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The goal of this paper is to develop energy-preserving variational integrators for time-dependent mechanical systems with forcing. We first present the Lagrange-d'Alembert principle in the extended Lagrangian mechanics framework and derive…

This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation…

We compare two approaches to the predictive modeling of dynamical systems from partial observations at discrete times. The first is continuous in time, where one uses data to infer a model in the form of stochastic differential equations,…

Energy methods for constructing time-stepping algorithms are of increased interest in application to nonlinear problems, since numerical stability can be inferred from the conservation of the system energy. Alternatively, symplectic…

Modifying the discrete mechanics proposed by T.D. Lee, we construct a class of discrete classical Hamiltonian systems, in which time is one of the dynamical variables. This includes a toy model of time machines which can travel forward and…

In this paper we present a general framework that allows one to study discretization of certain dynamical systems. This generalizes earlier work on discretization of Lagrangian and Hamiltonian systems on tangent bundles and cotangent…

We analyze the convergence rate of various momentum-based optimization algorithms from a dynamical systems point of view. Our analysis exploits fundamental topological properties, such as the continuous dependence of iterates on their…

We propose a finite element discretisation approach for the incompressible Euler equations which mimics their geometric structure and their variational derivation. In particular, we derive a finite element method that arises from a…

We develop in this paper a new framework for discrete calculus of variations when the actions have densities involving an arbitrary discretization operator. We deduce the discrete Euler-Lagrange equations for piecewise continuous critical…

A canonical formalism and constraint analysis for discrete systems subject to a variational action principle are devised. The formalism is equivalent to the covariant formulation, encompasses global and local discrete time evolution moves…

Discretizing variational principles, as opposed to discretizing differential equations, leads to discrete-time analogues of mechanics, and, systematically, to geometric numerical integrators. The phase space of such variational…

Recently, continuous-time dynamical systems have proved useful in providing conceptual and quantitative insights into gradient-based optimization, widely used in modern machine learning and statistics. An important question that arises in…

In two papers we proposed a continuum model for the dynamics of systems of self propelling particles with kinematic constraints on the velocities and discussed some of its properties. The model aims to be analogous to a discrete algorithm…

The article introduces a method to learn dynamical systems that are governed by Euler--Lagrange equations from data. The method is based on Gaussian process regression and identifies continuous or discrete Lagrangians and is, therefore,…

Discrete gradient methods are a powerful tool for the time discretization of dynamical systems, since they are structure-preserving regardless of the form of the total energy. In this work, we discuss the application of discrete gradient…

In this paper we introduce discrete gradient methods to discretize irreversible port-Hamiltonian systems showing that the main qualitative properties of the continuous system are preserved using this kind discretizations methods.

We study approximation of non-autonomous linear differential equations with variable delay over infinite intervals. We use piecewise constant argument to obtain a corresponding discrete difference equation. The study of numerical…

We propose a variational symplectic numerical method for the time integration of dynamical systems issued from the least action principle. We assume a quadratic internal interpolation of the state between two time steps and we approximate…

The optimal control of a mechanical system is of crucial importance in many realms. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion…

First order optimization algorithms play a major role in large scale machine learning. A new class of methods, called adaptive algorithms, were recently introduced to adjust iteratively the learning rate for each coordinate. Despite great…