researchseminars.org

M. Graham, A. Linot (Univ. of Wisconsin-Madison)
2021-10-20 19:00 - 20:00 UTC | https://researchseminars.org/talk/CNSwebinar/1/
Title: Data-driven dimension reduction, dynamic modeling, and control of complex chaotic systems (https://researchseminars.org/talk/CNSwebinar/1/)
by M. Graham, A. Linot (Univ. of Wisconsin-Madison) as part of Georgia Tech CNS Nonlinear Webinar

Abstract
Our overall aim is to combine ideas from dynamical systems theory and machine learning to develop and apply reduced-order models of flow processes with complex chaotic dynamics. A particular aim is a minimal description of dynamics on manifolds of dimension much less than the nominal state dimension and use of these models to develop effective control strategies for reducing energy dissipation.

Alec Linot: Modeling chaotic spatiotemporal dynamics with a minimal representation using Neural ODEs

Solutions to dissipative partial differential equations that exhibit chaotic dynamics often evolve to attractors that exist on finite-dimensional manifolds. We describe a data-driven reduced order modelling (ROM) method to find the coordinates on this manifold and find an ordinary differential equation (ODE) in these coordinates. The manifold coordinates are found by reducing the system dimension via an undercomplete autoencoder – a neural network that reduces then expands dimension – and an ODE is learned in this coordinate system with a Neural ODE. Learning an ODE, instead of a discrete time-map, allows us to evolve trajectories arbitrarily far forward, and allows for training on unevenly and/or widely spaced data in time. We test on the Kuramoto-Sivashinsky equation for domain sizes that exhibit spatiotemporally chaos, and find the ROM gives accurate short- and long-time statistics with training data separated up to 0.7 Lyapunov times.
https://arxiv.org/abs/2109.00060

K. Zeng, Carlo Perez de Jesus, Daniel Floryan (Univ. of Wisconsin-Madison, Univ. of Houston)
2021-10-27 19:00 - 20:00 | https://researchseminars.org/talk/CNSwebinar/2/
Title: Charting dynamics from data (https://researchseminars.org/talk/CNSwebinar/2/)
by K. Zeng, Carlo Perez de Jesus, Daniel Floryan (Univ. of Wisconsin-Madison, Univ. of Houston) as part of Georgia Tech CNS Nonlinear Webinar

Abstract
Kevin Zeng: Deep Reinforcement Learning Using Data-Driven Reduced-Order Models Discovers and Stabilizes Low Dissipation Equilibria

Deep reinforcement learning (RL), a data-driven method capable of discovering complex control strategies for high-dimensional systems, requires substantial interactions with the target system, making it costly when the system is computationally or experimentally expensive (e.g. flow control). We mitigate this challenge by combining dimension reduction via an autoencoder with a neural ODE framework to learn a low-dimensional dynamical model, which we substitute in place of the true system during RL training to efficiently estimate the control policy. We apply our method to data from the Kuramoto-Sivashinsky equation. With a goal of minimizing dissipation, we extract control policies from the model using RL and show that the model-based strategies perform well on the full dynamical system and highlight that the RL agent discovers and stabilizes a forced equilibrium solution, despite never having been given explicit information about this state’s existence.
https://arxiv.org/abs/2104.05437

Carlo Perez de Jesus
Dept. of Chemical and Biological Engineering, Univ. of Wisconsin-Madison

Data-driven estimation of inertial manifold dimension for chaotic Kolmogorov flow and time evolution on the manifold

Model reduction techniques have previously been applied to evolve the Navier-Stokes equations in time, however finding the minimal dimension needed to correctly capture the key dynamics is not a trivial task. To estimate this dimension we trained an undercomplete autoencoder on weakly chaotic vorticity data (32x32 grid) from Kolmogorov flow simulations, tracking the reconstruction error as a function of dimension. We also trained a discrete time stepper that evolves the reduced order model with a nonlinear dense neural network. The trajectory travels in the vicinity of relative periodic orbits (RPOs) followed by sporadic bursting events. At a dimension of five (as opposed to the full state dimension of 1024), power input-dissipation probability density function is well-approximated; Fourier coefficient evolution shows that the trajectory correctly captures the heteroclinic connections (bursts) between the different RPOs, and the prediction and true data track each other for approximately a Lyapunov time.

Daniel Floryan https://dfloryan.github.io/
Mechanical Engineering at the University of Houston

Charting dynamics from data

We often find ourselves working with systems for which governing equations are unknown, or if they are known, they may be high-dimensional to the point of being difficult to analyze and prohibitively expensive to make predictions with. These difficulties, together with the ever-increasing availability of data, have led to the new paradigm of data-driven model discovery. I will present recent work that fruitfully combines a classical idea from applied mathematics with modern methods of machine learning to learn minimal dynamical models directly from time series data. In full analogy with cartography, we learn a representation of a system as an atlas of charts. This approach allows us to obtain dynamical models of the lowest possible dimension, leads to computational benefits, and can separate state space into regions of distinct behaviors.
https://arxiv.org/abs/2108.05928

Vladimir Rosenhaus
2022-04-12 15:00 - 16:00 | https://researchseminars.org/talk/CNSwebinar/3/
Title: Feynman rules for wave turbulence (https://researchseminars.org/talk/CNSwebinar/3/)
by Vladimir Rosenhaus as part of Georgia Tech CNS Nonlinear Webinar

Abstract
It has long been known that weakly nonlinear field theories can have a late-time stationary state that is not the thermal state, but a wave turbulent state (the Kolmogorov-Zakharov state) with a far-from-equilibrium cascade of energy. We go beyond the existence of the wave turbulent state, studying fluctuations about the wave turbulent state. Specifically, we take a classical field theory with an arbitrary quartic interaction and add dissipation and Gaussian-random forcing. Employing the path integral relation between stochastic classical field theories and quantum field theories, we give a prescription, in terms of Feynman diagrams, for computing correlation functions in this system. We explicitly compute the two-point and four-point functions of the field to next-to-leading order in the coupling. Through an appropriate choice of forcing and dissipation, these correspond to correlation functions in the wave turbulent state. As a check, we reproduce the next-to-leading order term in the kinetic equation. The correlation functions and corrections to the KZ state that we compute should, in principle, be experimentally measurable quantities.

Zeb Rocklin
2022-04-19 15:00 - 16:00 | https://researchseminars.org/talk/CNSwebinar/4/
Title: Rigidity percolation in a random tensegrity via analytic graph theory (https://researchseminars.org/talk/CNSwebinar/4/)
by Zeb Rocklin as part of Georgia Tech CNS Nonlinear Webinar

Abstract
Tensegrities are mechanical structures that include cable-like elements that are strong and lightweight relative to rigid rods yet support only extensile stress. From suspension bridges to the musculoskeletal system to individual biological cells, humanity makes excellent use of tensegrities, yet the sharply nonlinear response of cables presents serious challenges to analytical theory. Here we consider large tensegrity structures with randomly placed cables (and struts) overlaid on a regular rigid backbone whose corresponding system of inequalities is reduced via analytic theory to an exact graph theory. We identify a novel coordination number that controls two rigidity percolation transitions: one in which global interactions between cables first support external loads and one in which the structure becomes fully rigid. We show that even the addition of a few cables strongly modifies conventional rigidity percolation, both by modifying the sharpness of the transition and by introducing avalanche effects in which a single constraint can eliminate multiple floppy modes.
