Hybrid soft sensor modeling for the wire section in tissue manufacturing: integrating first-principles and machine learning models

2.1 Papermaking and the role of the wire section

Papermaking is a continuous industrial process in which a dilute pulp suspension is progressively dewatered, formed, and dried into a continuous web of paper. A typical paper machine consists of several main sections: the headbox, wire section, press section, dryer section, creping unit (in tissue production), and the reel (Gullichsen and Paulapuro 1998). Among these, the wire (forming) section governs the initial development of key sheet characteristics – most notably basis weight and formation-that directly drive downstream quality, runnability, and energy efficiency. Recent industrial guidance and modeling studies emphasize that early dewatering dynamics and wet-end stability are decisive levers for overall performance (ABB Ltd 2025; Sjöstrand 2025; Sjöstrand and Bergström 2024; Valmet 2025). In the wire section, the stock delivered by the headbox is uniformly distributed onto a continuously moving wire, where rapid water removal occurs via gravitational drainage, foils, vacuum boxes, and suction elements. Contemporary models capture these mechanisms at engineering scale (e.g., compact relations for vacuum dewatering and TAD solids development), enabling their integration with estimation and control frameworks (Sjöstrand 2025; Sjöstrand and Bergström 2024). The process simultaneously sets fiber orientation and initial retention, both sensitive to mechanics (jet-to-wire, turbulence, vacuum loading) and chemistry (retention aids, charge/ash balance), which are monitored and stabilized in practice through retention and ash consistency measurement and control (ABB Ltd 2025; Valmet 2025). Modeling this section remains challenging due to limited direct sensors in the forming zone, strong multiphase interactions, and variability in furnish properties. Consequently, recent work increasingly combines physics-based relations with data-driven soft sensors to improve early quality observability and robustness under transients, leveraging temporal deep learning and attention mechanisms (Stanišić et al. 2024; Zhang et al. 2025). Tissue-focused machine learning studies further demonstrate how such hybrid and ML approaches can translate into tangible efficiency gains at machine level (Viitala et al. 2025; Zhang et al. 2021).

In this work, real industrial data from a tissue manufacturing plant was used to build and validate models for predicting basis weight in the wire section. The machine is equipped with a twin-wire former and operates without external vacuum during dewatering. The headbox delivers thick stock at a flow rate of 100,000 L/min, and the consistency of the suspension leaving the headbox ranges from 0.1 % to 1 %. After the wire section, the paper passes through a pressing section and a Yankee dryer, with the basis weight measured at the reel.

The available dataset spans seven months of production and includes five key process variables originally recorded at one-second intervals, but aggregated to 30 min intervals for analysis. Among them, the basis weight at the reel serves as the prediction target, while thick stock flow, fines content, moisture content, and wire speed act as key input variables. Due to measurement constraints and noise, the raw data underwent extensive cleaning using thresholding and median filtering methods.

Figure 1 illustrates the layout of a typical paper machine, with emphasis on the wire section and key dewatering components.

Figure 1:

Schematic overview of a tissue machine highlighting the wire section where basis weight and formation are initially developed.

2.2 First-principle modeling in papermaking

First-principle models describe process behavior based on fundamental conservation laws and physical principles. In the wire section, most FPMs use simplified mass balance equations to model water removal and estimate basis weight development over time (Cho 2005). These models are valued for their interpretability and alignment with physical reality, making them attractive for control and simulation tasks. However, their predictive accuracy is often limited by simplifying assumptions such as steady-state conditions, constant material properties, or ideal drainage behavior (Lennartsson et al. 2013). As a result, FPMs may struggle to capture nonlinear dynamics and variations caused by furnish composition or operational transients.

In the wire section of a paper machine, water removal and basis weight development are primarily governed by the dynamics of flow rate and consistency. The basis weight BW (in g/m²) can be approximated by:

Here, m ˙ s is the solid mass flow rate entering the press section, SR is the shrinkage rate of the wet web, v is the machine speed, w is the web width, and ϵ is a lumped efficiency factor accounting for retention and sheet consolidation.

