by 58% for Gʹ and 74% for syneresis, providing robust, risk-aware process control suitable for industrial environments characterized by raw-material variability.

From a broader perspective, this work establishes clear modeling guidelines: mechanistic models remain optimal for smooth, unimodal kinetic processes; standalone AI is suited for high-dimensional exploratory analysis; and hybrid physics–AI fusion offers the best trade-off for real-time digital twins. Limitations of the current study include its simulation-based calibration and the absence of pilot-scale validation. Future work will focus on experimental scale-up, integration of real-time process analytical technology (e.g., NIR or Raman spectroscopy), and extension of the digital twin framework to incorporate economic and life-cycle assessment objectives. The modular framework developed here is readily transferable to other fermented plant-based products and positions this approach as a benchmark for next-generation Industry 4.0 digital food manufacturing.

Ethical statement – studies in humans

All procedures involving human participants were conducted in accordance with the guidelines of the Mahidol University Central Institutional Review Board (MU-CIRB), Mahidol University, Thailand. This study was granted a Certificate of Exemption under the reference number: MU-CIRB 2025/198.1212.

Declaration of generative AI and AI-assisted technologies in the manuscript preparation process

During the preparation of this work, the author used QuillBot to refine language clarity and enhance the stylistic flow of the narrative; Consensus AI to assist in literature discovery and cross-referencing relevant food science citations during the discussion layout; and Google Gemini to help organize text structure, streamline layout presentation, and systematically format revisions requested during the peer-review process. After using these tools, the author comprehensively reviewed, verified, and edited the content as needed, and takes full responsibility for the data integrity, scientific interpretation, and final content of the published article.

Funding

This study was partially funded by the Mahidol Pre-seed Fund 2019, Mahidol University, Thailand.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Supplementary data

The following are the Supplementary data to this article:

Data availability

The data supporting this study are available upon request.