⚡ electrical engineering

Interpretable Hybrid Surrogate Modeling for Diameter Deviation Prediction in Injection Molding Using Symbolic Regression and Gaussian Process Residual Correction

This study presents an interpretable hybrid surrogate model that combines symbolic regression with Gaussian Process residual correction to accurately predict diameter deviations in injection molding under limited experimental data, achieving improved accuracy and providing a transparent framework for process optimization.

Original authors: Amine Aboutaib, Hanaa OUAOMAR, Souad Touairi, Mourad KADIRI

Published 2026-10-27

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Amine Aboutaib, Hanaa OUAOMAR, Souad Touairi, Mourad KADIRI

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of manufacturing, making plastic parts is a bit like baking a cake, but with far less room for error. When a machine injects molten plastic into a mold, it must hold the pressure just right, cool the material at the perfect speed, and time the cycle precisely. If the machine settings are even slightly off, the final part might be the wrong size. This is a problem of dimensional quality. Engineers have long tried to predict exactly how changing a machine's settings—like how hard they push the plastic or how fast it flows—will change the final size of the part. The challenge is that these factors do not act alone; they tangle together in complex ways. A change in pressure might have a different effect depending on how fast the plastic was injected or how long the machine waited before opening the mold. Usually, to understand these relationships, companies run hundreds of experiments, but time and money often limit them to just a few dozen. When data is scarce, standard computer models often become too complicated to understand or fail to predict accurately.

A team of researchers from Morocco set out to solve this puzzle for a specific type of plastic part, focusing on a defect called diameter deviation, which is simply how much the part's width differs from the intended size. They wanted to build a computer model that could predict this error using very little data, but with a crucial twist: the model had to be understandable. Instead of hiding the logic inside a black box, they wanted a clear mathematical rule that an engineer could read and trust. To do this, they combined two different approaches. First, they used a technique called symbolic regression, which is essentially a search for the simplest possible equation that fits the data. Then, they added a second layer, a statistical tool known as a Gaussian process, to catch the small, messy details that the first equation missed. The goal was to create a hybrid system that was both transparent and precise, even when working with only twenty-seven distinct operating conditions.

The researchers began by gathering data from two separate experimental campaigns, which together produced 127 runs. Because some of these runs repeated the same conditions, they averaged the results to create a clean set of twenty-seven representative scenarios. From these, they selected four main controls to study: the pressure used to hold the plastic in the mold, the counter-pressure pushing back against it, the total time of the cycle, and the speed at which the plastic was injected. Rather than feeding these four raw numbers directly into the computer, the team first transformed them into more meaningful combinations. They created new inputs that represented things like the balance between pressures or the interaction between speed and time. This step was vital because it gave the computer a better vocabulary to describe the physical process, allowing it to find a simpler and more accurate rule.

Using this prepared data, the researchers let the symbolic regression algorithm search for the best mathematical expression. The result was a compact equation that acted as the main predictor. This equation successfully captured the dominant trend of how the machine settings affected the part's size. When tested against the data it had not seen during its creation, this equation predicted the diameter deviation with an error of about 10.19 micrometers. To put that in perspective, a human hair is roughly 70 to 100 micrometers thick, so the model was predicting errors smaller than a tenth of a hair's width. While impressive, the researchers knew that no single equation could capture every tiny fluctuation in the complex manufacturing process. There were always small, unexplained wiggles left over.

To address these remaining errors, the team trained a second model, a Gaussian process, specifically to learn the mistakes made by the first equation. Think of this second model as a fine-tuner that looks at the gap between what the first equation predicted and what actually happened, learning to correct it. When they combined the two, the hybrid model performed even better, reducing the prediction error to 9.32 micrometers. This improvement, while seemingly small in absolute terms, was statistically significant. More importantly, the hybrid approach kept the core logic of the first equation visible and understandable, while the second part handled the messy details. This meant that an engineer could still look at the main rule to understand the general behavior of the machine, while relying on the correction layer for higher precision.

The researchers then used these models to ask a practical question: what machine settings would produce the smallest possible error? They ran an optimization search to find the ideal combination of pressure, speed, and time within the range they had tested. The simple equation suggested a best-case scenario with an error of about 33.25 micrometers. However, the hybrid model, with its ability to correct for subtle errors, pointed to a different set of settings that could theoretically reduce the error to just 23.73 micrometers. This shift showed that adding the correction layer didn't just improve the numbers; it actually changed the recommendation for how to run the machine. The researchers noted that these were predictions based on their models and had not yet been tested on a real machine, but the difference was large enough to warrant a real-world trial.

Throughout the study, the team was careful to be honest about the limits of their work. They emphasized that their model was a local guide, valid only for the specific machine, material, and mold they tested, and not a universal law for all plastic molding. They also pointed out that while their model provided a measure of uncertainty, showing how confident it was in its predictions, this confidence had not been fully verified with new, independent experiments. The true test would come when engineers tried the recommended settings on the factory floor. The study concluded that this hybrid approach offered a practical middle ground for industrial problems where data is scarce. It provided a way to get the best of both worlds: the clarity of a simple equation and the flexibility of a complex statistical correction, giving engineers a tool that is both smart and explainable.

