Accurate physical modeling and accelerated numerical simulation of turbulent flows remain primary challenges in CFD for aerospace engineering and related fields. This dissertation tackles these challenges with a focus on Reynolds-Averaged Navier--Stokes (RANS) models, which will continue to serve as the backbone for many practical aircraft applications. Specifically, in RANS turbulence modeling, the challenges include developing more efficient ensemble filters to learn nonlinear eddy viscosity models from observation data that move beyond the classical Boussinesq hypothesis, as well as developing non-equilibrium models that break away from the weak equilibrium assumption while maintaining computational efficiency. For accelerating RANS simulations, the challenges include leveraging existing simulation data to optimize the computational workflow while maintaining the method's adaptability to various computational settings. From a fundamental and mathematical perspective, we view these challenges as problems of modeling and learning complex nonlinear and nonlocal mappings, which we categorize into three types: field-to-point, field-to-field, and ensemble-to-ensemble. To model and resolve these mappings, we build up on recent advancements in machine learning and develop novel neural operator-based methods that not only possess strong representational capabilities but also preserve critical physical and mathematical principles. With the developed tools, we have demonstrated promising preliminary results in addressing these challenges and have the potential to significantly advance the state of the art in RANS turbulence modeling and simulation acceleration. / Doctor of Philosophy / Understanding and accurately predicting turbulent flows, such as those around airplanes or ships, are among the biggest challenges in computational fluid dynamics (CFD). This research aims to improve Reynolds-Averaged Navier--Stokes (RANS) models, which are widely used in practical engineering applications. Traditional RANS turbulence models are based on simplified assumptions that are linear and local, making it difficult to capture the true complexity of turbulent flows. My work addresses this limitation by developing new models that leverage advanced machine learning techniques to better represent turbulence. Specifically, I have focused on developing methods that extend beyond conventional approaches by learning more accurate local nonlinear constitutive relations and incorporating nonlocal effects---an important step toward improving simulation accuracy. In addition, I have explored strategies to accelerate RANS simulations by making more effective use of existing data, providing better initial conditions for simulations, and ultimately reducing computational costs. Preliminary results indicate that these new methods have the potential to push the boundaries of RANS turbulence modeling, enabling more accurate and efficient simulations.

In this thesis three different approaches for solving partial differential equa-tions with neural networks will be explored; namely Physics-Informed NeuralNetworks, Fourier Neural Operators and the Deep Ritz method. Physics-Informed Neural Networks and the Deep Ritz Method are unsupervised machine learning methods, while the Fourier Neural Operator is a supervised method. The Physics-Informed Neural Network is implemented on Burger’s equation,while the Fourier Neural Operator is implemented on Poisson’s equation and Darcy’s law and the Deep Ritz method is applied to several variational problems. The Physics-Informed Neural Network is also used for the inverse problem; given some data on a solution, the neural network is trained to determine what the underlying partial differential equation is whose solution is given by the data. Apart from this, importance sampling is also implemented to accelerate the training of physics-informed neural networks. The contributions of this thesis are to implement a slightly different form of importance sampling on the physics-informed neural network, to show that the Deep Ritz method can be used for a larger class of variational problems than the original publication suggests and to apply the Fourier Neural Operator on an application in geophyiscs involving Darcy’s law where the coefficient factor is given by exponentiated two-dimensional pink noise.