DDPS | Generative Machine Learning Approaches for Data-Driven Modeling and Reductions

Generative Machine Learning Approaches for Data-Driven Modeling and Reductions of Non-Linear Dynamics in Scientific Simulation by Paul Atzberger

Abstract: Scientific simulations arising from the statistical mechanics of soft materials, complex fluids, and biophysical systems present significant challenges given the wide range of spatial-temporal scales and roles of fluctuations. Recent emerging data-driven generative approaches are presenting new opportunities for developing reduced-order models and multi-scale methods. A current challenge is to develop ways to incorporate into learning methods inductive biases including physical principles and other prior scientific knowledge. To learn reductions for modeling non-linear dynamics, we discuss Geometric Variational Autoencoders (GD-VAEs) for obtaining representations incorporating topological information, smoothness, and adherence to physical principles. We then show results for how GD-VAEs can be used for data-driven modeling and reductions of high dimensional dynamical systems and non-linear PDEs. We also discuss ways to enhance interpretability of the learned representations. We then discuss Stochastic Dynamic Generative Adversarial Networks (SDYN-GANs) for data-driven learning of probabilistic models from observations of stochastic systems. SDYN-GANs learns dynamical representations in terms of SDEs and stable m-step stochastic numerical integrators for use in simulations. We show how SDYN-GANs can be used for inertial stochastic systems arising in statistical mechanics to learn parameters both of the drift and diffusive contributions. We then discuss how SDYN-GANs can be used to learn unknown non-linear force-laws from observations of the trajectories of the stochastic dynamics. The discussed methods and results show a few strategies toward developing more robust and interpretable machine learning methods for scientific simulations.

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.

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