545 research outputs found
Constructing Neural Network-Based Models for Simulating Dynamical Systems
Dynamical systems see widespread use in natural sciences like physics,
biology, chemistry, as well as engineering disciplines such as circuit
analysis, computational fluid dynamics, and control. For simple systems, the
differential equations governing the dynamics can be derived by applying
fundamental physical laws. However, for more complex systems, this approach
becomes exceedingly difficult. Data-driven modeling is an alternative paradigm
that seeks to learn an approximation of the dynamics of a system using
observations of the true system. In recent years, there has been an increased
interest in data-driven modeling techniques, in particular neural networks have
proven to provide an effective framework for solving a wide range of tasks.
This paper provides a survey of the different ways to construct models of
dynamical systems using neural networks. In addition to the basic overview, we
review the related literature and outline the most significant challenges from
numerical simulations that this modeling paradigm must overcome. Based on the
reviewed literature and identified challenges, we provide a discussion on
promising research areas
Towards Complex Dynamic Physics System Simulation with Graph Neural ODEs
The great learning ability of deep learning models facilitates us to
comprehend the real physical world, making learning to simulate complicated
particle systems a promising endeavour. However, the complex laws of the
physical world pose significant challenges to the learning based simulations,
such as the varying spatial dependencies between interacting particles and
varying temporal dependencies between particle system states in different time
stamps, which dominate particles' interacting behaviour and the physical
systems' evolution patterns. Existing learning based simulation methods fail to
fully account for the complexities, making them unable to yield satisfactory
simulations. To better comprehend the complex physical laws, this paper
proposes a novel learning based simulation model- Graph Networks with
Spatial-Temporal neural Ordinary Equations (GNSTODE)- that characterizes the
varying spatial and temporal dependencies in particle systems using a united
end-to-end framework. Through training with real-world particle-particle
interaction observations, GNSTODE is able to simulate any possible particle
systems with high precisions. We empirically evaluate GNSTODE's simulation
performance on two real-world particle systems, Gravity and Coulomb, with
varying levels of spatial and temporal dependencies. The results show that the
proposed GNSTODE yields significantly better simulations than state-of-the-art
learning based simulation methods, which proves that GNSTODE can serve as an
effective solution to particle simulations in real-world application.Comment: 12 pages,5 figures, 6 tables, 49 reference
Nonlinear proper orthogonal decomposition for convection-dominated flows
Autoencoder techniques find increasingly common use in reduced order modeling as a means to create a latent space. This reduced order representation offers a modular data-driven modeling approach for nonlinear dynamical systems when integrated with a time series predictive model. In this Letter, we put forth a nonlinear proper orthogonal decomposition (POD) framework, which is an end-to-end Galerkin-free model combining autoencoders with long short-term memory networks for dynamics. By eliminating the projection error due to the truncation of Galerkin models, a key enabler of the proposed nonintrusive approach is the kinematic construction of a nonlinear mapping between the full-rank expansion of the POD coefficients and the latent space where the dynamics evolve. We test our framework for model reduction of a convection-dominated system, which is generally challenging for reduced order models. Our approach not only improves the accuracy, but also significantly reduces the computational cost of training and testing.
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research under Award Number DE-SC0019290. O.S. gratefully acknowledges the Early Career Research Program (ECRP) support of the U.S. Department of Energy. O.S. also gratefully acknowledges the financial support of the National Science Foundation under Award No. DMS-2012255. T.I. acknowledges support through National Science Foundation Grant No. DMS-2012253.acceptedVersio
Neural-Network Approach to Dissipative Quantum Many-Body Dynamics
In experimentally realistic situations, quantum systems are never perfectly
isolated and the coupling to their environment needs to be taken into account.
Often, the effect of the environment can be well approximated by a Markovian
master equation. However, solving this master equation for quantum many-body
systems, becomes exceedingly hard due to the high dimension of the Hilbert
space. Here we present an approach to the effective simulation of the dynamics
of open quantum many-body systems based on machine learning techniques. We
represent the mixed many-body quantum states with neural networks in the form
of restricted Boltzmann machines and derive a variational Monte-Carlo algorithm
for their time evolution and stationary states. We document the accuracy of the
approach with numerical examples for a dissipative spin lattice system
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