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Deep Hamiltonian Networks Based on Symplectic Integrators Arxiv
Deep Hamiltonian networks based on symplectic integrators Aiqing Zhu1,2, Pengzhan Jin1,2, and Yifa Tang1,2,* 1LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China 2School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China *Email address: [email protected] Abstract HNets is a class of neural networks on grounds of physical prior for learning Hamiltonian systems. This paper explains the influences of different integrators as hyper-parameters on the HNets through error analysis. If we define the network target as the map with zero empirical loss on arbitrary training data, then the non-symplectic integrators cannot guarantee the existence of the network targets of HNets. We introduce the inverse modified equations for HNets and prove that the HNets based on symplectic integrators possess network targets and the differences between the network targets and the original Hamiltonians depend on the accuracy orders of the integrators. Our numerical experiments show that the phase flows of the Hamiltonian systems obtained by symplectic HNets do not exactly preserve the original Hamiltonians, but preserve the network targets calculated; the loss of the network target for the training data and the test data is much less than the loss of the original Hamiltonian; the symplectic HNets have more powerful generalization ability and higher accuracy than the non-symplectic HNets in addressing predicting issues. Thus, the symplectic integrators are of critical importance for HNets. Key words. Neural networks, HNets, Network target, Inverse modified equations, Symplectic integrator, Error analysis. arXiv:2004.13830v1 [math.NA] 23 Apr 2020 1 Introduction Dynamical systems play a critical role in shaping our understanding of the physical world. -
Arxiv:1108.0322V1 [Physics.Comp-Ph] 1 Aug 2011 Hamiltonian Phase Space Structure Be Preserved
Symplectic integrators with adaptive time steps A S Richardson and J M Finn T-5, Applied Mathematics and Plasma Physics, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA E-mail: [email protected], [email protected] Abstract. In recent decades, there have been many attempts to construct symplectic integrators with variable time steps, with rather disappointing results. In this paper we identify the causes for this lack of performance, and find that they fall into two categories. In the first, the time step is considered a function of time alone, D = D(t). In this case, backwards error analysis shows that while the algorithms remain symplectic, parametric instabilities arise because of resonance between oscillations of D(t) and the orbital motion. In the second category the time step is a function of phase space variables D = D(q; p). In this case, the system of equations to be solved is analyzed by introducing a new time variable t with dt = D(q; p)dt. The transformed equations are no longer in Hamiltonian form, and thus are not guaranteed to be stable even when integrated using a method which is symplectic for constant D. We analyze two methods for integrating the transformed equations which do, however, preserve the structure of the original equations. The first is an extended phase space method, which has been successfully used in previous studies of adaptive time step symplectic integrators. The second, novel, method is based on a non-canonical mixed-variable generating function. Numerical trials for both of these methods show good results, without parametric instabilities or spurious growth or damping. -
Numerical Solution of Ordinary Differential Equations
NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS Kendall Atkinson, Weimin Han, David Stewart University of Iowa Iowa City, Iowa A JOHN WILEY & SONS, INC., PUBLICATION Copyright c 2009 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created ore extended by sales representatives or written sales materials. The advice and strategies contained herin may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. -
Some Robust Integrators for Large Time Dynamics
Razafindralandy et al. Adv. Model. and Simul. in Eng. Sci. (2019)6:5 https://doi.org/10.1186/s40323-019-0130-2 R E V I E W Open Access Some robust integrators for large time dynamics Dina Razafindralandy1* , Vladimir Salnikov2, Aziz Hamdouni1 and Ahmad Deeb1 *Correspondence: dina.razafi[email protected] Abstract 1Laboratoire des Sciences de l’Ingénieur pour l’Environnement This article reviews some integrators particularly suitable for the numerical resolution of (LaSIE), UMR 7356 CNRS, differential equations on a large time interval. Symplectic integrators are presented. Université de La Rochelle, Their stability on exponentially large time is shown through numerical examples. Next, Avenue Michel Crépeau, 17042 La Rochelle Cedex, France Dirac integrators for constrained systems are exposed. An application on chaotic Full list of author information is dynamics is presented. Lastly, for systems having no exploitable geometric structure, available at the end of the article the Borel–Laplace integrator is presented. Numerical experiments on Hamiltonian and non-Hamiltonian