<p dir="ltr">Machine learning and deep learning techniques now provide innovative tools for addressing problems in biological, engineering, and physical systems. Physics-informed neural networks (PINNs) are a type of neural network that incorporate physical laws described by partial differential equations (PDEs) into their supervised learning tasks. This dissertation aims to enhance PINNs with improved training techniques and loss functions to tackle the complex physics of viscoelastic flow and rheology more effectively. The focus areas of the dissertation are listed as follows: i) Assigning relative weights to loss terms in training physics-informed neural networks (PINNs) is complex. We propose a solution using numerical integration via backward Euler discretization to leverage statistical properties of data for determining loss weights. Our study focuses on two and three-dimensional Navier-Stokes equations, using spatio-temporal velocity and pressure data to ascertain kinematic viscosity. We examine two-dimensional flow past a cylinder and three-dimensional flow within an aneurysm. Our method, tested for sensitivity and robustness against various factors, converges faster and more accurately than traditional PINNs, especially for three-dimensional Navier-Stokes equations. We validated our approach with experimental data, using the velocity field from PIV channel flow measurements to generate a reference pressure field and determine water viscosity at room temperature. Results showed strong performance with experimental datasets. Our proposed method is a promising solution for ’stiff’ PDEs and scenarios requiring numerous constraints where traditional PINNs struggle. ii) Machine learning algorithms are valuable for fluid mechanics, but high data costs limit their practicality. To address this, we present viscoelasticNet, a Physics-Informed Neural Network (PINN) framework that selects the appropriate viscoelastic constitutive model and learns the stress field from a given velocity flow field. We incorporate three non-linear viscoelastic models: Oldroyd-B, Giesekus, and Linear PTT. Our framework uses neural networks to represent velocity, pressure, and stress fields and employs the backward Euler method to construct PINNs for the viscoelastic model. The approach is multistage: first, it solves for stress, then uses stress and velocity fields to solve for pressure. ViscoelasticNet effectively learned the parameters of the viscoelastic constitutive model on noisy and sparse datasets. Applied to a two-dimensional stenosis geometry and cross-slot flow, our framework accurately learned constitutive equation parameters, though it struggled with peak stress at cross-slot corners. We suggest addressing this by exploring smaller domains. ViscoelasticNet can extend to other rheological models like FENE-P and extended Pom-Pom and learn entire equations, not just parameters. Future research could explore more complex geometries and three-dimensional cases. Complementing Particle Image Velocimetry (PIV), our method can determine pressure and stress fields once the constitutive equation is learned, allowing the modeling of future fluid applications. iii) Physics-Informed Neural Networks (PINNs) are widely used for solving inverse and forward problems in various scientific and engineering fields. However, most PINNs frameworks operate within the Eulerian domain, where physical quantities are described at fixed points in space. We explore coupling Eulerian and Lagrangian domains using PINNs. By tracking particles in the Lagrangian domain, we aim to learn the velocity field in the Eulerian domain. We begin with a sensitivity analysis, focusing on the time-step size of particle data and the number of particles. Initial tests with external flow past a cylinder show that smaller time-step sizes yield better results, while the number of particles has little effect on accuracy. We then extend our analysis to a real-world scenario: the interior of an airplane cabin. Here, we successfully reconstruct the velocity field by tracking passive particles. Our findings suggest that this coupled Eulerian-Lagrangian PINNs framework is a promising tool for enhancing traditional experimental techniques like particle tracking. It can be extended to learn additional flow properties, such as the pressure field for three-dimensional internal flows, and infer viscosity from passive particle tracking, providing deeper insights into complex fluids and their constitutive models. iv) Time-fractional differential equations are widely used across various fields but often present computational and stability challenges, especially in inverse problems. Leveraging Physics-Informed Neural Networks (PINNs) offers a promising solution for these issues. PINNs efficiently compute fractional time derivatives using finite differences and handle other derivatives via automatic differentiation. This study addresses two inverse problems: (1) anomalous diffusion and (2) fractional viscoelasticity. Our approach defines residual loss scaled with the standard deviation of observed data, using numerically generated and experimental datasets to learn fractional coefficients and calibrate parameters for the fractional Maxwell model. Our framework demonstrated robust performance for anomalous diffusion, maintaining less than 10% relative error in predicting the generalized diffusion coefficient and the fractional derivative order, even with 25% Gaussian noise added to the dataset. This highlights the framework’s resilience and accuracy in noisy conditions. We also validated our approach by predicting relaxation moduli for pig tissue samples, achieving relative errors below 10% compared to literature values. This underscores the efficacy of our fractional model with fewer parameters. Our method can be extended to model non-linear fractional viscoelasticity, incorporate experimental data for anomalous diffusion, and apply it to three-dimensional scenarios, broadening its practical applications.</p>
| Identifer | oai:union.ndltd.org:purdue.edu/oai:figshare.com:article/26357113 |
| Date | 25 July 2024 |
| Creators | Sukirt (8828960) |
| Source Sets | Purdue University |
| Detected Language | English |
| Type | Text, Thesis |
| Rights | CC BY 4.0 |
| Relation | https://figshare.com/articles/thesis/PHYSICS-INFORMED_NEURAL_NETWORKS_FOR_NON-NEWTONIAN_FLUIDS/26357113 |
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