EDUCATIONAL ACTIVITIES

This thesis presents a computational framework that combines PDE-constrained optimal control and reduced order modeling for accelerated blood flow simulation. The objective is to reconstruct unknown inlet boundary conditions from velocity data available on internal cross-sections. The problem is formulated using the stationary incompressible Navier–Stokes equations, and solved via a forward–backward optimal control strategy. To reduce computational cost, Proper Orthogonal Decomposition (POD) is used to construct reduced-order models for both state and adjoint variables. Numerical experiments show significant computational speedups (up to one order of magnitude) while maintaining high accuracy. The framework provides a foundation for future extensions to nonlinear, time-dependent, and patient-specific hemodynamic simulations.

This project investigates the application of Physics-Informed Neural Networks (PINNs) to the numerical solution of incompressible fluid flow problems. PINNs provide a mesh-free alternative to classical computational fluid dynamics methods by embedding the governing partial differential equations directly into the training process of a neural network.

The first chapter introduces the mathematical formulation of a multiphysics problem involving biomembranes, with emphasis on level set methods, geometric quantities, and the Helfrich bending energy. The associated fourth-order derivatives highlight the major computational challenges of such problems.

The second chapter presents the fundamental concepts of neural networks and their extension to physics-informed learning. The PINN framework is then applied to the incompressible Navier–Stokes equations using a mixed-variable formulation that improves stability and reduces derivative order in the loss function.

Finally, the methodology is validated through some benchmark simulations of steady and transient flow in a circular cylinder. The results show good qualitative agreement with reference solutions from the literature, while also highlighting some limitations mainly in the computational cost.

Although the full coupling between fluid flow and membrane dynamics is not implemented, this work establishes preliminary tools for future work on applying PINNs to complex multiphysics problems involving biomembranes.

This report investigates deep learning methods for semantic segmentation of petrographic textures related to carbonate rock imaging. Carbonate rocks contain highly heterogeneous pore systems and texture distributions that strongly affect properties such as porosity and permeability. Manual analysis of X-ray microtomography (μCT) images is time-consuming and subjective, which has encouraged the development of automated segmentation methods.

Two semantic segmentation architectures were implemented and compared: U-Net and Attention U-Net. The study used nine grayscale petrographic texture images integrated into the Randen texture segmentation framework, where each texture corresponds to a distinct class label. A predefined mask specifies the spatial regions of these classes, and a composite image is generated by assigning each texture to its corresponding region. The mask therefore serves as the ground-truth segmentation map used to evaluate the model predictions.

Training samples were generated as 32 × 32 labeled patches derived from the texture images integrated into the Randen framework, while evaluation was performed on a composite image with known ground-truth segmentation. Systematic experiments were conducted by varying the optimizer, the number of training epochs, and the network depth.

The results show that the best U-Net configuration achieved an accuracy of 0.9093 using a 15-layer architecture with Adam and 24 epochs, while the best Attention U-Net configuration achieved an accuracy of 0.8920 using a 15-layer architecture with SGDM and 50 epochs. Timing experiments also showed that U-Net required substantially less training time.

Overall, the findings suggest that encoder–decoder deep learning models are capable of effectively segmenting petrographic textures, with U-Net offering the best trade-off between accuracy and computational efficiency in this study.

This project aims to numerically simulate the deformations of red blood cell (RBC) membranes and to study their stationary shapes. Our approach simplifies the structure of the RBC membrane as a vesicle membrane due to its similarity to a phospholipid bilayer structure.

We adopt the well-known Helfrich model, in which the behavior of the membrane is described by minimizing a bending energy depending on the square of the membrane curvature. This results in a highly nonlinear problem, presenting significant challenges in computational fluid dynamics.

To address this, we propose a simplified modeling approach using tools from differential geometry. The problem is formulated as an optimization problem where the membrane minimizes its bending energy while preserving fixed surface area and perimeter constraints.

This leads to a saddle-point problem involving two Lagrange multipliers. The optimality conditions are derived and the resulting initial value problem is solved using various numerical schemes, including a shooting method.