Georgia Institute of Technology College of Sciences

Supervised operator learning is an emerging machine learning paradigm with applications to modeling the evolution of spatio-temporal dynamical systems and approximating general black-box relationships between functional data. We propose a novel operator learning method, LOCA (Learning Operators with Coupled Attention), motivated from the recent success of the attention mechanism. In our architecture, the input functions are mapped to a finite set of features which are then averaged with attention weights that depend on the output query locations. By coupling these attention weights together with an integral transform, LOCA is able to explicitly learn correlations in the target output functions, enabling us to approximate nonlinear operators even when the number of output function measurementsin the training set is very small. Our formulation is accompanied by rigorous approximation theoretic guarantees on the universal expressiveness of the proposed model. Empirically, we evaluate the performance of LOCA on several operator learning scenarios involving systems governed by ordinary and partial differential equations, as well as a black-box climate prediction problem. Through these scenarios we demonstrate state of the art accuracy, robustness with respect to noisy input data, and a consistently small spread of errors over testing data sets, even for out-of-distribution prediction tasks.
 

Parallel server systems have received a lot of attention since their introduction about 20 years ago. They are commonly used to model a situation where different type of jobs can be treated by servers with different specialties like data and call centers. Exact optimal policies are often not tractable for those systems. Instead, part of the literature was focused on finding policies that are asymptotically optimal as the load of the network approaches a value critical for stability (heavy traffic approximations). This is done by obtaining a weak convergence to a brownian control problem that is linked to a non-linear differential equation (Hamilton-Jacobi-Bellman). Asymptotically optimal policies have been analyzed for a long time under a restrictive assumption that is not natural for practical applications. This talk will present recent developments that allow for a more general asymptotic optimality result by focusing on the simplest non-trivial example.

Consider two volume-preserving, smooth diffeomorphisms f and g of a compact manifold M.  Define the random walk on M by selecting either f or g (i.i.d.) at each iterate.  A number of questions arise in this setting:

1. What are the closed subsets of M invariant under both f and g?
2. What are the stationary measures on M for the random walk.  In particular, are the stationary measures invariant under f and g?

Conjecturally, for a generic pair of f and g we should be able to answer the above.  I will describe one sufficient criteria on f and g underwhich we can give some partial answers to the above questions.  Such a criteria is expected to be generic amoung pairs of (volume-preserving) diffeomorphisms and should be able to be verified in a number of naturally occurring geometric settings where the above questions are not fully answered.  

We consider the 2-dim capillary gravity water wave problem -- the free boundary problem of the Euler equation with gravity and surface tension -- of finite depth x2 \in (-h,0) linearized at a uniformly monotonic shear flow U(x2). Our main results consist of two aspects, eigenvalue distribution and inviscid damping. We first prove that in contrast to finite channel flow and gravity wave, the linearized capillary gravity wave has two unbounded branches of eigenvalues for high wave numbers. Under certain conditions, we provide a complete picture of the eigenvalue distribution. Assuming there are no singular modes, we obtain the linear inviscid damping. We also identify the leading asymptotic terms of velocity and obtain the stronger decay for the remainders.