Deep Learning Approaches for Non-linear Dynamics: Generative Approaches for Data-Driven Simulations

Abstract: Recent emerging deep learning methods combined with more traditional numerical analysis are presenting new opportunities for developing data-driven approaches for modeling and simulation.  A central 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 partial differential equations.  We also discuss ways to enhance the 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 stochastic differential equations 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 for developing more robust and interpretable machine learning methods for scientific simulations.

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