The underlying mass and consistency balances are derived from the dynamic model proposed by (Cho 2005), they developed a comprehensive retention and dewatering model based on pilot paper machine data. The key equations used in our adapted model are summarized in Table 1.

To apply this model to a real industrial setting, modifications were necessary. Our plant features a twin-wire roll former operating without external vacuum, no filler or retention aid dosage, and a significantly higher machine speed (2,000 m/min vs. 40 m/min). Therefore, delay elements were removed, empirical retention functions simplified, and machine-specific constants updated to reflect current operating conditions. These changes preserve the core mass balance structure while increasing applicability to real-time industrial data.

2.3 Data-driven modeling approaches

Data-driven modeling, particularly with machine learning, has gained momentum for complex process modeling in papermaking. With the availability of high-resolution sensor data, ML techniques such as decision trees and ensemble methods can infer relationships between process variables and product properties without requiring explicit physical models. Gradient boosting approaches like XGBoost are particularly effective in industrial settings due to their ability to handle nonlinearities, missing data, and variable importance evaluation (Chen and Guestrin 2016).

Nevertheless, these models often lack physical interpretability and require large amounts of representative training data. Their reliability deteriorates when extrapolating beyond trained conditions or under sensor noise and drift, limiting trust in safety-critical applications.

2.4 Hybrid modeling approaches

Hybrid modeling approaches aim to combine the interpretability of FPMs with the flexibility and accuracy of ML. In papermaking and other process industries, such models are used when partial physical knowledge is available but insufficient to fully describe system dynamics (Hotvedt et al. 2021). Hybrid models are particularly beneficial in scenarios with nonlinear behaviors, unmeasured disturbances, or incomplete instrumentation.

Common hybrid model structures include:

1. Parallel hybrid models: The FPM and ML models run independently on the same inputs, and their outputs are fused via a fixed or dynamically adjusted weighting mechanism (Psichogios and Ungar 1992; Willard et al. 2022). This structure enables balancing physical consistency and data-driven correction under variable conditions.

2. Series hybrid models: One model (typically the FPM) provides intermediate predictions which serve as additional input features for the ML model (Sharma and Liu 2021). This cascaded approach lets the ML model learn context-aware corrections based on physical predictions.

3. Residual hybrid models: The ML model is trained on the residual error between the FPM output and the actual measurement. It acts as a corrective block to compensate for model mismatch or missing physics (Chaffart and Ricardez-Sandoval 2019).

Figures 2–4 illustrate these hybrid modeling architectures.

Figure 2:

Parallel hybrid model: the outputs of FPM and ML are combined via a (possibly learned) weighting function.

Figure 3:

Series hybrid model: the output of the FPM is passed as input to the ML model.

Figure 4:

Residual hybrid model: the ML model learns the error between FPM predictions and real output to provide a corrective term.

In this work, a parallel hybrid model is developed to predict basis weight in the wire section. This approach was chosen for several reasons: First, the parallel hybrid architecture offers high flexibility by allowing both the first-principles model (FPM) and the data-driven model (ML) to independently contribute to the prediction. Unlike residual models, which require the FPM to closely approximate the target signal to avoid biasing the residual, or series models, which introduce dependency and error propagation between models, the parallel structure enables robust fusion without imposing strict constraints on either component. Second, the weighting mechanism – realized via an auxiliary ML model – enables dynamic adaptation to changing operating conditions and sensor drift. This is particularly important in papermaking processes where regime shifts (e.g., grade changes, startup phases) can significantly affect model reliability. By adjusting the contribution of FPM and ML based on recent performance, the hybrid model maintains both physical consistency and predictive accuracy across varying process regimes. A simplified mass balance model is combined with a temporal convolutional neural network (Conv1D) trained on industrial data. An auxiliary recurrent neural network dynamically computes the weighting factor between the two models’ outputs, allowing the hybrid model to adapt to changing process conditions while maintaining physical consistency (Rudolph et al. 2024).

