BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:M. Graham\, A. Linot (Univ. of Wisconsin-Madison) DTSTART:20211020T190000Z DTEND:20211020T200000Z DTSTAMP:20260404T111214Z UID:CNSwebinar/1 DESCRIPTION:Title: Data-driven dimension reduction\, dynamic modeling\, and control of complex chaotic systems\nby M. Graham\, A. Linot (Univ. of Wiscons in-Madison) as part of Georgia Tech CNS Nonlinear Webinar\n\n\nAbstract\nO ur overall aim is to combine ideas from dynamical systems theory and machi ne learning to develop and apply reduced-order models of flow processes wi th complex chaotic dynamics. A particular aim is a minimal description of dynamics on manifolds of dimension much less than the nominal state dimens ion and use of these models to develop effective control strategies for re ducing energy dissipation.\n\nAlec Linot: Modeling chaotic spatiotemporal dynamics with a minimal representation using Neural ODEs\n\nSolutions to d issipative partial differential equations that exhibit chaotic dynamics of ten evolve to attractors that exist on finite-dimensional manifolds. We de scribe a data-driven reduced order modelling (ROM) method to find the coor dinates on this manifold and find an ordinary differential equation (ODE) in these coordinates. The manifold coordinates are found by reducing the s ystem dimension via an undercomplete autoencoder – a neural network that reduces then expands dimension – and an ODE is learned in this coordina te system with a Neural ODE. Learning an ODE\, instead of a discrete time- map\, allows us to evolve trajectories arbitrarily far forward\, and allow s for training on unevenly and/or widely spaced data in time. We test on t he Kuramoto-Sivashinsky equation for domain sizes that exhibit spatiotempo rally chaos\, and find the ROM gives accurate short- and long-time statist ics with training data separated up to 0.7 Lyapunov times.\nhttps://arxiv. org/abs/2109.00060\n LOCATION:https://stable.researchseminars.org/talk/CNSwebinar/1/ END:VEVENT BEGIN:VEVENT SUMMARY:K. Zeng\, Carlo Perez de Jesus\, Daniel Floryan (Univ. of Wisconsi n-Madison\, Univ. of Houston) DTSTART:20211027T190000Z DTEND:20211027T200000Z DTSTAMP:20260404T111214Z UID:CNSwebinar/2 DESCRIPTION:Title: Charting dynamics from data\nby K. Zeng\, Carlo Perez de Jes us\, Daniel Floryan (Univ. of Wisconsin-Madison\, Univ. of Houston) as par t of Georgia Tech CNS Nonlinear Webinar\n\n\nAbstract\nKevin Zeng: Deep Re inforcement Learning Using Data-Driven Reduced-Order Models Discovers and Stabilizes Low Dissipation Equilibria\n\nDeep reinforcement learning (RL)\ , a data-driven method capable of discovering complex control strategies f or high-dimensional systems\, requires substantial interactions with the t arget system\, making it costly when the system is computationally or expe rimentally expensive (e.g. flow control). We mitigate this challenge by co mbining 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 equatio n. With a goal of minimizing dissipation\, we extract control policies fro m 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 an d stabilizes a forced equilibrium solution\, despite never having been giv en explicit information about this state’s existence.\nhttps://arxiv.org /abs/2104.05437\n\nCarlo Perez de Jesus\nDept. of Chemical and Biological Engineering\, Univ. of Wisconsin-Madison\n\nData-driven estimation of iner tial manifold dimension for chaotic Kolmogorov flow and time evolution on the manifold\n\nModel 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 ta sk. To estimate this dimension we trained an undercomplete autoencoder on weakly chaotic vorticity data (32x32 grid) from Kolmogorov flow simulation s\, 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 o f relative periodic orbits (RPOs) followed by sporadic bursting events. At a dimension of five (as opposed to the full state dimension of 1024)\, po wer 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 th e prediction and true data track each other for approximately a Lyapunov t ime.\n\nDaniel Floryan https://dfloryan.github.io/\nMechanical Engineering at the University of Houston\n\nCharting dynamics from data\n\nWe often f ind ourselves working with systems for which governing equations are unkno wn\, or if they are known\, they may be high-dimensional to the point of b eing 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 w ill present recent work that fruitfully combines a classical idea from app lied mathematics with modern methods of machine learning to learn minimal dynamical models directly from time series data. In full analogy with cart ography\, we learn a representation of a system as an atlas of charts. Thi s approach allows us to obtain dynamical models of the lowest possible dim ension\, leads to computational benefits\, and can separate state space in to regions of distinct behaviors.\n https://arxiv.org/abs/210 8.05928\n LOCATION:https://stable.researchseminars.org/talk/CNSwebinar/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Vladimir Rosenhaus DTSTART:20220412T150000Z DTEND:20220412T160000Z DTSTAMP:20260404T111214Z UID:CNSwebinar/3 DESCRIPTION:Title: Feynman rules for wave turbulence\nby Vladimir Rosenhaus as part of Georgia Tech CNS Nonlinear Webinar\n\n\nAbstract\nIt has long bee n known that weakly nonlinear field theories can have a late-time stationa ry state that is not the thermal state\, but a wave turbulent state (the K olmogorov-Zakharov state) with a far-from-equilibrium cascade of energy. W e go beyond the existence of the wave turbulent state\, studying fluctuati ons about the wave turbulent state. Specifically\, we take a classical fie ld theory with an arbitrary quartic interaction and add dissipation and Ga ussian-random forcing. Employing the path integral relation between stoch astic classical field theories and quantum field theories\, we give a pres cription\, in terms of Feynman diagrams\, for computing correlation funct ions in this system. We explicitly compute the two-point and four-point f unctions of the field to next-to-leading order in the coupling. Through an appropriate choice of forcing and dissipation\, these correspond to corre lation functions in the wave turbulent state. As a check\, we reproduce t he next-to-leading order term in the kinetic equation. The correlation fu nctions and corrections to the KZ state that we compute should\, in princi ple\, be experimentally measurable quantities.\n LOCATION:https://stable.researchseminars.org/talk/CNSwebinar/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Zeb Rocklin DTSTART:20220419T150000Z DTEND:20220419T160000Z DTSTAMP:20260404T111214Z UID:CNSwebinar/4 DESCRIPTION:Title: Rigidity percolation in a random tensegrity via analytic graph t heory\nby Zeb Rocklin as part of Georgia Tech CNS Nonlinear Webinar\n\ n\nAbstract\nTensegrities are mechanical structures that include cable-lik e elements that are strong and lightweight relative to rigid rods yet supp ort only extensile stress. From suspension bridges to the musculoskeletal system to individual biological cells\, humanity makes excellent use of te nsegrities\, yet the sharply nonlinear response of cables presents serious challenges to analytical theory. Here we consider large tensegrity struct ures with randomly placed cables (and struts) overlaid on a regular rigid backbone whose corresponding system of inequalities is reduced via analyti c theory to an exact graph theory. We identify a novel coordination number that controls two rigidity percolation transitions: one in which global i nteractions between cables first support external loads and one in which t he structure becomes fully rigid. We show that even the addition of a few cables strongly modifies conventional rigidity percolation\, both by modi fying the sharpness of the transition and by introducing avalanche effects in which a single constraint can eliminate multiple floppy modes.\n LOCATION:https://stable.researchseminars.org/talk/CNSwebinar/4/ END:VEVENT END:VCALENDAR