In this article, we propose a symplectic algorithm for the stable manifolds of the Hamilton-Jacobi equations combined with an iterative procedure in [N. Sakamoto and A. J. van der Schaft, "Analytical approximatio...
详细信息
In this article, we propose a symplectic algorithm for the stable manifolds of the Hamilton-Jacobi equations combined with an iterative procedure in [N. Sakamoto and A. J. van der Schaft, "Analytical approximation methods for the stabilizing solution of the Hamilton-Jacobi equation," IEEE Trans. Autom. Control, 2008]. Our algorithm includes two key aspects. The first one is to prove a precise estimate for radius of convergence and the errors of local approximate stable manifolds. The second one is to extend the local approximate stable manifolds to larger ones by symplectic algorithms, which have better long-time behaviors than general-purpose schemes. Our approach avoids the case of divergence of the iterative sequence of approximate stable manifolds and reduces the computation cost. We illustrate the effectiveness of the algorithm by an optimal control problem with exponential nonlinearity.
When solving Hamiltonian systems using numerical integrators, preserving the symplectic structure may be crucial for many problems. At the same time, solving chaotic or stiff problems requires integrators to approxima...
详细信息
When solving Hamiltonian systems using numerical integrators, preserving the symplectic structure may be crucial for many problems. At the same time, solving chaotic or stiff problems requires integrators to approximate the trajectories with extreme precision. So, integrating Hamilton's equations to a level of scientific reliability such that the answer can be used for scientific interpretation, may be computationally expensive. However, a neural network can be a viable alternative to numerical integrators, offering high-fidelity solutions orders of magnitudes faster. To understand whether it is also important to preserve the symplecticity when neural networks are used, we analyze three well-known neural network architectures that are including the symplectic structure inside the neural network's topology. Between these neural network architectures many similarities can be found. This allows us to formulate a new, generalized framework for these architectures. In the generalized framework symplectic Recurrent Neural Networks, SympNets and H & eacute;nonNets are included as special cases. Additionally, this new framework enables us to find novel neural network topologies by transitioning between the established ones. We compare new Generalized Hamiltonian Neural Networks (GHNNs) against the already established SympNets, H & eacute;nonNets and physics-unaware multilayer perceptrons. This comparison is performed with data for a pendulum, a double pendulum and a gravitational 3-body problem. In order to achieve a fair comparison, the hyperparameters of the different neural networks are chosen such that the prediction speeds of all four architectures are the same during inference. A special focus lies on the capability of the neural networks to generalize outside the training data. The GHNNs outperform all other neural network architectures for the problems considered.
Structure-preserving algorithms and algorithms with uniform error bound have constituted two interesting classes of numerical methods. In this paper, we blend these two kinds of methods for solving nonlinear systems w...
详细信息
Structure-preserving algorithms and algorithms with uniform error bound have constituted two interesting classes of numerical methods. In this paper, we blend these two kinds of methods for solving nonlinear systems with highly oscillatory solution, and the blended algorithms inherit and respect the advantage of each method. Two kinds of algorithms are presented to preserve the symplecticity and energy of the Hamiltonian systems, respectively. Long time energy conservation is analysed for symplectic algorithms and the proposed algorithms are shown to have uniform error bound in the position for the highly oscillatory structure. Moreover, some methods with uniform error bound in the position and in the velocity are derived and analysed. Two numerical experiments are carried out to support all the theoretical results established in this paper by showing the performance of the blended algorithms.
Although a complete unified theory for elasticity, plasticity, and damage does not yet exist, an approach on the basis of thermomechanical principles may be able to serve as the foundation for such a theory. With this...
详细信息
Although a complete unified theory for elasticity, plasticity, and damage does not yet exist, an approach on the basis of thermomechanical principles may be able to serve as the foundation for such a theory. With this in mind, as a first step, a mixed formulation is developed for fully coupled, spatially discretized linear thermoelasticity under the Lagrangian formalism by using Hamilton's principle. A variational integration scheme is then proposed for the temporal discretization of the resulting Euler-Lagrange equations. With this discrete numerical time-step solution, it becomes possible, for proper choices of state variables, to restate the problem in the form of an optimization. Ultimately, this allows the formulation of a principle of minimum generalized complementary potential energy for the discrete-time thermoelastic system. DOI: 10.1061/(ASCE)EM.1943-7889.0000346. (C) 2012 American Society of Civil Engineers.
In this paper we look at the performance of trigonometric integrators applied to highly oscillatory differential equations. It is widely known that sonic of the trigonometric integrators suffer from low-order resonanc...
详细信息
In this paper we look at the performance of trigonometric integrators applied to highly oscillatory differential equations. It is widely known that sonic of the trigonometric integrators suffer from low-order resonances for particular step sizes. We show here that, in general, trigonometric integrators also suffer from higher-order resonances which can lead to loss of nonlinear stability. We illustrate this with the Fermi-Pasta-Ulam problem, a highly oscillatory Hamiltonian system. We also show that in some cases trigonometric integrators preserve invariant or adiabatic quantities but at the wrong values. We use statistical properties such as time averages to further evaluate the performance of the trigonometric methods and compare the performance with that of the mid-point rule.
暂无评论