arXiv subjects

Yifa Tang

Publications and source records attributed to Yifa Tang.

34 records · Page 2Linked to original sources

arXiv2022

Explicit K-symplectic methods for nonseparable non-canonical Hamiltonian systems

We propose efficient numerical methods for nonseparable non-canonical Hamiltonian systems which are explicit, K-symplectic in the extended phase space with long time energy conservation properties. They are based on extending the original phase space to several copies of the phase space and imposing a mechanical restraint on the copies of the phase space. Explicit K-symplectic methods are constructed for three non-canonical Hamiltonian systems. Numerical results show that they outperform the higher order Runge-Kutta methods in preserving the phase orbit and the energy of the system over long time.

math.NA↗

arXiv2022

VPNets: Volume-preserving neural networks for learning source-free dynamics

We propose volume-preserving networks (VPNets) for learning unknown source-free dynamical systems using trajectory data. We propose three modules and combine them to obtain two network architectures, coined R-VPNet and LA-VPNet. The distinct feature of the proposed models is that they are intrinsic volume-preserving. In addition, the corresponding approximation theorems are proved, which theoretically guarantee the expressivity of the proposed VPNets to learn source-free dynamics. The effectiveness, generalization ability and structure-preserving property of the VP-Nets are demonstrated by numerical experiments.

cs.LG↗

arXiv2022

On Numerical Integration in Neural Ordinary Differential Equations

The combination of ordinary differential equations and neural networks, i.e., neural ordinary differential equations (Neural ODE), has been widely studied from various angles. However, deciphering the numerical integration in Neural ODE is still an open challenge, as many researches demonstrated that numerical integration significantly affects the performance of the model. In this paper, we propose the inverse modified differential equations (IMDE) to clarify the influence of numerical integration on training Neural ODE models. IMDE is determined by the learning task and the employed ODE solver. It is shown that training a Neural ODE model actually returns a close approximation of the IMDE, rather than the true ODE. With the help of IMDE, we deduce that (i) the discrepancy between the learned model and the true ODE is bounded by the sum of discretization error and learning loss; (ii) Neural ODE using non-symplectic numerical integration fail to learn conservation laws theoretically. Several experiments are performed to numerically verify our theoretical analysis.

cs.LG↗

arXiv2022

Poisson Integrators based on splitting method for Poisson systems

We propose Poisson integrators for the numerical integration of separable Poisson systems. We analyze three situations in which the Poisson systems are separated in three ways and the Poisson integrators can be constructed by using the splitting method. Numerical results show that the Poisson integrators outperform the higher order non-Poisson integrators in phase orbit tracking, long-term energy conservation and efficiency.

math.NA↗

arXiv2022

A modified EM method and its fast implementation for multi-term Riemann-Liouville stochastic fractional differential equations

In this paper, a modified Euler-Maruyama (EM) method is constructed for a kind of multi-term Riemann-Liouville stochastic fractional differential equations and the strong convergence order min{1-α_m, 0.5} of the proposed method is proved with Riemann-Liouville fractional derivatives' orders 0<α_1<α_2<...<α_m <1. Then, based on the sum-of-exponentials approximation, a fast implementation of the modified EM method which is called a fast EM method is derived to greatly improve the computational efficiency. Finally, some numerical examples are carried out to support the theoretical results and show the powerful computational performance of the fast EM method.

math.NA↗

arXiv2022

Symplectic Integrators in Corotating Coordinates

The dynamic equation of mass point in rotating coordinates is governed by Coriolis and centrifugal force, besides a corotating potential relative to frame. Such a system is no longer a canonical Hamiltonian system so that the construction of symplectic integrator is problematic. In this paper, we present three integrators for this question. It is significant that those schemes have the good property of near-conservation of energy. We proved that the discrete symplectic map of $(p_n, x_n) \mapsto (p_{n+1}, x_{n+1})$ in corotating coordinates exists and the two integrators are variational symplectic. Two groups of numerical experiments demonstrates the precision and long-term convergence of these integrators in the examples of corotating top-hat density and circular restricted three-body system.

astro-ph.IM↗

arXiv2022

Approximation capabilities of measure-preserving neural networks

Measure-preserving neural networks are well-developed invertible models, however, their approximation capabilities remain unexplored. This paper rigorously analyses the approximation capabilities of existing measure-preserving neural networks including NICE and RevNets. It is shown that for compact $U \subset \R^D$ with $D\geq 2$, the measure-preserving neural networks are able to approximate arbitrary measure-preserving map $ψ: U\to \R^D$ which is bounded and injective in the $L^p$-norm. In particular, any continuously differentiable injective map with $\pm 1$ determinant of Jacobian are measure-preserving, thus can be approximated.

cs.LG↗

arXiv2021

Inverse modified differential equations for discovery of dynamics

The combination of numerical integration and deep learning, i.e., ODE-net, has been successfully employed in a variety of applications. In this work, we introduce inverse modified differential equations (IMDE) to contribute to the behaviour and error analysis of discovery of dynamics using ODE-net. It is shown that the difference between the learned ODE and the truncated IMDE is bounded by the sum of learning loss and a discrepancy which can be made sub exponentially small. In addition, we deduce that the total error of ODE-net is bounded by the sum of discrete error and learning loss. Furthermore, with the help of IMDE, theoretical results on learning Hamiltonian system are derived. Several experiments are performed to numerically verify our theoretical results.

