Neural learning by geometric integration of reduced ‘rigid-body’ equations

UNIVERSITET Neural Learning by Geometric Integration of Reduced ‘ Rigid - Body ’ Equations

In previous contributions we presented a new class of algorithms for orthonormal learning of linear neural networks with p inputs and m outputs, based on the equations describing the dynamics of a massive rigid frame in a submanifold of IR. While exhibiting interesting features, such as good numerical stability, strongly binding to the orthonormal manifolds, and good controllability of the lear...

Geometric Numerical Integration of Differential Equations

Geometric Numerical Integration of Differential Equations

What is geometric integration? ‘Geometric integration’ is the term used to describe numerical methods for computing the solution of differential equations, while preserving one or more physical/mathematical properties of the system exactly (i.e. up to round–off error)1. What properties can be preserved in this way? A first aspect of a dynamical system that is important to preserve is its phase ...

Algorithm for numerical integration of the rigid-body equations of motion

The method of molecular dynamics (MD) plays a prominent role in studying molecular liquids. All existing techniques appropriate to simulate such systems can be categorized in dependence on what type of parameters are chosen to represent the rotational degrees of freedom and what kind of numerical algorithm is applied to integrate the corresponding equations of motion. In the molecular approach,...

Geometric Structure-preserving Optimal Control of a Rigid Body

In this paper, we study a discrete variational optimal control problem for a rigid body. The cost to be minimized is the external torque applied to move the rigid body from an initial condition to a pre-specified terminal condition. Instead of discretizing the equations of motion, we use the discrete equations obtained from the discrete Lagrange–d’Alembert principle, a process that better appro...