Physics-Informed Neural Networks for Inverse Modeling of Viscosity in Injection Molding

Abstract

Physics-informed neural networks integrate physical laws into machine learning models to solve differential equations. This thesis investigates an inverse physics-informed neural network framework for estimating viscosity in Tetra Pak’s injection molding process, where direct measurements are not possible. By enforcing the Navier-Stokes equations and using sparse velocity and pressure data, a virtual sensor is developed to infer the viscosity profile.

The inverse formulation treats viscosity as an additional learnable parameter, optimized jointly with the NN’s free parameters. This allows the model to identify the viscosity value that best explains the observed data while remaining consistent with the governing physics. A key aim is to detect variations in viscosity, which is critical for quality control in production. This is evaluated using datasets from different materials and pressure settings, and the model successfully distinguishes the corresponding viscosities in all cases.

The model accurately reconstructs the velocity and pressure fields, indicating that the underlying PDE system is correctly solved. While absolute viscosity estimates are biased, likely due to mismatch between the constitutive model and the data, the predicted values maintain correct relative order compared to reference data and form distinct clusters by material and pressure. Loss balancing is shown to be a critical factor for convergence and accuracy, while reducing the number of physics enforcement points is less sensitive and can significantly lower computational cost. These results support the potential of inverse physics-informed neural networks for virtual sensing and enhanced process control in manufacturing.

Popular Abstract

Injection molding is a manufacturing process where molten plastic is injected into molds to form components. At Tetra Pak, this method is used to produce closures for their packaging solutions. A key factor in this process is viscosity, which measures how easily the material flows. However, viscosity can vary between batches due to factors like machine settings or material differences, leading to issues that may affect product quality. Currently, viscosity cannot be directly measured during production, meaning variations cannot be detected or adjusted in real time. But what if we could monitor viscosity by estimating it from other available measurements in the machine?

This thesis explores that idea by developing a virtual sensor: a machine learning model that predicts viscosity from indirect measurements like pressure and velocity. The model uses Physics-Informed Neural Networks (PINNs), which integrate machine learning with scientific physics.

PINNs are a recent development in scientific machine learning. Unlike traditional models that fit data, PINNs incorporate physics into the training process, ensuring the model adheres to physical laws. This approach helps avoid physically implausible solutions and improves generalization in data-sparse regions.

The physical laws are embedded directly into the model’s training, utilizing automatic differentiation, a technique that calculates derivatives by following the steps inside the network. This allows the model to evaluate how well its predictions align with physical equations at specific points, penalizing violations of these laws to maintain realistic behavior.

The goal is to estimate the unknown viscosity from observed velocity and pressure data. Since the governing physics behind the data is only partially known, viscosity is treated as a hidden parameter in an inverse problem. In the context of PINNs, this is naturally addressed by extending the framework to treat unknown parameters, such as viscosity, as additional learnable variables. Unlike traditional models, which require repeated simulations for each guess of viscosity, the PINN framework jointly optimizes these hidden parameters along with the network parameters. This allows the model to learn the viscosity efficiently, directly from the data, while also ensuring that the solution respects the underlying physics.

The model focuses on a simplified part of the injection molding process: a heated hose through which molten plastic flows. Using simulated sensor-like inputs that replicate sparse real-world data, the model estimates viscosity, though with slight underestimation. Still, it successfully differentiates between materials with varying viscosities and pressures, preserving the correct relative order of values. This suggests the model captures the underlying physics effectively. Additionally, the sparse data highlights PINNs' ability to extract valuable insights from limited information by leveraging scientific knowledge.

Training PINNs presents challenges, as the model must balance fitting data, respecting boundary conditions, and satisfying physical laws. This balance is crucial for stable training. Without it, the model may overemphasize one objective at the expense of the others, undermining the core benefit of PINNs.

Future work is needed to evaluate the model’s performance with noisy or imperfect data and to determine whether it can detect subtle viscosity variations in real-world production environments, where even small changes are critical.

Ultimately, this thesis takes a step toward smarter process control in manufacturing by showing how physics-informed machine learning can uncover hidden insights from existing data.

• author: Lindqvist, Ellen
• supervisor: Alexandros Sopasakis
• organization: Mathematics (Faculty of Engineering)
• course: FMAM05 20251
• year: 2025
• type: H2 - Master's Degree (Two Years)
• subject: Mathematics and Statistics
• keywords: Physics-Informed Neural Networks, Inverse Problems, Viscosity Estimation, Injection Molding, Scientific Machine Learning, Virtual Sensor, Shear-Thinning Fluids
• publication/series: Master's Theses in Mathematical Sciences
• report number: LUTFMA-3595-2025
• ISSN: 1404-6342
• other publication id: 2025:E64
• language: English
• id: 9197590
• date added to LUP: 2025-06-27 12:38:34
• date last changed: 2025-06-27 12:38:34