systems are carried out, as well as on a partial differential equation. Keywords: Symplectic integrators, Dirac integrators, Long-time stability, Borel summation, Divergent series Introduction In many domains of mechanics, simulations over a large time interval are crucial. This is, for instance, the case in molecular dynamics, in weather forecast or in astronomy. While many time integrators are available in literature, only few of them are suitable for large time simulations. Indeed, many numerical schemes fail to correctly predict the expected physical phenomena such as energy preservation, as the simulation time grows. For equations having an underlying geometric structure (Hamiltonian systems, varia- tional problems, Lie symmetry group, Dirac structure, ...), geometric integrators appear to be very robust for large time simulation. -
Some Robust Integrators for Large Time Dynamics
Razafindralandy et al. Adv. Model. and Simul. in Eng. Sci. (2019)6:5 https://doi.org/10.1186/s40323-019-0130-2 R E V I E W Open Access Some robust integrators for large time dynamics Dina Razafindralandy1* , Vladimir Salnikov2, Aziz Hamdouni1 and Ahmad Deeb1 *Correspondence: dina.razafi[email protected] Abstract 1Laboratoire des Sciences de l’Ingénieur pour l’Environnement This article reviews some integrators particularly suitable for the numerical resolution of (LaSIE), UMR 7356 CNRS, differential equations on a large time interval. Symplectic integrators are presented. Université de La Rochelle, Their stability on exponentially large time is shown through numerical examples. Next, Avenue Michel Crépeau, 17042 La Rochelle Cedex, France Dirac integrators for constrained systems are exposed. An application on chaotic Full list of author information is dynamics is presented. Lastly, for systems having no exploitable geometric structure, available at the end of the article the Borel–Laplace integrator is presented. Numerical experiments on Hamiltonian and non-Hamiltonian systems are carried out, as well as on a partial differential equation. Keywords: Symplectic integrators, Dirac integrators, Long-time stability, Borel summation, Divergent series Introduction In many domains of mechanics, simulations over a large time interval are crucial. This is, for instance, the case in molecular dynamics, in weather forecast or in astronomy. While many time integrators are available in literature, only few of them are suitable for large time simulations. Indeed, many numerical schemes fail to correctly predict the expected physical phenomena such as energy preservation, as the simulation time grows. For equations having an underlying geometric structure (Hamiltonian systems, varia- tional problems, Lie symmetry group, Dirac structure, ...), geometric integrators appear to be very robust for large time simulation. -
A High-Order Boris Integrator
A high-order Boris integrator Mathias Winkela,∗, Robert Speckb,a, Daniel Ruprechta aInstitute of Computational Science, University of Lugano, Switzerland. bJ¨ulichSupercomputing Centre, Forschungszentrum J¨ulich,Germany. Abstract This work introduces the high-order Boris-SDC method for integrating the equations of motion for electrically charged particles in an electric and magnetic field. Boris-SDC relies on a combination of the Boris-integrator with spectral deferred corrections (SDC). SDC can be considered as preconditioned Picard iteration to com- pute the stages of a collocation method. In this interpretation, inverting the preconditioner corresponds to a sweep with a low-order method. In Boris-SDC, the Boris method, a second-order Lorentz force integrator based on velocity-Verlet, is used as a sweeper/preconditioner. The presented method provides a generic way to extend the classical Boris integrator, which is widely used in essentially all particle-based plasma physics simulations involving magnetic fields, to a high-order method. Stability, convergence order and conserva- tion properties of the method are demonstrated for different simulation setups. Boris-SDC reproduces the expected high order of convergence for a single particle and for the center-of-mass of a particle cloud in a Penning trap and shows good long-term energy stability. Keywords: Boris integrator, time integration, magnetic field, high-order, spectral deferred corrections (SDC), collocation method 1. Introduction Often when modeling phenomena in plasma physics, for example particle dynamics in fusion vessels or particle accelerators, an externally applied magnetic field is vital to confine the particles in the physical device [1, 2]. In many cases, such as instabilities [3] and high-intensity laser plasma interaction [4], the magnetic field even governs the microscopic evolution and drives the phenomena to be studied. -
Vibration Odes 1
