Published in Advanced Modeling and Simulation in Engineering Sciences, the study introduces a Reduced-Order Model (ROM) based on a modified Constitutive-Relation Error (mCRE) formulation. This approach aims to overcome the limitations of standard data-driven methods, which often struggle with measurement noise and mesh dependency in Computational Fluid Dynamics (CFD) simulations. The model is designed to extract physically consistent viscosity values from steady, incompressible flows, serving as a robust tool for inverse parameter identification in fluid mechanics.
The central challenge addressed by the study is the difficulty of identifying rheological properties, particularly viscosity, from experimental or simulation data that contains noise. Viscosity is a key parameter describing a fluid’s resistance to flow, but its identification is often an ill-posed problem, especially in the presence of strong shear effects. Standard regression-based models, such as those using Mean Squared Error (MSE) loss functions, tend to over-learn numerical artifacts associated with specific mesh discretizations. These models can amplify high-frequency noise when calculating the gradients required for viscosity identification, leading to unreliable results when applied to new or noisy datasets.
To address these issues, the authors propose a hybrid approach that combines Deep Learning with physical constraints. The core of the method is a Convolutional Neural Network (CNN) augmented with a Physics-Informed Layer (PIL). Unlike purely data-driven models that learn statistical correlations, this Physics-Informed Neural Network (PINN) incorporates the fundamental principles of fluid mechanics directly into its training process. The mCRE framework acts as a physical constraint within the network, enforcing structural consistency between the predicted velocity fields and the governing Cauchy equations. This ensures that the model does not merely fit the noisy data but adheres to the underlying laws of mass and momentum conservation.
The formulation is specifically adapted to the Finite Volume Method (FVM), which is widely used in CFD. In this framework, the mCRE loss function is defined by integrating fluxes through the faces of control volumes rather than relying on point-wise values at nodes. This integral formulation allows the model to preserve local force balances and mass conservation across the domain. The loss function comprises several terms, including those for incompressibility, momentum equilibrium, and constitutive law consistency. By minimizing the error in these physical balances, the network learns to filter out data noise while identifying the viscosity parameter that satisfies the fluid’s rheological behavior.
A key innovation in the study is the use of Multiple Mesh Training (MMT) to ensure the model’s robustness across different spatial resolutions. By training the network on velocity fields generated from various mesh refinements, the model is forced to learn resolution-invariant features rather than memorizing grid-specific patterns. The PIL normalizes the input data using local geometric metrics, such as cell volumes and face areas, allowing the network to interpret physical fluxes consistently regardless of the mesh density. This mesh-agnostic capability is crucial for practical applications, where the optimal mesh resolution may vary or be unknown at the time of inference.
The study evaluated the performance of the proposed mCRE-CNN against a standard MSE-CNN using synthetic data from steady Poiseuille flows. The test cases included both Newtonian and non-Newtonian fluids, with Gaussian noise levels ranging from 5% to 60% applied to the velocity profiles. For Newtonian fluids, the mCRE-CNN demonstrated superior robustness, showing a shallower increase in relative error as noise levels increased compared to the MSE-CNN. In specific tests at a Reynolds number of 200 with 30% noise, the mCRE-CNN predicted a Reynolds number of 199, whereas the MSE-CNN predicted 195, indicating a closer approximation to the true value.
The model’s ability to generalize to unseen noise levels was also tested. When evaluated on noise levels of 70% to 100%, which were not included in the training dataset, the mCRE-CNN maintained a stable error slope, while the MSE-CNN’s error increased significantly. This suggests that the physics-informed approach provides a form of implicit regularization that prevents overfitting to the specific noise statistics of the training data. The model effectively distinguishes between physical signals and random perturbations, relying on the consistency of the flow field with the governing equations rather than just data matching.
For non-Newtonian fluids, the mCRE-CNN also outperformed the MSE-CNN, particularly for shear-thinning behaviors characterized by a flow behavior index n less than 1. In configurations with n = 0.2, 0.5, and 0.8, the mCRE-CNN consistently produced lower relative errors in viscosity identification across various noise levels. The study highlights that the mCRE loss function, by emphasizing relative errors and physical consistency, is better suited for capturing the complex, non-linear variations in velocity profiles associated with variable viscosity. This is in contrast to MSE, which penalizes absolute deviations and may neglect accuracy in regions with smaller but physically significant velocity changes.