3.1 Industrial dataset and preprocessing

The dataset was collected from a tissue production facility operating a twin-wire former. Five key process variables (PVs) were recorded over a period of seven months at 30-min intervals: basis weight, thick stock flow, fines content, moisture content, and wire speed.

To ensure data reliability, the following preprocessing steps were applied:

1. Threshold filtering: Removal of measurements outside physically plausible bounds.

2. Time alignment: Compensation for seasonal time shifts and ensuring uniform sampling.

As the dataset represents a chronological time series, random shuffling was avoided to preserve temporal dependencies. Instead, the dataset was split sequentially in a hindcasting-like manner: the first 40 % were used for training, followed by 20 % for validation, 20 % for weight-model training, and the final 20 % for testing. This setup ensures that each model is evaluated only on future, unseen process behavior, reflecting realistic deployment scenarios.

3.2 First-principle modeling approach

The FPM was adapted from the wire section model originally proposed (Cho 2005). It is based on mass conservation principles and simulates short circulation dynamics, consistency evolution, and initial basis weight formation. The model was implemented in MATLAB/Simulink using a block-diagram structure.

Several simplifications were made to adapt the model to a modern hydraulic headbox configuration:

1. No headbox volume (ideal hydraulic behavior)

2. Absence of filler or retention aid dosing

3. Updated operating parameters for flow rate and machine speed

These modifications improved model generality and enabled realistic simulation using actual plant input data.

3.3 Machine learning architecture

To model the nonlinear relationship between process variables and basis weight, temporal neural network architectures capable of capturing sequential dependencies and dynamic process behavior were investigated. The motivation for this choice lies in the temporal nature of the process: basis weight at the reel is influenced by upstream conditions with time delays and smoothing effects.

Three neural network architectures were considered:

1. Conv1D: Captures local temporal patterns using convolutional filters.

2. Long Short-Term Memory (LSTM): Recurrent network suitable for modeling long-term dependencies.

3. Gated Recurrent Unit (GRU): A simplified recurrent model with fewer parameters than LSTM.

To justify the architecture selection, a structured benchmarking study was conducted. A hyperparameter search over model depth, number of units, window size, and learning rate was performed using a combination of grid search and automated tuning with Keras Tuner. Each model was trained and validated using the same training pipeline and input features, with early stopping based on validation RMSE.

The Conv1D model consistently outperformed LSTM and GRU in both training time and predictive accuracy on the validation set. This is attributed to the relatively short effective memory required for the prediction task (less than 4 h) and the stationary structure of the input data after preprocessing.

The final configuration for the selected architecture was:

1. Model: 1D Convolutional Neural Network (Conv1D)

2. Window size: 7 time steps (equivalent to 3.5 h)

3. Number of filters: 64

4. Kernel size: 3

5. Learning rate: 0.001

6. Loss function: Mean Squared Error (MSE)

Model performance was evaluated based on Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) on the validation set. The Conv1D model demonstrated better generalization and robustness under regime shifts compared to the recurrent models.

3.4 Hybrid model integration

To fuse the predictions from the FPM and ML models, a weighted parallel model was implemented. The hybrid prediction is computed as:

Two strategies were used to define the weight w _fpm:

1. Static weighting: A fixed weight computed to minimize RMSE over a tuning dataset.

2. Dynamic weighting: A secondary ML model was trained to predict the optimal weight at each time step, using the recent history of FPM and ML outputs as input features.

Dynamic weighting was realized using a secondary GRU model. The GRU weighting model was implemented using a stacked architecture. It consisted of two GRU layers with 64 units each. Between the recurrent layers, a dropout rate of 0.2 was applied to reduce overfitting. After the recurrent blocks, the network included two fully connected layers (64 units, ReLU activation and 8 units, ReLU activation) followed by a final dense layer with linear activation for the output weight prediction. The model was trained with the Adam optimizer using learning rates of 0.001 and 0.0001, with mean squared error as the loss function. Early stopping (patience = 10) and ReduceLROnPlateau callbacks were employed to prevent overfitting and adapt learning rates during training. The input was based on 50-step time windows including recent feature and target histories, and the dataset was split chronologically into 40 % training, 10 % validation, and 50 % testing.

