Georgia Institute of Technology College of Sciences

In many applications across science and engineering it is common to have access to disparate types of data or models with different levels of fidelity. In general, low-fidelity data are easier to obtain in greater quantities, but it may be too inaccurate or not dense enough to accurately train a machine learning model. High-fidelity data is costly to obtain, so there may not be sufficient data to use in training, however, it is more accurate.  A small amount of high-fidelity data, such as from measurements or simulations, combined with low fidelity data, can improve predictions when used together. The important step in such constructions is the representation of the correlations between the low- and high-fidelity data. In this talk, we will present two frameworks for multifidelity machine learning. The first one puts particular emphasis on operator learning, building on the Deep Operator Network (DeepONet). The second one is inspired by the concept of model reduction. We will present the main constructions along with applications to closure for multiscale systems and continual learning. Moreover, we will discuss how multifidelity approaches fit in a broader framework which includes ideas from deep learning, stochastic processes, numerical methods, computability theory and renormalization of complex systems.

A fully nonlinear surface dynamics of the time dependent potential flow of ideal incompressible fluid with a free surface is considered in two dimensional geometry. Arbitrary large surface waves can be efficiently characterized through a time-dependent conformal mapping of a fluid domain into the lower complex half-plane. We reformulate the exact Eulerian dynamics through a non-canonical nonlocal Hamiltonian system for the pair of new conformal variables. We also consider a generalized hydrodynamics for two components of superfluid Helium which has the same non-canonical Hamiltonian structure. In both cases the fluid dynamics is fully characterized by the complex singularities in the upper complex half-plane of the conformal map and the complex velocity. Analytical continuation through the branch cuts generically results in the Riemann surface with infinite number of sheets including Stokes wave, An infinite family of solutions with moving poles are found on the Riemann surface. Residues of poles are the constants of motion. These constants commute with each other in the sense of underlying non-canonical Hamiltonian dynamics which provides an argument in support of the conjecture of complete Hamiltonian integrability of surface dynamics. If we consider initial conditions with short branch cuts then fluid dynamics is reduced to the complex Hopf equation for the complex velocity coupled with the complex transport equation for the conformal mapping. These equations are fully integrable by characteristics producing the infinite family of solutions, including the pairs of moving square root branch points. The solutions are compared with the simulations of the full Eulerian dynamics giving excellent agreement.

TBA