Georgia Institute of Technology College of Sciences

One essential challenge in the computational modeling of multiscale systems is the availability of reliable and interpretable closures that faithfully encode the micro-dynamics. For systems without clear scale separation, there generally exists no such a simple set of macro-scale field variables that allow us to project and predict the dynamics in a self-determined way. We introduce a machine-learning (ML) based approach that enables us to reduce high-dimensional multi-scale systems to reliable macro-scale models with low-dimensional variational structures that preserve canonical degeneracies and symmetry constraints. The non-Newtonian hydrodynamics of polymeric fluids is used as an example to illustrate the essential idea. Unlike our conventional wisdom about ML modeling that focuses on learning the PDE form, the present approach directly learns the energy variational structure from the micro-model through an end-to-end process via the joint learning of a set of micro-macro encoder functions. The final model, named the deep non-Newtonian model (DeePN2), retains a multi-scale nature with clear physical interpretation and strictly preserves the frame-indifference constraints. We show that DeePN2 can capture the broadly overlooked viscoelastic differences arising from the specific molecular structural mechanics without human intervention.

The study of nodal statistics provides insight into the spectral properties of graphs and matrices, drawing strong parallels with classical results in analysis. In this talk, we will give an overview of the field, covering key results on nodal domains and nodal counts for graphs and their connection to known results in the continuous setting. In addition, we will discuss some recent progress towards a complete understanding of the extremal properties of the nodal statistics of a matrix.

This thesis presents several contributions to the fields of image and geometry processing. In 2D image processing, we propose a method that not only computes the relative depth of objects in bitmap format but also inpaints occluded regions using a PDE-based model and vector representation. Our approach demonstrates both qualitative and quantitative advantages over the state-of-the-art depth-aware bitmap-to-vector conversion models.

In the area of 3D point cloud processing, we introduce a method for generating a robust normal vector field that preserves first order discontinuity while being resistant to noise, supported by a degree of theoretical guarantee. This technique has potential applications in solving PDEs on point clouds, detecting sharp features, and reconstructing surfaces from incomplete and noisy data.

Additionally, we present a dedicated work on surface reconstruction from point cloud data. While many existing models can reconstruct implicit surfaces and some include denoising capabilities, a common drawback is the loss of sharp features: edges and corners are often smoothed out in the process. To address this limitation, we propose a method that not only denoises but also preserves sharp edges and corners during surface reconstruction from noisy data.