scholarly journals Modeling Trajectories with Neural Ordinary Differential Equations

2021 ◽  
Author(s):  
Yuxuan Liang ◽  
Kun Ouyang ◽  
Hanshu Yan ◽  
Yiwei Wang ◽  
Zekun Tong ◽  
...  
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Real World ◽  
Continuous Time ◽  
Trajectory Data ◽  
Time Dynamics ◽  
Trajectory Classification ◽  
Novel Method ◽  
Spatial Trajectory ◽  
Hidden States

Recent advances in location-acquisition techniques have generated massive spatial trajectory data. Recurrent Neural Networks (RNNs) are modern tools for modeling such trajectory data. After revisiting RNN-based methods for trajectory modeling, we expose two common critical drawbacks in the existing uses. First, RNNs are discrete-time models that only update the hidden states upon the arrival of new observations, which makes them an awkward fit for learning real-world trajectories with continuous-time dynamics. Second, real-world trajectories are never perfectly accurate due to unexpected sensor noise. Most RNN-based approaches are deterministic and thereby vulnerable to such noise. To tackle these challenges, we devise a novel method entitled TrajODE for more natural modeling of trajectories. It combines the continuous-time characteristic of Neural Ordinary Differential Equations (ODE) with the robustness of stochastic latent spaces. Extensive experiments on the task of trajectory classification demonstrate the superiority of our framework against the RNN counterparts.

AUTOMATIC PARALLELIZATION OF ELECTRO-MECHANICAL SYSTEMS GOVERNED BY ODEs

2002 ◽  
Vol 26 (3) ◽  
pp. 347-365
Author(s):  
C.A. Rabbath ◽  
A. Ait El Cadi ◽  
M. Abdonne ◽  
N. Lechevin ◽  
S. Lapierre ◽  
...  
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Block Diagram ◽  
Time Integration ◽  
Mechanical Systems ◽  
Automatic Parallelization ◽  
Iteration Step ◽  
Control Laws ◽  
Loop Control ◽  
Novel Method

The paper proposes an effective approach for the automatic parallelization of models of electro-mechanical systems governed by ordinary differential equations. The novel method takes a nominal mathematical model, expressed in block diagram language, and portions in parallel the code to be executed on a set of standard microprocessors. The integrity of the simulations is preserved, the computing resources available are efficiently used, and the simulations are compliant with real-time constraints; that is, the time integration of the ordinary differential equations is performed within restricted time limits at each iteration step. The proposed method is applied to a two-degree-of-freedom revolute joint robotic system that includes an induction motor and two inner-outer loop control laws. Numerical simulations validate the proposed approach.

scholarly journals Numerical Solution of Stieltjes Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1571
Author(s):  
Francisco J. Fernández ◽  
F. Adrián F. Tojo
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Explicit Solution ◽  
Numerical Scheme ◽  
Linear Ordinary Differential Equation ◽  
Stieltjes Integral ◽  
Novel Method ◽  
Theoretical Results

This work is devoted to the obtaining of a new numerical scheme based on quadrature formulae for the Lebesgue–Stieltjes integral for the approximation of Stieltjes ordinary differential equations. This novel method allows us to numerically approximate models based on Stieltjes ordinary differential equations for which no explicit solution is known. We prove several theoretical results related to the consistency, convergence, and stability of the numerical method. We also obtain the explicit solution of the Stieltjes linear ordinary differential equation and use it to validate the numerical method. Finally, we present some numerical results that we have obtained for a realistic population model based on a Stieltjes differential equation and a system of Stieltjes differential equations with several derivators.

scholarly journals Dynamics of the Oxygen, Carbon Dioxide, and Water Interaction across the Insect Spiracle

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
S. M. Simelane ◽  
S. Abelman ◽  
F. D. Duncan
Keyword(s):  
Carbon Dioxide ◽  
Water Vapor ◽  
Differential Equations ◽  
Numerical Simulations ◽  
Ordinary Differential Equations ◽  
Continuous Time ◽  
Water Interaction ◽  
Stability And Instability ◽  
Respiratory Gases

This paper explores the dynamics of respiratory gases interactions which are accompanied by the loss of water through an insect’s spiracle. Here we investigate and analyze this interaction by deriving a system of ordinary differential equations for oxygen, carbon dioxide, and water vapor. The analysis is carried out in continuous time. The purpose of the research is to determine bounds for the gas volumes and to discuss the complexity and stability of the equilibria. Numerical simulations also demonstrate the dynamics of our model utilizing the new conditions for stability and instability.

