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Toward Robust Machine Learning Methods for Data-Driven Modeling and Simulation

Program in Applied Mathematics Colloquium

Toward Robust Machine Learning Methods for Data-Driven Modeling and Simulation
Series: Program in Applied Mathematics Colloquium
Location: MATH 501
Presenter: Paul Atzberger, Mechanical Engineering Department, UC Santa Barbara

Recent emerging data-driven methods combined with more traditional numerical analysis are presenting new opportunities for model development and for performing simulations. Motivating applications include reduced-order and sub-grid modeling in fluid mechanics, model selection and solvers in biophysics, and surrogates in engineering for optimization and design. A central challenge for recent emerging "black box" techniques, such as deep neural networks, is to develop ways to incorporate into learning methods known scientific and technical knowledge as constraints or as other types of inductive biases. Motivated by biophysical modeling of membrane proteins and the roles played by geometry and transport equations on curved surfaces, we show how hybrid data-driven solvers can be developed for partial differential equations on manifolds.  We then discuss how these methods can be used to study membrane protein interactions and drift-diffusion dynamics taking into account the roles of hydrodynamic coupling, geometry, and thermal fluctuations.  We then discuss another class of problems and general methods for learning representations of non-linear stochastic dynamics leveraging recent data-driven approaches related to Variational Autoencoders (VAEs) and Generative Adversarial Networks (GANs).  We show how these approaches can be used to obtain inductive biases incorporating physical principles when developing reduced-order models, dimension reductions, or learning unknown force-laws.  We present results for partial differential equations in fluid mechanics, reaction-diffusion processes, and particle systems.  Throughout, we aim to highlight opportunities for combining recent emerging machine learning methods with more traditional numerical approaches to develop practical and more robust computational methods for scientific modeling and simulation.

Bio: Paul J. Atzberger studied mathematics at the Courant Institute at New York University where he received his PhD. Subsequently, he was a postdoctoral fellow at Rensselaer Polytechnic Institute. He joined the faculty at the University of California Santa Barbara in the Department of Mathematics.  He works on research in scientific computation, machine learning, and stochastic analysis with applications in the sciences and engineering.

 Paul Atzberger Research Group:  https://web.math.ucsb.edu/~atzberg/pmwiki_intranet/index.php?n=AtzbergerHomePage.Homepage?setskin=atzbergerHomepage4

 Place: Math Building, Room 402  https://map.arizona.edu/89