Spring 2024 

April 4 , 2024: Boeing Colloquium - Diane Henderson, Penn State

Location and time: SMI 205 at 4pm

Title: Understanding nonlinear surface water waves on deep water

Abstract: Oceanographers in the 60s conducted an ambitious experiment (1) in which they tracked waves that were generated by large storms near New Zealand across the Pacific Ocean until they hit the beaches at Alaska. Paradoxically, at about the same time, mathematicians in the Soviet Union, the U.S., and England (2) independently developed mathematical models that predicted such waves to be unstable, meaning that they could not survive to be tracked all the way across the Pacific. In the 70s experimentalists (3) conducted laboratory experiments on these types of waves. They generated waves with a given frequency that propagated down a wavetank, but at the end of the wavetank, the waves had a slightly lower frequency. The mathematical model did not explain this observation. In this talk, we consider these observations and our experiments on modulated wavetrains within the framework of the mathematical models: the scalar and vector nonlinear Schroedinger equations with and without dissipation and/or higher order terms. We examine the data within the context of conserved quantities of these equations to determine when the models are likely to be valid or not. We present recent results from our quest, motivated by recent stability analyses of Stokes waves (4), to observe subharmonic instabilities of waves in deep, finite and shallow water.

(1) Snodgrass, F. E., G. W. Groves, K. F. Hasselmann, G. R. Miller, W. H. Munk, and W. H. Powers (1966), Propagation of ocean swell across the Pacific, Philos. Trans. R. Soc. London A, 259, 431–497.

(2a) Benney, D. J. & Newell, A. C. 1967 The propagation of nonlinear wave envelopes. Stud. Appl.Maths 46, 133–139.

(2b) Lighthill, M. J. 1965 Contribution to the theory of waves in nonlinear dispersive systems. J. Inst. Math. Applics. 1, 269–306.

(2c) Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wavetrains in deep water. Part 1. J. Fluid Mech. 27, 417–430.

(2d) Ostrovsky, L. A. 1967 Propagation of wave packets and space-time self-focussing in a nonlinearmedium. Sov. Phys. J. Exp. Theor. Phys. 24, 797–800.

(2e) Whitham, G. B. 1967 Nonlinear dispersion of water waves. J. Fluid Mech. 27, 399–412.

(2f) Zakharov, V. E. 1967 Instability of self-focusing of light. Sov. Phys. J. Exp. Theor. Phys. 24, 455–459.

(2g) Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 2, 190–194.

(3a) Lake, B. M. & Yuen, H. C. 1977 A note on some water-wave experiments and the comparison ofdata with theory. J. Fluid Mech. 83, 75–81.

(3b) Lake, B. M., Yuen, H. C., Rungaldier, H. & Ferguson, W. E. 1977 Nonlinear deep-water waves:theory and experiment. Part 2. Evolution of a continuous wave train. J. Fluid Mech. 83, 49–74.

(4) B. Deconinck, S. A. Dyachenko, P. M. Lushnikov, A. Semenova, The instability of near-extreme Stokes waves, Proceedings of the National Academy of Sciences, 120(32):p.e2308935120 (2023)

April 18, 2024: Nicolas Garcia Trillos, University of Wisconsin at Madison

Location: SMI 205 at 4pm

Title: FedCBO: Reaching Group Consensus in Clustered Federated Learning and Robustness to Backdoor Adversarial Attacks

Abstract: Federated learning is an important framework in modern machine learning that seeks to integrate the training of learning models from multiple users, each with their own local data set, in a way that is sensitive to the users’ data privacy and to communication cost constraints. In clustered federated learning, one assumes an additional unknown group structure among users, and the goal is to train models that are useful for users in each group, rather than to train a single global model for all users.  

In the first part of this talk, I will present a novel solution to the problem of clustered federated learning that is inspired by ideas in consensus-based optimization (CBO). Our new CBO-type method is based on a system of interacting particles that is oblivious to group memberships. Our algorithm is accompanied by theoretical justification and tested on real data experiments. I will then discuss an additional issue of concern in federated learning: the vulnerability of federated learning protocols to “backdoor” adversarial attacks. This discussion will motivate the introduction of a second, improved particle system with enhanced robustness properties and that, at an abstract level, can be interpreted as a bi-level optimization algorithm based on interacting particle dynamics. 

This talk is based on joint works with Sixu Li, Yuhua Zhu, Konstantin Riedl, and Jose Carrillo.

May 2, 2024: Boeing Colloquium - Andrew Fowler, University of Limerick

Location: SMI 205 at 4pm

Title: Drumlins

Abstract: Drumlins are small rounded hills which occur in swarms, and which are formed under ice sheets. They are ubiquitous  in North America and Northern Europe due to the former presence of the great ice sheets of the last ice age. The enigma of their formation has generated a rich literature over hte last two hundred years. The instability theory of drumlin formation has its roots in the work of Richard Hindmarsh in the late 1990s. His basic idea was that drumlins form through an instability due to the shearing motion of ice flowing over a deformable subglacial till. The ingredients of the theory are thus ice flow and till flow. Later, water flow was added. The development of the theory beyond the basic linear instability result was initially hampered by a catacomb of difficulties: these include till rheology, two-dimensionality and cavitation.The numerical solution of the model is also fraught with complication. In this talk,  I will describe and illustrate their efforts to resolve these and other difficulties as they arose over the course of the last twenty years, and a summary of the way in which the theory needs to progress will be outlined.

May 16, 2024: Haizhao Yang, Maryland

Location: SMI 205 at 4pm

Title: Finite Expression Method: A Symbolic Approach for Scientific Machine Learning 

Abstract: Machine learning has revolutionized computational science and engineering with impressive breakthroughs, e.g., making the efficient solution of high-dimensional computational tasks feasible and advancing domain knowledge via scientific data mining. This leads to an emerging field called scientific machine learning. In this talk, we introduce a new method for a symbolic approach to solving scientific machine learning problems. This method seeks interpretable learning outcomes via combinatorial optimization in the space of functions with finitely many analytic expressions and, hence, this methodology is named the finite expression method (FEX). It is proved in approximation theory that FEX can avoid the curse of dimensionality in discovering high-dimensional complex systems. As a proof of concept, a deep reinforcement learning method is proposed to implement FEX for learning the solution of high-dimensional PDEs and learning the governing equations of raw data