Nonlinear Dynamics of Two Discrete-Time Versions of the Continuous-Time Brusselator Model

2019 ◽  
Vol 29 (10) ◽  
pp. 1950142
Author(s):  
Paulo C. Rech
Keyword(s):  
Differential Equations ◽  
Mathematical Models ◽  
Ordinary Differential Equations ◽  
Discrete Time ◽  
Continuous Time ◽  
Periodic Structures ◽  
Parameter Spaces ◽  
First Order ◽  
Brusselator Model ◽  
Dynamical Behaviors

In this paper, we report results related with the dynamics of two discrete-time mathematical models, which are obtained from a same continuous-time Brusselator model consisting of two nonlinear first-order ordinary differential equations. Both discrete-time mathematical models are derived by integrating the set of ordinary differential equations, but using different methods. Such results are related, in each case, with parameter-spaces of the two-dimensional map which results from the respective discretization process. The parameter-spaces obtained using both maps are then compared, and we show that the occurrence of organized periodic structures embedded in a quasiperiodic region is verified in only one of the two cases. Bifurcation diagrams, Lyapunov exponents plots, and phase-space portraits are also used, to illustrate different dynamical behaviors in both discrete-time mathematical models.

scholarly journals Deductive Stability Proofs for Ordinary Differential Equations

2021 ◽  
pp. 181-199
Author(s):  
Yong Kiam Tan ◽  
André Platzer
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Real World ◽  
Formal Analysis ◽  
Dynamic Logic ◽  
Logical Structure ◽  
Theorem Prover ◽  
Continuous Systems ◽  
Stability Properties ◽  
Controlled Systems

AbstractStability is required for real world controlled systems as it ensures that those systems can tolerate small, real world perturbations around their desired operating states. This paper shows how stability for continuous systems modeled by ordinary differential equations (ODEs) can be formally verified in differential dynamic logic (). The key insight is to specify ODE stability by suitably nesting the dynamic modalities of with first-order logic quantifiers. Elucidating the logical structure of stability properties in this way has three key benefits: i) it provides a flexible means of formally specifying various stability properties of interest, ii) it yields rigorous proofs of those stability properties from ’s axioms with ’s ODE safety and liveness proof principles, and iii) it enables formal analysis of the relationships between various stability properties which, in turn, inform proofs of those properties. These benefits are put into practice through an implementation of stability proofs for several examples in KeYmaera X, a hybrid systems theorem prover based on .

WHAT CONNECTION TOPOLOGY PROHIBIT CHAOS IN CONTINUOUS TIME NETWORKS?

2007 ◽  
Vol 10 (04) ◽  
pp. 449-461 ◽  
Author(s):  
XIAO-SONG YANG ◽  
QUAN YUAN ◽  
LIN WANG
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Continuous Time ◽  
Chaotic Behavior ◽  
Three Dimensional ◽  
Direct Connection ◽  
The Third

In this paper, we are concerned with two interesting problems in the dynamics of neural networks. What connection topology will prohibit chaotic behavior in a continuous time neural network (NN). To what extent is a continuous time neural network (NN) described by continuous ordinary differential equations simple enough yet still able to exhibit chaos? We study these problems in the context of the classical neural networks with three neurons, which can be described by three-dimensional autonomous ordinary differential equations. We first consider the case where there is no direct interconnection between the first neuron and the third neuron. We then discuss the case where each pair of neurons has a direct connection. We show that the existence of the directed loop in connection topology is necessary for chaos to occur.

scholarly journals Construction of Interval Wavelet Based on Restricted Variational Principle and Its Application for Solving Differential Equations

2008 ◽  
Vol 2008 ◽  
pp. 1-14 ◽  
Author(s):  
Shu-Li Mei ◽  
Hong-Liang Lv ◽  
Qin Ma
Keyword(s):  
Variational Principle ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Local Property ◽  
Wavelet Method ◽  
Shannon Wavelet ◽  
Novel Method ◽  
Numerical Performance

Based on restricted variational principle, a novel method for interval wavelet construction is proposed. For the excellent local property of quasi-Shannon wavelet, its interval wavelet is constructed, and then applied to solve ordinary differential equations. Parameter choices for the interval wavelet method are discussed and its numerical performance is demonstrated.

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