10 research outputs found
Coarse Stability and Bifurcation Analysis Using Stochastic Simulators: Kinetic Monte Carlo Examples
We implement a computer-assisted approach that, under appropriate conditions,
allows the bifurcation analysis of the coarse dynamic behavior of microscopic
simulators without requiring the explicit derivation of closed macroscopic
equations for this behavior. The approach is inspired by the so-called
time-step per based numerical bifurcation theory. We illustrate the approach
through the computation of both stable and unstable coarsely invariant states
for Kinetic Monte Carlo models of three simple surface reaction schemes. We
quantify the linearized stability of these coarsely invariant states, perform
pseudo-arclength continuation, detect coarse limit point and coarse Hopf
bifurcations and construct two-parameter bifurcation diagrams.Comment: 26 pages, 5 figure
Tipping Points of Evolving Epidemiological Networks: Machine Learning-Assisted, Data-Driven Effective Modeling
We study the tipping point collective dynamics of an adaptive
susceptible-infected-susceptible (SIS) epidemiological network in a
data-driven, machine learning-assisted manner. We identify a
parameter-dependent effective stochastic differential equation (eSDE) in terms
of physically meaningful coarse mean-field variables through a deep-learning
ResNet architecture inspired by numerical stochastic integrators. We construct
an approximate effective bifurcation diagram based on the identified drift term
of the eSDE and contrast it with the mean-field SIS model bifurcation diagram.
We observe a subcritical Hopf bifurcation in the evolving network's effective
SIS dynamics, that causes the tipping point behavior; this takes the form of
large amplitude collective oscillations that spontaneously -- yet rarely --
arise from the neighborhood of a (noisy) stationary state. We study the
statistics of these rare events both through repeated brute force simulations
and by using established mathematical/computational tools exploiting the
right-hand-side of the identified SDE. We demonstrate that such a collective
SDE can also be identified (and the rare events computations also performed) in
terms of data-driven coarse observables, obtained here via manifold learning
techniques, in particular Diffusion Maps. The workflow of our study is
straightforwardly applicable to other complex dynamics problems exhibiting
tipping point dynamics.Comment: 22 pages, 12 figure
Learning effective stochastic differential equations from microscopic simulations: combining stochastic numerics and deep learning
We identify effective stochastic differential equations (SDE) for coarse
observables of fine-grained particle- or agent-based simulations; these SDE
then provide coarse surrogate models of the fine scale dynamics. We approximate
the drift and diffusivity functions in these effective SDE through neural
networks, which can be thought of as effective stochastic ResNets. The loss
function is inspired by, and embodies, the structure of established stochastic
numerical integrators (here, Euler-Maruyama and Milstein); our approximations
can thus benefit from error analysis of these underlying numerical schemes.
They also lend themselves naturally to "physics-informed" gray-box
identification when approximate coarse models, such as mean field equations,
are available. Our approach does not require long trajectories, works on
scattered snapshot data, and is designed to naturally handle different time
steps per snapshot. We consider both the case where the coarse collective
observables are known in advance, as well as the case where they must be found
in a data-driven manner.Comment: 19 pages, includes supplemental materia
Measurement of Mutual Coulomb Dissociation in GeV Au+Au collisions at RHIC
We report on the first measurement of Mutual Coulomb Dissociation in heavy
ion collisions. We employ forward calorimeters to measure neutron multiplicity
at beam rapidity in peripheral collisions. The cross-section for simultaneous
electromagnetic breakup of Au nuclei at GeV is
barns in good agreement with calculations.Comment: This paper has been submitted for publication in Phys. Rev. Let
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Kinetic Monte Carlo simulations of travelling pulses and spiral waves in the lattice Lotka-Volterra model
Kinetic Monte Carlo simulations are used to study the stochastic two-species Lotka-Volterra model on a square lattice. For certain values of the model parameters, the system constitutes an excitable medium: travelling pulses and rotating spiral waves can be excited. Stable solitary pulses travel with constant (modulo stochastic fluctuations) shape and speed along a periodic lattice. The spiral waves observed persist sometimes for hundreds of rotations, but they are ultimately unstable and break-up (because of fluctuations and interactions between neighboring fronts) giving rise to complex dynamic behavior in which numerous small spiral waves rotate and interact with each other. It is interesting that travelling pulses and spiral waves can be exhibited by the model even for completely immobile species, due to the non-local reaction kinetics. © 2012 American Institute of Physics
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A common approach to the computation of coarse-scale steady states and to consistent initialization on a slow manifold
We present a simple technique for the computation of coarse-scale steady states of dynamical systems with time scale separation in the form of a " wrapper" around a fine-scale simulator. We discuss how this approach alleviates certain problems encountered by comparable existing approaches, and illustrate its use by computing coarse-scale steady states of a lattice Boltzmann fine scale code. Interestingly, in the same context of multiple time scale problems, the approach can be slightly modified to provide initial conditions on the slow manifold with prescribed coarse-scale observables. The approach is based on appropriately designed short bursts of the fine-scale simulator whose results are used to track changes in the coarse variables of interest, a core component of the equation-free framework. © 2011