The computational cost of the mCRE-CNN is higher per epoch than that of the MSE-CNN due to the complexity of calculating the physical residuals. However, the study notes that this cost stabilizes after initial training epochs. The additional computational effort is justified by the model’s enhanced generalization capabilities and the elimination of the need for retraining on new mesh configurations. The authors argue that the long-term efficiency gains from a mesh-agnostic model outweigh the initial training costs, particularly in scenarios involving diverse CFD solvers or experimental data with varying resolutions.
The study acknowledges several limitations, noting that the current work focuses on steady, incompressible flows in two-dimensional configurations. The training dataset, while sufficient for demonstrating feasibility, is not optimized for all possible parameter combinations. The authors state that further enrichment of the dataset and investigation of optimal weighting factors for the loss function terms are necessary to improve accuracy and convergence speed. Additionally, the model’s performance at higher Reynolds numbers and in turbulent flows remains to be explored, potentially requiring the integration of turbulence models or Large Eddy Simulation (LES) data.
Future work, outlined as Part II of the study, will extend the method to unsteady flows using a hybrid CNN-LSTM architecture. This extension aims to address the time-dependent dynamics of fluid flows, which are critical for applications in real-time monitoring and control. The authors also plan to investigate the application of the mCRE framework to fluid-structure interactions and porous media, where the interaction between fluids and solids introduces additional complexity. By combining the robustness of physics-informed learning with the flexibility of deep neural networks, this research provides a pathway toward more reliable and interpretable surrogate models in computational fluid dynamics.
In their detailed mathematical formulation, the researchers structured the Finite Volume modified Constitutive-Relation Error by decomposing the loss into several explicit mechanical constraints. Beyond standard velocity divergence and momentum balance, the composite framework integrates specific terms to address cell-level flux dynamics and wall boundary conditions. Additionally, because collocated grids can induce decoupling between velocity and pressure, the formulation adapts Rhie-Chow interpolation principles directly into the loss calculation. This modification corrects face fluxes against jumping pressure gradients, actively penalizing unphysical checkerboard oscillations without requiring staggered spatial arrangements.
The underlying neural architecture processes one-dimensional velocity profiles using three sequential convolutional blocks configured with 16, 32, and 64 filters alongside rectified linear unit activation and max pooling operations. These convolutional layers feed into a flatten layer and two dense layers to output the predicted viscosity values. Optimization relies on the Adam algorithm to update network weights iteratively across training epochs. During training, the Physics-Informed Layer serves as a fixed convolution filter where weights are established entirely by local mesh geometry rather than adjusted through machine learning backpropagation. By feeding the network multi-channel tensors containing cell volumes, face areas, and normal vectors, the model normalizes differential operators so that coarse and fine control volumes remain physically equivalent.
Statistical evaluations across 100 numerical experiments confirmed the model’s consistency when identifying parameters under heavy noise corruption. In tests conducted on a shear-thinning configuration with a behavior index of 0.2 under 80% added Gaussian noise, statistical comparisons of error distributions demonstrated lower mean errors and tighter quartile spreads for the physics-informed model relative to standard regression. A parallel series of 100 experiments performed at an index of 0.5 under 60% noise revealed the same stability. Furthermore, examining mesh refinement without multi-mesh training showed that relative error naturally decreased across resolutions of 32, 64, and 128 nodes, while multi-mesh training eliminated the requirement to calibrate individual networks for each discrete resolution.
Subject of Research: Virtual viscometer based on Physics-Informed mCRE formulation using Deep-Learning with Finite-Volume Multiple-Mesh Training. Part I: steady flow CNN-augmented Reduced-Order Model
Article Title: Virtual viscometer based on Physics-Informed mCRE formulation using Deep-Learning with Finite-Volume Multiple-Mesh Training. Part I: steady flow CNN-augmented Reduced-Order Model
Article References: Lacazedieu, E. (2026). Virtual viscometer based on Physics-Informed mCRE formulation using Deep-Learning with Finite-Volume Multiple-Mesh Training. Part I: steady flow CNN-augmented Reduced-Order Model. Advanced Modeling and Simulation in Engineering Sciences, 13(1), Article 24. https://doi.org/10.1186/s40323-026-00343-8
Image Credits: AI Generated
DOI: 10.1186/s40323-026-00343-8
Keywords: Virtual, viscometer, based, Physics-Informed, mCRE, formulation, Deep-Learning, Finite-Volume, Multiple-Mesh, Training, Part, steady, flow, CNN-augmented, Reduced-Order
News Source: Katie Riggs. (November 13, 2026). Physics-Informed Neural Network Enhances Viscosity Identification in Noisy Fluid Flows. Scienmag.
