Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

To treat the multiple time scales of ocean dynamics in an efficient manner, the baroclinic-barotropic splitting technique has been widely used for solving the primitive equations for ocean modeling. In this paper, we propose second and third-order multirate explicit time-stepping schemes for such split systems based on the strong stability-preserving Runge-Kutta (SSPRK) framework. Our method allows for a large time step to be used for advancing the three-dimensional (slow) baroclinic mode and a small time step for the two-dimensional (fast) barotropic mode, so that each of the two mode solves only need satisfy their respective CFL condition to maintain numerical stability. It is well known that the SSPRK method achieves high-order temporal accuracy by utilizing a convex combination of forward-Euler steps. At each time step of our method, the baroclinic velocity is first computed by using the SSPRK scheme to advance the baroclinic-barotropic system with the large time step, then the barotropic velocity is specially corrected by using the same SSPRK scheme with the small time step to advance the barotropic subsystem with a barotropic forcing interpolated based on values from the preceding baroclinic solves. Finally, the fluid thickness and the sea surface height perturbation is updated by coupling the predicted baroclinic and barotropic velocities. Two benchmark tests drawn from the ``MPAS-Ocean" platform are used to numerically demonstrate the accuracy and parallel performance of the proposed schemes.

The bluejeans link for the seminar is https://bluejeans.com/457724603/4379

While deep learning has been used for dynamics learning, limited physical accuracy and an inability to generalize under distributional shift limit its applicability to real world. In this talk, I will demonstrate how to incorporate symmetries into deep neural networks and significantly improve the physical consistency, sample efficiency, and generalization in learning dynamics. I will showcase the applications of these models to challenging problems such as turbulence forecasting and trajectory prediction for autonomous vehicles.

We construct integrators to be used in Hamiltonian (or Hybrid) Monte Carlo sampling. The new integrators are easily implementable and, for a given computational budget, may deliver five times as many accepted proposals as standard leapfrog/Verlet without impairing in any way the quality of the samples. They are based on a suitable modification of the   processing technique first introduced by J.C. Butcher. The idea of modified processing may also be useful for other purposes, like the construction of high-order splitting integrators with positive coefficients.

Joint work with Mari Paz Calvo, Fernando Casas, and Jesús M. Sanz-Serna

Many scientific problems involve invariant structures, and learning functions that rely on a much lower dimensional set of features than the data itself.   Incorporating these invariances into a parametric model can significantly reduce the model complexity, and lead to a vast reduction in the number of labeled examples required to estimate the parameters.  We display this benefit in two settings.  The first setting concerns ReLU networks, and the size of networks and number of points required to learn certain functions and classification regions.  Here, we assume that the target function has built in invariances, namely that it only depends on the projection onto a very low dimensional, function defined manifold (with dimension possibly significantly smaller than even the intrinsic dimension of the data).  We use this manifold variant of a single or multi index model to establish network complexity and ERM rates that beat even the intrinsic dimension of the data.  We should note that a corollary of this result is developing intrinsic rates for a manifold plus noise data model without needing to assume the distribution of the noise decays exponentially, and we also discuss implications in two-sample testing and statistical distances.  The second setting for building invariances concerns linearized optimal transport (LOT), and using it to build supervised classifiers on distributions.  Here, we construct invariances to families of group actions (e.g., shifts and scalings of a fixed distribution), and show that LOT can learn a classifier on group orbits using a simple linear separator.   We demonstrate the benefit of this on MNIST by constructing robust classifiers with only a small number of labeled examples.  This talk covers joint work with Timo Klock, Xiuyuan Cheng, and Caroline Moosmueller.