The Experts below are selected from a list of 7104 Experts worldwide ranked by ideXlab platform
Eli Hershkovitz - One of the best experts on this subject based on the ideXlab platform.
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fast Numerical Integrator for stochastic differential equations with nonstationary multiplicative noise
Journal of Physical Chemistry A, 2001Co-Authors: Eli Hershkovitz, Rigoberto HernandezAbstract:I. Introduction Whenever a chemical process may be described classically, the solution of Hamilton’s equations of motion formally provides all the necessary dynamical information. However, for very large systems, such a solution is not generally available. In such systems, reduced-dimensionalsprojectivesapproaches allow the problem to be tractable, though the dynamics will no longer be Hamiltonian. In the present work, a fast Numerical algorithm is developed for the Numerical integration of reduced-dimensional systems whose projected environment may include nonstationary responses in time or space. Specifically, the dynamics of a generalized variable x may be written in a multidimensional closed differential form,
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a fourth order Numerical Integrator for stochastic langevin equations
Journal of Chemical Physics, 1998Co-Authors: Eli HershkovitzAbstract:Stochastic differential equations (SDE’s) can be Numerically integrated using second-order accuracy methods. Higher order schemes are not in use because of the complexity of the algorithm and because of the difficulties in producing non-Gaussian noises. Yet for the case of the Langevin equation (LE) which is a subclass of SDE’s, high order Integrators can be developed. A fast fourth-order Integrator is presented here. The improved efficiency of the new Integrator allows for solution of systems which could not be integrated accurately with the standard second-order methods.
David N Spergel - One of the best experts on this subject based on the ideXlab platform.
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a bayesian neural network predicts the dissolution of compact planetary systems
Proceedings of the National Academy of Sciences of the United States of America, 2021Co-Authors: Miles Cranmer, Daniel Tamayo, Hanno Rein, Peter W Battaglia, Samuel Hadden, Philip J Armitage, David N SpergelAbstract:We introduce a Bayesian neural network model that can accurately predict not only if, but also when a compact planetary system with three or more planets will go unstable. Our model, trained directly from short N-body time series of raw orbital elements, is more than two orders of magnitude more accurate at predicting instability times than analytical estimators, while also reducing the bias of existing machine learning algorithms by nearly a factor of three. Despite being trained on compact resonant and near-resonant three-planet configurations, the model demonstrates robust generalization to both nonresonant and higher multiplicity configurations, in the latter case outperforming models fit to that specific set of integrations. The model computes instability estimates up to 10 5 times faster than a Numerical Integrator, and unlike previous efforts provides confidence intervals on its predictions. Our inference model is publicly available in the SPOCK (https://github.com/dtamayo/spock) package, with training code open sourced (https://github.com/MilesCranmer/bnn_chaos_model).
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a bayesian neural network predicts the dissolution of compact planetary systems
arXiv: Earth and Planetary Astrophysics, 2021Co-Authors: Miles Cranmer, Daniel Tamayo, Hanno Rein, Peter W Battaglia, Samuel Hadden, Philip J Armitage, David N SpergelAbstract:Despite over three hundred years of effort, no solutions exist for predicting when a general planetary configuration will become unstable. We introduce a deep learning architecture to push forward this problem for compact systems. While current machine learning algorithms in this area rely on scientist-derived instability metrics, our new technique learns its own metrics from scratch, enabled by a novel internal structure inspired from dynamics theory. Our Bayesian neural network model can accurately predict not only if, but also when a compact planetary system with three or more planets will go unstable. Our model, trained directly from short N-body time series of raw orbital elements, is more than two orders of magnitude more accurate at predicting instability times than analytical estimators, while also reducing the bias of existing machine learning algorithms by nearly a factor of three. Despite being trained on compact resonant and near-resonant three-planet configurations, the model demonstrates robust generalization to both non-resonant and higher multiplicity configurations, in the latter case outperforming models fit to that specific set of integrations. The model computes instability estimates up to five orders of magnitude faster than a Numerical Integrator, and unlike previous efforts provides confidence intervals on its predictions. Our inference model is publicly available in the SPOCK package, with training code open-sourced.
Melvin Leok - One of the best experts on this subject based on the ideXlab platform.
