4,809 research outputs found
Investigating ultrafast quantum magnetism with machine learning
We investigate the efficiency of the recently proposed Restricted Boltzmann
Machine (RBM) representation of quantum many-body states to study both the
static properties and quantum spin dynamics in the two-dimensional Heisenberg
model on a square lattice. For static properties we find close agreement with
numerically exact Quantum Monte Carlo results in the thermodynamical limit. For
dynamics and small systems, we find excellent agreement with exact
diagonalization, while for systems up to N=256 spins close consistency with
interacting spin-wave theory is obtained. In all cases the accuracy converges
fast with the number of network parameters, giving access to much bigger
systems than feasible before. This suggests great potential to investigate the
quantum many-body dynamics of large scale spin systems relevant for the
description of magnetic materials strongly out of equilibrium.Comment: 18 pages, 5 figures, data up to N=256 spins added, minor change
Spatio-temporal spike trains analysis for large scale networks using maximum entropy principle and Monte-Carlo method
Understanding the dynamics of neural networks is a major challenge in
experimental neuroscience. For that purpose, a modelling of the recorded
activity that reproduces the main statistics of the data is required. In a
first part, we present a review on recent results dealing with spike train
statistics analysis using maximum entropy models (MaxEnt). Most of these
studies have been focusing on modelling synchronous spike patterns, leaving
aside the temporal dynamics of the neural activity. However, the maximum
entropy principle can be generalized to the temporal case, leading to Markovian
models where memory effects and time correlations in the dynamics are properly
taken into account. In a second part, we present a new method based on
Monte-Carlo sampling which is suited for the fitting of large-scale
spatio-temporal MaxEnt models. The formalism and the tools presented here will
be essential to fit MaxEnt spatio-temporal models to large neural ensembles.Comment: 41 pages, 10 figure
Learning Dynamic Boltzmann Distributions as Reduced Models of Spatial Chemical Kinetics
Finding reduced models of spatially-distributed chemical reaction networks
requires an estimation of which effective dynamics are relevant. We propose a
machine learning approach to this coarse graining problem, where a maximum
entropy approximation is constructed that evolves slowly in time. The dynamical
model governing the approximation is expressed as a functional, allowing a
general treatment of spatial interactions. In contrast to typical machine
learning approaches which estimate the interaction parameters of a graphical
model, we derive Boltzmann-machine like learning algorithms to estimate
directly the functionals dictating the time evolution of these parameters. By
incorporating analytic solutions from simple reaction motifs, an efficient
simulation method is demonstrated for systems ranging from toy problems to
basic biologically relevant networks. The broadly applicable nature of our
approach to learning spatial dynamics suggests promising applications to
multiscale methods for spatial networks, as well as to further problems in
machine learning
Model reduction for stochastic CaMKII reaction kinetics in synapses by graph-constrained correlation dynamics.
A stochastic reaction network model of Ca(2+) dynamics in synapses (Pepke et al PLoS Comput. Biol. 6 e1000675) is expressed and simulated using rule-based reaction modeling notation in dynamical grammars and in MCell. The model tracks the response of calmodulin and CaMKII to calcium influx in synapses. Data from numerically intensive simulations is used to train a reduced model that, out of sample, correctly predicts the evolution of interaction parameters characterizing the instantaneous probability distribution over molecular states in the much larger fine-scale models. The novel model reduction method, 'graph-constrained correlation dynamics', requires a graph of plausible state variables and interactions as input. It parametrically optimizes a set of constant coefficients appearing in differential equations governing the time-varying interaction parameters that determine all correlations between variables in the reduced model at any time slice
Statistical Physics and Representations in Real and Artificial Neural Networks
This document presents the material of two lectures on statistical physics
and neural representations, delivered by one of us (R.M.) at the Fundamental
Problems in Statistical Physics XIV summer school in July 2017. In a first
part, we consider the neural representations of space (maps) in the
hippocampus. We introduce an extension of the Hopfield model, able to store
multiple spatial maps as continuous, finite-dimensional attractors. The phase
diagram and dynamical properties of the model are analyzed. We then show how
spatial representations can be dynamically decoded using an effective Ising
model capturing the correlation structure in the neural data, and compare
applications to data obtained from hippocampal multi-electrode recordings and
by (sub)sampling our attractor model. In a second part, we focus on the problem
of learning data representations in machine learning, in particular with
artificial neural networks. We start by introducing data representations
through some illustrations. We then analyze two important algorithms, Principal
Component Analysis and Restricted Boltzmann Machines, with tools from
statistical physics
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