The GRU receives as input:

1. recent prediction errors (residuals) of FPM and ML,

2. model confidence indicators (e.g., moving average of residuals),

3. current values of key process variables (PV2–PV5).

This enables the system to adjust the weight w _fpm at every time step based on the local reliability of each model. The weight model is trained to minimize the hybrid RMSE on a separate validation set. This improves adaptivity under regime shifts and transient startup behavior.

The dynamic weighting approach yielded the best performance, particularly in scenarios with transient process behavior or regime shifts.

4.1 First-principle model performance

The FPM was evaluated using normalized industrial data for the target variable (basis weight). While the FPM captured key trends and general system dynamics, its accuracy was limited by structural simplifications and fixed parameter settings (see Figure 5).

Figure 5:

Normalized comparison of industrial data versus FPM prediction for basis weight.

4.2 Machine learning model performance

The Conv1D model outperformed the FPM in both RMSE and MAE, producing smoother and more responsive predictions. However, its performance deteriorated in high-variance regions and during rapid transitions, suggesting challenges in capturing short-term dynamics (see Figure 6).

Figure 6:

Normalized comparison of industrial data versus ML prediction (Conv1D) for basis weight.

4.3 Hybrid model with static weighting

Two static weighting strategies were evaluated:

1. Per-row selection: At each data point, the model (FPM or ML) with the lower squared error was selected.

2. RMSE minimization: A fixed weight was tuned to minimize RMSE over the validation set.

Both strategies yielded modest improvements over the individual models, with RMSE-based weighting performing slightly better (see Figure 7).

Figure 7:

Normalized comparison of industrial data, static hybrid predictions (per-row and RMSE-based), ML predictions, and FPM predictions for basis weight.

4.4 Hybrid model with dynamic weighting – sliding window

A dynamic hybrid model using a sliding window to compute local performance-based weights outperformed static strategies. The optimal window size was selected based on a grid search using the designated validation set for the weighting model (20 % of the data). As the data represent a time series, the split was performed chronologically to preserve temporal dependencies and avoid data leakage. Randomized cross-validation was not applied, as it is unsuitable for time-dependent datasets (see Figure 8).

Figure 8:

Comparison of static and dynamic sliding window hybrid predictions for basis weight.

4.5 Hybrid model with dynamic weighting – additional ML model

The highest-performing model integrated a secondary ML model (GRU) to dynamically predict the weighting between FPM and ML predictions based on recent performance history. This model provided robust and adaptive behavior across various operating conditions (see Figure 9).

Figure 9:

Comparison of hybrid predictions using ML-based dynamic weighting.

4.6 Evaluation summary

The evaluation of different modeling strategies highlights the trade-offs between physical interpretability and predictive accuracy. The first-principles model (FPM) provides transparent results but suffers from structural simplifications that limit its precision. The data-driven Conv1D model, by contrast, delivers more accurate predictions but lacks physical consistency and shows reduced robustness under regime shifts. Hybrid approaches combine the complementary strengths of both paradigms. Static weighting improves prediction accuracy moderately, while dynamic weighting enables adaptation to transient and non-stationary conditions. The highest-performing configuration is the hybrid model with ML-based dynamic weighting, which integrates a GRU-based mechanism to dynamically adjust the contributions of FPM and ML outputs.

Table 2 summarizes the performance of all tested models in terms of both normalized error metrics and absolute values expressed in g/m². While normalized values facilitate direct comparison across modeling strategies, absolute metrics are more relevant for practical industrial assessment. The ML-based hybrid model achieves the lowest error with an RMSE of 2.12 g/m² and MAE of 1.53 g/m², significantly outperforming both the standalone FPM (RMSE = 3.51 g/m²) and Conv1D ML model (RMSE = 2.58 g/m²). These results confirm that the hybrid framework combines physical consistency with adaptability, providing a scalable foundation for real-time quality monitoring and control in papermaking (see Table 2).