Georgia Institute of Technology College of Sciences

This dissertation focuses on neural network approximations of systems that have low dimensional structure, specifically for generative modeling and latent dynamics approximation. 

First, we establish an approximation framework for push-forward deep generative models under the manifold hypothesis. Given samples from a target probability measure supported on a low-dimensional manifold embedded in Euclidean space, we construct a neural network such that the push forward of an easy-to-sample measure by that network is close in Wasserstein metric to the target measure. The construction decomposes the target measure into local measures supported on local charts of the manifold and generates these local measures using optimal transportation theory. Crucially, the constructed network size scales with the intrinsic dimension of the manifold rather than the ambient dimension.

Next, we move on to latent dynamics learning, particularly for surrogate modeling. Given example trajectories, our goal is to construct a neural network which can autoregressively predict the evolution of a given unseen initial condition. To do so, we developed WELDNet (which stands for Windowed autoEncoders for Learning Dynamics with Neural Networks). In this approach, the time domain is segmented into overlapping regions called windows, upon which autoencoder networks are trained to compress the data to low dimensional latent space and propagator networks are trained to learn the induced time stepping map on latent space. The different windows are connected by transcoder neural networks which translate between two latent space representations of the same data. We established an approximation theory for WELDNet and performed numerical experiments on one- and two-dimensional evolutionary PDEs to show the advantage of this windowed approach compared to state-of-the-art baselines.

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