math.NA↗

arXiv2020

Unit triangular factorization of the matrix symplectic group

In this work, we prove that any symplectic matrix can be factored into no more than 9 unit triangular symplectic matrices. This structure-preserving factorization of the symplectic matrices immediately reveals two well-known features that, (i) the determinant of any symplectic matrix is one, (ii) the matrix symplectic group is path connected, as well as a new feature that (iii) all the unit triangular symplectic matrices form a set of generators of the matrix symplectic group. Furthermore, this factorization yields effective methods for the unconstrained parametrization of the matrix symplectic group as well as its structured subsets. The unconstrained parametrization enables us to apply faster and more efficient unconstrained optimization algorithms to the problems with symplectic constraints under certain circumstances.

math.SG↗

arXiv2020

SympNets: Intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems

We propose new symplectic networks (SympNets) for identifying Hamiltonian systems from data based on a composition of linear, activation and gradient modules. In particular, we define two classes of SympNets: the LA-SympNets composed of linear and activation modules, and the G-SympNets composed of gradient modules. Correspondingly, we prove two new universal approximation theorems that demonstrate that SympNets can approximate arbitrary symplectic maps based on appropriate activation functions. We then perform several experiments including the pendulum, double pendulum and three-body problems to investigate the expressivity and the generalization ability of SympNets. The simulation results show that even very small size SympNets can generalize well, and are able to handle both separable and non-separable Hamiltonian systems with data points resulting from short or long time steps. In all the test cases, SympNets outperform the baseline models, and are much faster in training and prediction. We also develop an extended version of SympNets to learn the dynamics from irregularly sampled data. This extended version of SympNets can be thought of as a universal model representing the solution to an arbitrary Hamiltonian system.

cs.LG↗

arXiv2020

Quantifying the generalization error in deep learning in terms of data distribution and neural network smoothness

The accuracy of deep learning, i.e., deep neural networks, can be characterized by dividing the total error into three main types: approximation error, optimization error, and generalization error. Whereas there are some satisfactory answers to the problems of approximation and optimization, much less is known about the theory of generalization. Most existing theoretical works for generalization fail to explain the performance of neural networks in practice. To derive a meaningful bound, we study the generalization error of neural networks for classification problems in terms of data distribution and neural network smoothness. We introduce the cover complexity (CC) to measure the difficulty of learning a data set and the inverse of the modulus of continuity to quantify neural network smoothness. A quantitative bound for expected accuracy/error is derived by considering both the CC and neural network smoothness. Although most of the analysis is general and not specific to neural networks, we validate our theoretical assumptions and results numerically for neural networks by several data sets of images. The numerical results confirm that the expected error of trained networks scaled with the square root of the number of classes has a linear relationship with respect to the CC. We also observe a clear consistency between test loss and neural network smoothness during the training process. In addition, we demonstrate empirically that the neural network smoothness decreases when the network size increases whereas the smoothness is insensitive to training dataset size.

stat.ML↗

arXiv2020

Deep Hamiltonian networks based on symplectic integrators

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.

math.NA↗

arXiv2020

Spectrally accurate space-time solution of Manakov systems

In this paper, we study the numerical solution of Manakov systems by using a spectrally accurate Fourier decomposition in space, coupled with a spectrally accurate time integration. This latter relies on the use of spectral Hamiltonian boundary Value Methods. The used approach allows to conserve all the physical invariants of the systems. Some numerical tests are reported.

math.NA↗

arXiv2016

Explicit symplectic algorithms based on generating functions for relativistic charged particle dynamics in time-dependent electromagnetic field

Relativistic dynamics of a charged particle in time-dependent electromagnetic fields has theoretical significance and a wide range of applications. It is often multi-scale and requires accurate long-term numerical simulations using symplectic integrators. For modern large-scale particle simulations in complex, time-dependent electromagnetic field, explicit symplectic algorithms are much more preferable. In this paper, we treat the relativistic dynamics of a particle as a Hamiltonian system on the cotangent space of the space-time, and construct for the first time explicit symplectic algorithms for relativistic charged particles of order 2 and 3 using the sum-split technique and generating functions.

physics.plasm-ph↗

arXiv2016

Explicit symplectic algorithms based on generating functions for charged particle dynamics

Dynamics of a charged particle in the canonical coordinates is a Hamiltonian system, and the well-known symplectic algorithm has been regarded as the de facto method for numerical integration of Hamiltonian systems due to its long-term accuracy and fidelity. For long-term simulations with high efficiency, explicit symplectic algorithms are desirable. However, it is widely accepted that explicit symplectic algorithms are only available for sum-separable Hamiltonians, and that this restriction severely limits the application of explicit symplectic algorithms to charged particle dynamics. To overcome this difficulty, we combine the familiar sum-split method and a generating function method to construct second and third order explicit symplectic algorithms for dynamics of charged particle. The generating function method is designed to generate explicit symplectic algorithms for product-separable Hamiltonian with form of $H(\mathbf{p},\mathbf{q})=\mathbf{p}_{i}f(\mathbf{q})$ or $H(\mathbf{p},\mathbf{q})=\mathbf{q}_{i}f(\mathbf{p})$. Applied to the simulations of charged particle dynamics, the explicit symplectic algorithms based on generating functions demonstrate superiorities in conservation and efficiency.

physics.plasm-ph↗

arXiv2015

Symplectic methods for non-canonical Hamiltonian systems

We show that, when applied to any non-canonical Hamiltonian system, any integrator that is symplectic for canonical Hamiltonian problems is actually conjugate symplectic for the non-canonical structure. This result is useful because it implies that canonically symplectic methods may be successfully applied to long-time integrations of non-canonical Hamiltonian problems, thus avoiding the need to construct ad hoc new methods. Numerical results for three non-canonical Hamiltonian systems demonstrate that (canonically) symplectic methods have significant advantages in numerical accuracy and near energy preservation over non-symplectic methods.

math.SG↗