[Paper Review] Physics-informed neural networks for non-Newtonian fluid thermo-mechanical problems: an application to rubber calendering process

Thi Nguyen Khoa Nguyen, Thibault Dairay+2 more|arXiv (Cornell University)|Jan 31, 2022

Model Reduction and Neural Networks46 references39 citations

TL;DR

This paper applies Physics-Informed Neural Networks (PINNs) to a non-Newtonian fluid thermo-mechanical problem in rubber calendering, using partial sensor data to infer full velocity, pressure, and temperature fields and identify unknown physical parameters. PINNs successfully reconstruct hidden physics and estimate parameters with high accuracy even under noisy measurements and ill-posed boundary conditions, demonstrating robustness and adaptability in industrial-scale simulations.

ABSTRACT

Physics-Informed Neural Networks (PINNs) have gained much attention in various fields of engineering thanks to their capability of incorporating physical laws into the models. However, the assessment of PINNs in industrial applications involving coupling between mechanical and thermal fields is still an active research topic. In this work, we present an application of PINNs to a non-Newtonian fluid thermo-mechanical problem which is often considered in the rubber calendering process. We demonstrate the effectiveness of PINNs when dealing with inverse and ill-posed problems, which are impractical to be solved by classical numerical discretization methods. We study the impact of the placement of the sensors and the distribution of unsupervised points on the performance of PINNs in a problem of inferring hidden physical fields from some partial data. We also investigate the capability of PINNs to identify unknown physical parameters from the measurements captured by sensors. The effect of noisy measurements is also considered throughout this work. The results of this paper demonstrate that in the problem of identification, PINNs can successfully estimate the unknown parameters using only the measurements on the sensors. In ill-posed problems where boundary conditions are not completely defined, even though the placement of the sensors and the distribution of unsupervised points have a great impact on PINNs performance, we show that the algorithm is able to infer the hidden physics from local measurements.

Motivation & Objective

• To demonstrate the application of PINNs to a complex industrial non-Newtonian fluid thermo-mechanical problem in rubber calendering.
• To investigate the impact of sensor placement and collocation point distribution on PINN performance in ill-posed inverse problems.
• To evaluate PINN robustness in identifying unknown physical parameters from noisy measurements.
• To compare PINN performance using different neural network architectures (e.g., Rowdy Net, adaptive activation functions) in solving coupled PDEs.
• To validate PINN predictions against high-fidelity finite element simulations for accuracy and convergence.

Proposed method

• PINNs are trained using a loss function combining supervised data (temperature sensor measurements), boundary/initial conditions, and PDE residual constraints.
• The governing equations include a generalized Stokes equation for incompressible non-Newtonian flow and a heat transfer equation, coupled via temperature-dependent viscosity.
• Collocation points are strategically placed using a finite element mesh to improve accuracy at high-gradient regions, rather than random sampling.
• Locally adaptive activation functions (L-LAAFs, N-LAAFs) and deep Kronecker networks (Rowdy Net) are employed to enhance model expressivity and convergence.
• Noise is introduced via uncorrelated Gaussian noise on sensor data to test robustness, with extended training (up to 200,000 epochs) to ensure convergence.
• A high-fidelity finite element solver provides reference solutions for quantitative comparison of PINN predictions.

Experimental results

Research questions

• RQ1 How does sensor placement affect PINN performance in inferring full thermo-mechanical fields from partial temperature measurements?
• RQ2 What is the impact of collocation point distribution—random vs. mesh-based—on PINN accuracy in ill-posed problems?
• RQ3 Can PINNs accurately identify unknown physical parameters (e.g., 1/Pe, Br/Pe, λ) from noisy sensor data?
• RQ4 How do advanced architectures like Rowdy Net and adaptive activation functions improve PINN performance in complex fluid problems?
• RQ5 To what extent can PINNs maintain accuracy when boundary conditions are incomplete and measurements are corrupted by noise?

Key findings

• PINNs successfully inferred full velocity, pressure, and temperature fields using only temperature sensor data, even with incomplete boundary conditions.
• Case 1 and Case 3 sensor configurations (with sensors at input/output and lateral boundaries) yielded slightly better performance than Case 2 (only input/output lines), though Case 2 still achieved high accuracy.
• Training with collocation points derived from a finite element mesh significantly improved prediction accuracy compared to purely random sampling, especially in high-gradient regions.
• Even with 10% Gaussian noise on sensor data, PINNs predicted the unknown parameters (1/Pe, Br/Pe, λ) within 3.77e-03, 1.16e-01, and 1.94e+01 respectively, close to the reference values (4.00e-03, 1.20e-01, 2.00e+01).
• The use of Rowdy Net and adaptive activation functions enhanced convergence and reduced relative error, improving overall model accuracy.
• Despite qualitative satisfaction in some cases, high relative errors were observed, suggesting the need for error metrics based on physical modes (e.g., POD/DMD) for better evaluation.