Hamiltonian systems are differential equations which describe systems in classical mechanics, plasma physics, and sampling problems. They exhibit many structural properties, such as a lack of attractors and the presence of conservation laws. To predict Hamiltonian dynamics based on discrete trajectory observations, incorporation of prior knowledge about Hamiltonian structure greatly improves predictions. This is typically done by learning the system's Hamiltonian and then integrating the Hamiltonian vector field with a symplectic integrator. For this, however, Hamiltonian data needs to be approximated based on the trajectory observations. Moreover, the numerical integrator introduces an additional discretisation error. In this paper, we show that an inverse modified Hamiltonian structure adapted to the geometric integrator can be learned directly from observations. A separate approximation step for the Hamiltonian data avoided. The inverse modified data compensates for the discretisation error such that the discretisation error is eliminated. The technique is developed for Gaussian Processes.
From efficient symplectic exponentiation of matrices to symplectic integration of high-dimensional Hamiltonian systems with slowly varying quadratic stiff potentials
In this paper, we are concerned with the construction and analysis of a new class of methods obtained as double jump compositions with complex coefficients and projection on the real axis. It is shown in particular that the new integrators are symmetric and symplectic up to high orders if one uses a symmetric and symplectic basic method. In terms of efficiency, the aforementioned technique requires fewer stages than standard compositions of the same orders and is thus expected to lead to faster methods.
Compared to other symplectic integrators (the Wisdom and Holman map and its higher order generalizations) that also take advantage of the hierarchical nature of the motion of the planets around the central star, our methods require solving implicit equations at each time-step. We claim that, despite this disadvantage, FCIRK16 is more efficient than explicit symplectic integrators for high precision simulations thanks to: (i) its high order of precision, (ii) its easy parallelization, and (iii) its efficient mixed-precision implementation which reduces the effect of round-off errors. In additio…
The well known 1956 unpublished report by De Vogelaere, motivated by the physics of particle accelerators, is the first demonstration of the existence of numerical integrators (now called symplectic integrators) that preserve a basic property of Hamiltonian systems. The purpose of this short note is to introduce René De Vogelaere, list his pioneering accomplishments, and relate them to other early work on symplectic integrators. The preprint itself, typset in LaTeX, is attached as an appendix.
DiscoTEX 1.0: Discontinuous collocation and implicit-turned-explicit (IMTEX) integration symplectic, symmetric numerical algorithms with high order jumps for differential equations II: extension to higher-orders of numerical convergence
It is well known that symplectic integrators lose their near energy preservation properties when variable step sizes are used. The most common approach to combine adaptive step sizes and symplectic integrators involves the Poincaré transformation of the original Hamiltonian. In this article, we provide a framework for the construction of variational integrators using the Poincaré transformation. Since the transformed Hamiltonian is typically degenerate, the use of Hamiltonian variational integrators based on Type II or Type III generating functions is required instead of the more traditional L…
This work presents a nonintrusive physics-preserving method to learn reduced-order models (ROMs) of canonical Hamiltonian systems. Traditional intrusive projection-based model reduction approaches utilize symplectic Galerkin projection to construct Hamiltonian ROMs by projecting Hamilton's equations of the full model onto a symplectic subspace. This symplectic projection requires complete knowledge about the full model operators and full access to manipulate the computer code. In contrast, the proposed Hamiltonian operator inference approach embeds the physics into the operator inference frame…
In this article, a brief description of Discrete Mechanics and Variational Integrators which preserve the symplectic structure of the flow will be provided and a Newton-Raphson algorithm that can be used to solve implicit equations on the SO(3) manifold will be developed. These techniques will be used to simulate the rotational dynamics of a rigid body evolving on the Lie Group SO(3).
Many force-gradient explicit symplectic integration algorithms have been designed for the Hamiltonian $H=T (\mathbf{p})+V(\mathbf{q})$ with kinetic energy $T(\mathbf{p})=\mathbf{p}^2/2$ in the existing references. When the force-gradient operator is appropriately adjusted as a new operator, they are still suitable for a class of Hamiltonian problems $H=K(\mathbf{p},\mathbf{q})+V(\mathbf{q})$ with \emph{integrable} part $K(\mathbf{p},\mathbf{q}) = \sum_{i=1}^{n} \sum_{j=1}^{n}a_{ij}p_ip_j+\sum_{i=1}^{n} b_ip_i$, where $a_{ij}=a_{ij}(\textbf{q})$ and $b_i=b_i(\textbf{q})$ are functions of coordi…
Maintaining conservation laws in the fully discrete setting is critical for accurate long-time behavior of numerical simulations and requires accounting for discrete conservation properties in both space and time. This paper derives arbitrary order finite element exterior calculus spatial discretizations for the two-dimensional (2D) Navier-Stokes and drift-reduced magnetohydrodynamic equations that conserve both energy and enstrophy to machine precision when coupled with generally symplectic time-integration methods. Both continuous and discontinuous-Galerkin (DG) weak formulations can ensure …
DiscoTEX 1.0: Discontinuous collocation and implicit-turned-explicit (IMTEX) integration symplectic, symmetric numerical algorithms with higher order jumps for differential equations I: numerical black hole perturbation theory applications
CSympNet-ID: conformal-symplectic map learning for linearly damped Hamiltonian systems
School of Mathematics (Zhuhai), Sun Yat-sen University（中山大学数学学院（珠海））；National Engineering Laboratory for Big Data Analysis and Applications, Peking University（北京大学大数据分析与应用国家工程实验室）；State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学国家重点实验室）
