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:20260422T225843Z UID:CNSwebinar/1 DESCRIPTION:Title: Data-driven dimension reduction\, dynamic modeling\, and control of com plex chaotic systems\nby M. Graham\, A. Linot (Univ. of Wisconsin-Madi son) as part of Georgia Tech CNS Nonlinear Webinar\n\n\nAbstract\nOur over all aim is to combine ideas from dynamical systems theory and machine lear ning to develop and apply reduced-order models of flow processes with comp lex chaotic dynamics. A particular aim is a minimal description of dynamic s 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.\n\nAlec Linot: Modeling chaotic spatiotemporal dynamic s with a minimal representation using Neural ODEs\n\nSolutions to dissipat ive partial differential equations that exhibit chaotic dynamics often evo lve 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 thes e coordinates. The manifold coordinates are found by reducing the system d imension via an undercomplete autoencoder – a neural network that reduce s then expands dimension – and an ODE is learned in this coordinate syst em with a Neural ODE. Learning an ODE\, instead of a discrete time-map\, a llows us to evolve trajectories arbitrarily far forward\, and allows for t raining on unevenly and/or widely spaced data in time. We test on the Kura moto-Sivashinsky equation for domain sizes that exhibit spatiotemporally c haos\, and find the ROM gives accurate short- and long-time statistics wit h training data separated up to 0.7 Lyapunov times.\nhttps://arxiv.org/abs /2109.00060\n LOCATION:https://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:20260422T225843Z UID:CNSwebinar/2 DESCRIPTION:Title: Charting dynamics from data\nby K. Zeng\, Carlo Perez de Jesus\, Da niel Floryan (Univ. of Wisconsin-Madison\, Univ. of Houston) as part of Ge orgia Tech CNS Nonlinear Webinar\n\n\nAbstract\nKevin Zeng: Deep Reinforce ment Learning Using Data-Driven Reduced-Order Models Discovers and Stabili zes Low Dissipation Equilibria\n\nDeep reinforcement learning (RL)\, a dat a-driven method capable of discovering complex control strategies for high -dimensional systems\, requires substantial interactions with the target s ystem\, making it costly when the system is computationally or experimenta lly expensive (e.g. flow control). We mitigate this challenge by combining dimension reduction via an autoencoder with a neural ODE framework to lea rn 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 m odel using RL and show that the model-based strategies perform well on the full dynamical system and highlight that the RL agent discovers and stabi lizes a forced equilibrium solution\, despite never having been given expl icit information about this state’s existence.\nhttps://arxiv.org/abs/21 04.05437\n\nCarlo Perez de Jesus\nDept. of Chemical and Biological Enginee ring\, Univ. of Wisconsin-Madison\n\nData-driven estimation of inertial ma nifold dimension for chaotic Kolmogorov flow and time evolution on the man ifold\n\nModel reduction techniques have previously been applied to evolve the Navier-Stokes equations in time\, however finding the minimal dimensi on 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\, tra cking the reconstruction error as a function of dimension. We also trained a discrete time stepper that evolves the reduced order model with a nonli near dense neural network. The trajectory travels in the vicinity of relat ive periodic orbits (RPOs) followed by sporadic bursting events. At a dime nsion of five (as opposed to the full state dimension of 1024)\, power inp ut-dissipation probability density function is well-approximated\; Fourier coefficient evolution shows that the trajectory correctly captures the he teroclinic connections (bursts) between the different RPOs\, and the predi ction and true data track each other for approximately a Lyapunov time.\n\ nDaniel Floryan https://dfloryan.github.io/\nMechanical Engineering at the University of Houston\n\nCharting dynamics from data\n\nWe often find our selves working with systems for which governing equations are unknown\, or if they are known\, they may be high-dimensional to the point of being di fficult to analyze and prohibitively expensive to make predictions with. T hese difficulties\, together with the ever-increasing availability of data \, have led to the new paradigm of data-driven model discovery. I will pre sent recent work that fruitfully combines a classical idea from applied ma thematics with modern methods of machine learning to learn minimal dynamic al models directly from time series data. In full analogy with cartography \, we learn a representation of a system as an atlas of charts. This appro ach allows us to obtain dynamical models of the lowest possible dimension\ , leads to computational benefits\, and can separate state space into regi ons of distinct behaviors.\n https://arxiv.org/abs/2108.05928 \n LOCATION:https://researchseminars.org/talk/CNSwebinar/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Vladimir Rosenhaus DTSTART:20220412T150000Z DTEND:20220412T160000Z DTSTAMP:20260422T225843Z UID:CNSwebinar/3 DESCRIPTION:Title: Feynman rules for wave turbulence\nby Vladimir Rosenhaus as part o f Georgia Tech CNS Nonlinear Webinar\n\n\nAbstract\nIt has long been known that weakly nonlinear field theories can have a late-time stationary stat e that is not the thermal state\, but a wave turbulent state (the Kolmogor ov-Zakharov state) with a far-from-equilibrium cascade of energy. We go be yond the existence of the wave turbulent state\, studying fluctuations abo ut the wave turbulent state. Specifically\, we take a classical field theo ry with an arbitrary quartic interaction and add dissipation and Gaussian- random forcing. Employing the path integral relation between stochastic c lassical field theories and quantum field theories\, we give a prescriptio n\, in terms of Feynman diagrams\, for computing correlation functions in this system. We explicitly compute the two-point and four-point function s of the field to next-to-leading order in the coupling. Through an approp riate 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\, b e experimentally measurable quantities.\n LOCATION:https://researchseminars.org/talk/CNSwebinar/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Zeb Rocklin DTSTART:20220419T150000Z DTEND:20220419T160000Z DTSTAMP:20260422T225843Z UID:CNSwebinar/4 DESCRIPTION:Title: Rigidity percolation in a random tensegrity via analytic graph theory\nby Zeb Rocklin as part of Georgia Tech CNS Nonlinear Webinar\n\n\nAbst ract\nTensegrities are mechanical structures that include cable-like eleme nts that are strong and lightweight relative to rigid rods yet support onl y extensile stress. From suspension bridges to the musculoskeletal system to individual biological cells\, humanity makes excellent use of tensegrit ies\, yet the sharply nonlinear response of cables presents serious challe nges to analytical theory. Here we consider large tensegrity structures wi th randomly placed cables (and struts) overlaid on a regular rigid backbon e whose corresponding system of inequalities is reduced via analytic theor y to an exact graph theory. We identify a novel coordination number that c ontrols two rigidity percolation transitions: one in which global interact ions between cables first support external loads and one in which the stru cture becomes fully rigid. We show that even the addition of a few cables strongly modifies conventional rigidity percolation\, both by modifying t he sharpness of the transition and by introducing avalanche effects in whi ch a single constraint can eliminate multiple floppy modes.\n LOCATION:https://researchseminars.org/talk/CNSwebinar/4/ END:VEVENT END:VCALENDAR