Georgia Institute of Technology College of Sciences

In this talk, we introduce and survey continuous-time deep learning approaches based on neural ordinary differential equations (neural ODEs) arising in supervised learning, generative modeling, and numerical solution of high-dimensional optimal control problems. We will highlight theoretical advantages and numerical benefits of neural ODEs in deep learning and their use to solve otherwise intractable PDE problems.

In this talk, we give an overview of recent work in gradient elasticity.  We first give a friendly introduction to gradient elasticity—a mathematical framework for understanding three-dimensional bodies that do not dissipate a form of energy during deformation. Compared to classical elasticity theory, gradient elasticity incorporates higher spatial derivatives that encode certain microstructural information and become significant at small spatial scales. We then discuss a recently introduced theory of three-dimensional Green-elastic bodies containing gradient elastic material boundary surfaces. We then indicate how the resulting model successfully eliminates pathological singularities inherent in classical linear elastic fracture mechanics, presenting a new and geometric alternative theory of fracture.

An optimal control problem in the space of probability measures, and the viscosity solutions of the corresponding dynamic programming equations defined using the intrinsic linear derivative are studied. The value function is shown to be Lipschitz continuous with respect to a novel smooth Fourier Wasserstein metric. A comparison result between the Lipschitz viscosity sub and super solutions of the dynamic programming equation is proved using this metric, characterizing the value function as the unique Lipschitz viscosity solution. This is joint work with Prof. H. Mete Soner. 

  It is known that the symplectic property is preserved by the mean curvature flow in a K\"ahler-Einstein surface which is called "symplectic mean curvature flow". It was proved that there is no finite time Type I singularities for the symplectic mean curvature flow. We will talk about recent progress on an important Type II singularity of symplectic mean curvature flow-symplectic translating soliton. We will show that a symplectic translating soliton must be a plane under some natural assumptions which are necessary by investigating some examples.