Technical Summary: Interpretable Hybrid Surrogate Modeling for Diameter Deviation Prediction in Injection Molding

Problem Statement
Dimensional quality in injection molding is difficult to model due to the strong coupling between process variables (pressure, flow, thermal history) and the limited experimental budgets typical of industrial settings. Traditional machine learning approaches often sacrifice interpretability for predictive power, while classical statistical methods may lack the flexibility to capture complex nonlinear interactions. This study addresses the challenge of developing a predictive model for diameter deviation that remains interpretable, handles small datasets effectively, and provides a transparent basis for process optimization.

Methodology
The authors propose a sequential hybrid surrogate modeling framework designed for small-data regimes (27 representative operating conditions derived from 127 experimental runs). The workflow consists of four main stages:

Process-Informed Feature Engineering: Instead of using raw process variables alone, the study constructs a 12-dimensional input space from four primary variables: holding pressure ( $P$ ), counter-pressure ( $C$ ), cycle time ( $T$ ), and injection speed ( $V$ ). This engineered representation includes pressure ratios, normalized pressure contrasts, multiplicative interaction terms (e.g., $C \cdot V \cdot T$ ), and rate-like combinations. These features are designed to expose specific process interactions to the symbolic search algorithm.
Symbolic Regression (SR) Baseline: Using the PySR library, the method searches for a compact, explicit mathematical expression that relates the engineered features to the diameter deviation. The selected equation serves as the deterministic, interpretable baseline. The final expression utilizes only three inputs: normalized holding pressure, normalized pressure contrast, and a normalized interaction term ( $C \cdot V \cdot T$ ).
Gaussian Process (GP) Residual Correction: A Gaussian Process with an RBF + White kernel is trained on the residuals (the difference between measured values and the SR baseline predictions) using the original four process variables. This GP acts as an adaptive correction layer, modeling local nonlinearities and discrepancies that the symbolic equation misses, without replacing the analytical core.
Surrogate Optimization: Both the standalone symbolic model and the hybrid (SR + GP) model are used as objective functions within a Differential Evolution (DE) optimizer to identify settings that minimize predicted diameter deviation within the experimentally explored ranges.

Key Results

Predictive Performance: The standalone symbolic model achieved an $R^2$ of 0.8665 and an RMSE of 9.37 µm on the full dataset. Under Leave-One-Out Cross-Validation (LOOCV), it yielded an $R^2$ of 0.8423 and an RMSE of 10.19 µm.
Hybrid Improvement: The addition of the GP residual layer improved the LOOCV performance to an $R^2$ of 0.8681 and reduced the RMSE to 9.32 µm. The authors characterize this gain as measurable but modest, noting that the primary value lies in the refinement of the local landscape rather than a dramatic global metric improvement.
Optimization Outcomes: When optimized via Differential Evolution, the symbolic baseline predicted a minimum diameter deviation of approximately 33.25 µm. The hybrid model predicted a lower minimum of 23.73 µm.
Uncertainty Quantification: The GP layer provides a predictive variance, allowing for the assessment of confidence in predictions. However, the study notes that with only 27 conditions, the uncertainty map appeared relatively uniform, limiting the ability to distinguish between high-confidence and low-confidence regions without further validation.

Significance and Claims
The paper positions its contribution primarily as methodological rather than as a discovery of a universal physical law for injection molding. The authors claim:

Interpretability with Flexibility: The framework successfully demonstrates that an explicit, inspectable analytical core (the symbolic equation) can be paired with a flexible probabilistic correction layer (the GP). This avoids the "black box" nature of deep learning while capturing more complexity than a simple linear or fixed-response surface model.
Small-Data Strategy: The study validates a workflow where process-informed feature engineering guides symbolic equation discovery, enabling the recovery of compact relationships from very limited data (27 conditions).
Optimization Landscape Sensitivity: A key finding is that the residual correction layer significantly alters the shape of the predicted objective landscape, shifting the recommended process settings. This highlights that even modest improvements in predictive accuracy can lead to different optimization recommendations.
Transparency: The authors emphasize that the model remains transparent; the dominant trend is defined by an equation, and the GP is restricted to modeling the residual error, making the model's behavior easier to communicate to process engineers.

Limitations and Scope
The authors explicitly state that the optimization results (predicted minima of ~33 µm and ~24 µm) are model-based predictions that have not been experimentally verified. The model is a local surrogate valid only within the explored operating domain and does not claim to be a universal law transferable to different machines, materials, or geometries. Furthermore, the study acknowledges that the feature engineering contribution was not isolated via ablation studies, and the uncertainty quantification lacks independent calibration. The work serves as a proof-of-concept for a transparent, hybrid approach to small-data industrial modeling.

Technical Summary: Interpretable Hybrid Surrogate Modeling for Diameter Deviation Prediction in Injection Molding

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