Point clouds and sets are ubiquitous, unusual and unstructured data-types which present unique problems to machine learning when used as inputs. Since sets are inherently unaffected by permutations, the input space for these problems naturally forms a non-Euclidean space which is oftentimes not even a manifold. Moreover, similar inputs can have wildly varying cardinalities and so the input space is in general infinite-dimensional. Despite these mathematical difficulties, PointNet (Qi et al.) and Deep Sets (Zaheer et al.) form two foundational contributions for deep learning in this area. In this thesis we study the expressive power of such networks. To that end, we prove theorems about their Lipschitz properties, extend them to well-studied infinite-dimensional spaces, and prove new cardinality-agnostic universality results for point clouds. These results completely characterize the approximable functions and so can be used to compare the representational strength of the underlying model classes. In particular, a normalized version of the DeepSets architecture cannot uniformly approximate the diameter function but can uniformly approximate the center-of-mass function whereas PointNet can uniformly approximate the former but not the latter. Additionally, even when limited to a fixed input cardinality, PointNet cannot uniformly approximate the average value of a continuous function over sets of more than two points. We additionally obtain explicit error lower-bounds for this error of approximation and a present a simple geometric method to produce arbitrarily many examples of this failure-mode. Along the way, we also prove various general purpose universal approximation theorems, one of which is for generalized neural networks whose input space is a set of unknown or nonexistent topology.

We discretize spatial domains into lattices. We provide the multivariate Fokker-Planck partial differential equation and its numerical solutions. We establish our convolutional neural network details, aiming to train these networks on our Fokker-Planck data with the goal of recovering Fokker-Planck coefficients. We break up our objective into different cases. For each case, we discuss results. First, we consider the simplified diffusion equation, then the advection-diffusion equation, where our networks learn the differential operators. We consider long-time integration methods. Finally, we consider finite difference and finite volume methods.

The investigation of soft materials poses many important challenges having implications for application areas that range from the design of better materials for use in engineering practice to gaining further insights into the functioning of biological organisms. In soft materials there is often an interplay between direct microstructure-level interactions and fluctuations in yielding observed bulk macroscopic material properties. As part of these interactions geometry and hydrodynamic interactions often play a central role. We shall investigate interfacial phenomena associated with soft materials, particularly relevant to the study of lipid bilayer membranes. We shall address the problem of how to formulate and numerically approximate continuum mechanics on 2-manifolds in the case of non-trivial geometries

having spherical topology. We shall be particularly interested in developing methods for investigating the case of hydrodynamic flow responses on curved surfaces. We shall present results for an initial model assuming that we have smooth, star-shaped (radial) membrane geometry. We show how spectral methods of approximation can be developed based on use of spherical harmonics expansions, Lebedev quadrature. We use these approaches to investigate how hydrodynamic flow responses depend on the surface geometry. We find that the surface curvature can significantly effect dissipation rates and augment flow responses. We then develop more general methods for the case of any smooth geometry having spherical topology using numerical approaches based on

Generalized Moving Least Squares (GMLS). We use these to further investigate hydrodynamic flows in this setting. We conclude by briefly discussing our current work to extend these numerical approaches to even more general smooth compact manifolds without the need for spherical topology.