Vibration ODEs 1 Vibration problems lead to differential equations with solutions that oscillate in time, typically in a damped or undamped sinusoidal fashion. Such solutions put certain demands on the numerical methods compared to other phenomena whose solutions are monotone or very smooth. Both the frequency and amplitude of the oscillations need to be accurately handled by the numerical schemes. The forthcom- ing text presents a range of different methods, from classical ones (Runge-Kutta and midpoint/Crank-Nicolson methods), to more modern and popular symplectic (ge- ometric) integration schemes (Leapfrog, Euler-Cromer, and Störmer-Verlet meth- ods), but with a clear emphasis on the latter. Vibration problems occur throughout mechanics and physics, but the methods discussed in this text are also fundamen- tal for constructing successful algorithms for partial differential equations of wave nature in multiple spatial dimensions. 1.1 Finite Difference Discretization Many of the numerical challenges faced when computing oscillatory solutions to ODEs and PDEs can be captured by the very simple ODE u00 C u D 0. This ODE is thus chosen as our starting point for method development, implementation, and analysis. 1.1.1 A Basic Model for Vibrations The simplest model of a vibrating mechanical system has the following form: u00 C !2u D 0; u.0/ D I; u0.0/ D 0; t 2 .0; T : (1.1) Here, ! and I are given constants. Section 1.12.1 derives (1.1) from physical prin- ciples and explains what the constants mean. The exact solution of (1.1)is u.t/ D I cos.!t/ : (1.2) © The Author(s) 2017 1 H.P. -
High-Order Regularised Symplectic Integrator for Collisional Planetary Systems Antoine C
A&A 628, A32 (2019) https://doi.org/10.1051/0004-6361/201935786 Astronomy & © A. C. Petit et al. 2019 Astrophysics High-order regularised symplectic integrator for collisional planetary systems Antoine C. Petit, Jacques Laskar, Gwenaël Boué, and Mickaël Gastineau ASD/IMCCE, CNRS-UMR8028, Observatoire de Paris, PSL University, Sorbonne Université, 77 avenue Denfert-Rochereau, 75014 Paris, France e-mail: [email protected] Received 26 April 2019 / Accepted 15 June 2019 ABSTRACT We present a new mixed variable symplectic (MVS) integrator for planetary systems that fully resolves close encounters. The method is based on a time regularisation that allows keeping the stability properties of the symplectic integrators while also reducing the effective step size when two planets encounter. We used a high-order MVS scheme so that it was possible to integrate with large time-steps far away from close encounters. We show that this algorithm is able to resolve almost exact collisions (i.e. with a mutual separation of a fraction of the physical radius) while using the same time-step as in a weakly perturbed problem such as the solar system. We demonstrate the long-term behaviour in systems of six super-Earths that experience strong scattering for 50 kyr. We compare our algorithm to hybrid methods such as MERCURY and show that for an equivalent cost, we obtain better energy conservation. Key words. celestial mechanics – planets and satellites: dynamical evolution and stability – methods: numerical 1. Introduction accuracy. Because it is only necessary to apply them when an output is desired, they do not affect the performance of the inte- Precise long-term integration of planetary systems is still a chal- grator. -
A Position Based Approach to Ragdoll Simulation
LINKÖPING UNIVERSITY Department of Electrical Engineering Master Thesis A Position Based Approach to Ragdoll Simulation Fiammetta Pascucci LITH-ISY-EX- -07/3966- -SE June 2007 LINKÖPING UNIVERSITY Department of Electrical Engineering Master Thesis A position Based Approach to Ragdoll Simulation Fiammetta Pascucci LITH-ISY-EX- -07/3966- -SE June 2007 Supervisor: Ingemar Ragnemalm Examinator: Ingemar Ragnemalm Abstract Create the realistic motion of a character is a very complicated work. This thesis aims to create interactive animation for characters in three dimen- sions using position based approach. Our character is pictured from ragdoll, which is a structure of system particles where all particles are linked by equidistance con- straints. The goal of this thesis is observed the fall in the space of our ragdoll after creating all constraints, as structure, contact and environment constraints. The structure constraint represents all joint constraints which have one, two or three Degree of Freedom (DOF). The contact constraints are represented by collisions between our ragdoll and other objects in the space. Finally, the environment constraints are represented by means of the wall con- straint. The achieved results allow to have a realist fall of our ragdoll in the space. Keywords: Particle System, Verlet’s Integration, Skeleton Animation, Joint Con- straint, Environment Constraint, Skinning, Collision Detection. Acknowledgments I would to thanks all peoples who aid on making my master thesis work. In particular, thanks a lot to the person that most of all has believed in me. This person has always encouraged to go forward, has always given a lot love and help me in difficult movement. -
Exponential Integration for Hamiltonian Monte Carlo