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computational geometric optimal control of connected rigid bodies in a perfect fluid
arXiv: Optimization and Control, 2009Co-Authors: Melvin Leok, Harris N McclamrochAbstract:This paper formulates an optimal control problem for a system of rigid bodies that are connected by ball joints and immersed in an irrotational and incompressible fluid. The rigid bodies can translate and rotate in three-dimensional space, and each joint has three rotational degrees of freedom. We assume that internal control moments are applied at each joint. We present a computational procedure for Numerically solving this optimal control problem, based on a geometric Numerical Integrator referred to as a Lie group variational Integrator. This computational approach preserves the Hamiltonian structure of the controlled system and the Lie group configuration manifold of the connected rigid bodies, thereby finding complex optimal maneuvers of connected rigid bodies accurately and efficiently. This is illustrated by Numerical computations.
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dynamics of connected rigid bodies in a perfect fluid
arXiv: Numerical Analysis, 2008Co-Authors: Melvin Leok, Harris N McclamrochAbstract:This paper presents an analytical model and a geometric Numerical Integrator for a system of rigid bodies connected by ball joints, immersed in an irrotational and incompressible fluid. The rigid bodies can translate and rotate in three-dimensional space, and each joint has three rotational degrees of freedom. This model characterizes the qualitative behavior of three-dimensional fish locomotion. A geometric Numerical Integrator, refereed to as a Lie group variational Integrator, preserves Hamiltonian structures of the presented model and its Lie group configuration manifold. These properties are illustrated by a Numerical simulation for a system of three connected rigid bodies.
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a lie group variational Integrator for the attitude dynamics of a rigid body with applications to the 3d pendulum
International Conference on Control Applications, 2005Co-Authors: Taeyoung Lee, N H Mcclamroch, Melvin LeokAbstract:A Numerical Integrator is derived for a class of models that describe the attitude dynamics of a rigid body in the presence of an attitude dependent potential. The Numerical Integrator is obtained from a discrete variational principle, and exhibits excellent geometric conservation properties. In particular, by performing computations at the level of the Lie algebra, and updating the solution using the matrix exponential, the attitude automatically evolves on the rotation group embedded in the space of matrices. The geometric conservation properties of the Numerical Integrator imply long time Numerical stability. We apply this variational Integrator to the uncontrolled 3D pendulum, that is a rigid asymmetric body supported at a frictionless pivot acting under the influence of uniform gravity. Interesting dynamics of the 3D pendulum are exposed
Rigoberto Hernandez - One of the best experts on this subject based on the ideXlab platform.
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fast Numerical Integrator for stochastic differential equations with nonstationary multiplicative noise
Journal of Physical Chemistry A, 2001Co-Authors: Eli Hershkovitz, Rigoberto HernandezAbstract:I. Introduction Whenever a chemical process may be described classically, the solution of Hamilton’s equations of motion formally provides all the necessary dynamical information. However, for very large systems, such a solution is not generally available. In such systems, reduced-dimensionalsprojectivesapproaches allow the problem to be tractable, though the dynamics will no longer be Hamiltonian. In the present work, a fast Numerical algorithm is developed for the Numerical integration of reduced-dimensional systems whose projected environment may include nonstationary responses in time or space. Specifically, the dynamics of a generalized variable x may be written in a multidimensional closed differential form,
Ali Jadbabaie - One of the best experts on this subject based on the ideXlab platform.
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direct runge kutta discretization achieves acceleration
Neural Information Processing Systems, 2018Co-Authors: Jingzhao Zhang, Aryan Mokhtari, Suvrit Sra, Ali JadbabaieAbstract:We study gradient-based optimization methods obtained by directly discretizing a second-order ordinary differential equation (ODE) related to the continuous limit of Nesterov's accelerated gradient method. When the function is smooth enough, we show that acceleration can be achieved by a stable discretization of this ODE using standard Runge-Kutta Integrators. Specifically, we prove that under Lipschitz-gradient, convexity and order-(s + 2) differentiability assumptions, the sequence of iterates generated by discretizing the proposed second-order ODE converges to the optimal solution at a rate of O(N-2 s/s+1), where s is the order of the Runge-Kutta Numerical Integrator. Furthermore, we introduce a new local flatness condition on the objective, under which rates even faster than O(N-2) can be achieved with low-order Integrators and only gradient information. Notably, this flatness condition is satisfied by several standard loss functions used in machine learning. We provide Numerical experiments that verify the theoretical rates predicted by our results.
