Departmental Colloquium

Questions such as this may be formulated as questions regarding solutions to nonlinear evolutionary partial differential equations having a small coefficient on the leading order derivative term. Evolutionary partial differential equations may be regarded as (semi-) dynamical systems in an infinite-dimensional space. An abstract theorem is proved giving the existence of an invariant manifold for a semi-dynamical system when an approximately invariant manifold exists with a certain topological nondegeneracy condition in a neighborhood. This is then used to prove the existence of eternal solutions to the nonlinear PDE and answer the question about the motion of a droplet on a curved manifold. The abstract theorem extends fundamental work of Hirsch-Pugh-Shub and Fenichel on the perturbation of invariant manifolds from the 1970's to infinite-dimensional semi-dynamical systems. This represents joint work with Kening Lu and Chongchun Zeng.

This talk aims to close the gap by developing new theories and scalable numerical algorithms for complex dynamical systems that can be realistically predicted and validated. We are creating new technologies that can be translated into more secure and reliable new trustworthy AI systems that can be deployed for real-time complex dynamical system prediction, surveillance, and defense applications to improve the stability and efficiency of complex dynamical systems and national security of the United States. We will present a novel neural homogenization-based physics-informed neural network (NN) for multiscale problems. We will also introduce new NNs that learn functionals and nonlinear operators from functions with simultaneous uncertainty estimates. In particular, we present a probabilistic neural operator network training procedure for solving partial differential equations with inhomogeneous boundary conditions. Using a light-weight extension of deep operator network (DeepONet) architecture, the trained networks are designed to provide rapid predictions along with simultaneous uncertainty estimates to help identify potential inaccuracies in the network predictions. We demonstrate that the novel probabilistic DeepONet can learn various explicit operators with predictive uncertainties.

We review the definition of Schwartz and tempered distributions and functions on Nash manifolds given by du Cloux and Aizenbud-Gourevitch. We then use these definitions, and some basic homological computations, to describe the solutions to some special systems of differential equations. We will then show how we can use these results to solve certain problems in representation theory.