Exponential Integration for Hamiltonian Monte Carlo Wei-Lun Chao1 [email protected] Justin Solomon2 [email protected] Dominik L. Michels2 [email protected] Fei Sha1 [email protected] 1Department of Computer Science, University of Southern California, Los Angeles, CA 90089 2Department of Computer Science, Stanford University, 353 Serra Mall, Stanford, California 94305 USA Abstract butions that generate random walks in sample space. A fraction of these walks aligned with the target distribution is We investigate numerical integration of ordinary kept, while the remainder is discarded. Sampling efficiency differential equations (ODEs) for Hamiltonian and quality depends critically on the proposal distribution. Monte Carlo (HMC). High-quality integration is For this reason, designing good proposals has long been a crucial for designing efficient and effective pro- subject of intensive research. posals for HMC. While the standard method is leapfrog (Stormer-Verlet)¨ integration, we propose For distributions over continuous variables, Hamiltonian the use of an exponential integrator, which is Monte Carlo (HMC) is a state-of-the-art sampler using clas- robust to stiff ODEs with highly-oscillatory com- sical mechanics to generate proposal distributions (Neal, ponents. This oscillation is difficult to reproduce 2011). The method starts by introducing auxiliary sam- using leapfrog integration, even with carefully pling variables. It then treats the negative log density of selected integration parameters and precondition- the variables’ joint distribution as the Hamiltonian of parti- ing. Concretely, we use a Gaussian distribution cles moving in sample space. Final positions on the motion approximation to segregate stiff components of trajectories are used as proposals. the ODE. -
Simple Quantitative Tests to Validate Sampling from Thermodynamic
Simple quantitative tests to validate sampling from thermodynamic ensembles Michael R. Shirts∗ Department of Chemical Engineering, University of Virginia, VA 22904 E-mail: [email protected] Abstract It is often difficult to quantitatively determine if a new molecular simulation algorithm or software properly implements sampling of the desired thermodynamic ensemble. We present some simple statistical analysis procedures to allow sensitive determination of whether a de- sired thermodynamic ensemble is properly sampled. We demonstrate the utility of these tests for model systems and for molecular dynamics simulations in a range of situations, includ- ing constant volume and constant pressure simulations, and describe an implementation of the tests designed for end users. 1 Introduction Molecular simulations, including both molecular dynamics (MD) and Monte Carlo (MC) tech- arXiv:1208.0910v1 [cond-mat.stat-mech] 4 Aug 2012 niques, are powerful tools to study the properties of complex molecular systems. When used to specifically study thermodynamics of such systems, rather than dynamics, the primary goal of molecular simulation is to generate uncorrelated samples from the appropriate ensemble as effi- ciently as possible. This simulation data can then be used to compute thermodynamic properties ∗To whom correspondence should be addressed 1 of interest. Simulations of several different ensembles may be required to simulate some thermo- dynamic properties, such as free energy differences between states. An ever-expanding number of techniques have been proposed to perform more and more sophisticated sampling from com- plex molecular systems using both MD and MC, and new software tools are continually being introduced in order to implement these algorithms and to take advantage of advances in hardware architecture and programming languages. -
A Spectral Deferred Correction Method Applied to the Shallow Water Equations on a Sphere
OCTOBER 2013 J I A E T A L . 3435 A Spectral Deferred Correction Method Applied to the Shallow Water Equations on a Sphere JUN JIA,JUDITH C. HILL,KATHERINE J. EVANS, AND GEORGE I. FANN Oak Ridge National Laboratory, Oak Ridge, Tennessee MARK A. TAYLOR Sandia National Laboratories, Albuquerque, New Mexico (Manuscript received 13 February 2012, in final form 7 December 2012) ABSTRACT Although significant gains have been made in achieving high-order spatial accuracy in global climate modeling, less attention has been given to the impact imposed by low-order temporal discretizations. For long-time simulations, the error accumulation can be significant, indicating a need for higher-order temporal accuracy. A spectral deferred correction (SDC) method is demonstrated of even order, with second- to eighth-order accuracy and A-stability for the temporal discretization of the shallow water equations within the spectral-element High-Order Methods Modeling Environment (HOMME). Because this method is stable and of high order, larger time-step sizes can be taken while still yielding accurate long-time simulations. The spectral deferred correction method has been tested on a suite of popular benchmark problems for the shallow water equations, and when compared to the explicit leapfrog, five-stage Runge–Kutta, and fully implicit (FI) second-order backward differentiation formula (BDF2) time-integration methods, it achieves higher accuracy for the same or larger time-step sizes. One of the benchmark problems, the linear advection of a Gaussian bell height anomaly, is extended to run for longer time periods to mimic climate-length simula- tions, and the leapfrog integration method exhibited visible degradation for climate length simulations whereas the second-order and higher methods did not.