Charting dynamics from data

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. 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 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. arxiv.org/abs/2108.05928

mathematical physicsnonlinear sciencescomputational physicsfluid dynamics

Audience: researchers in the topic

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Georgia Tech CNS Nonlinear Webinar

Series comments: Contact predrag.cvitanovic@physics.gatech.edu for livestream link . Archive of past webinars: smartech.gatech.edu/handle/1853/63906

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