Related papers: Second-order variational equations for N-body simu…
First-order stochastic methods are the state-of-the-art in large-scale machine learning optimization owing to efficient per-iteration complexity. Second-order methods, while able to provide faster convergence, have been much less explored…
We discuss an elementary derivation of variational symmetries and corresponding integrals of motion for the Lagrangian systems depending on acceleration. Providing several examples, we make the manuscript accessible to a wide range of…
Model reduction for fluid flow simulation continues to be of great interest across a number of scientific and engineering fields. Here, we explore the use of Neural Ordinary Differential Equations, a recently introduced family of…
The nuclear many-body problem for medium-mass systems is commonly addressed using wave-function expansion methods that build upon a second-quantized representation of many-body operators with respect to a chosen computational basis. While…
An ever-growing observational aggregate of extrasolar planets has revealed that systems of planets that reside in or near mean-motion resonances are relatively common. While the origin of such systems is attributed to protoplanetary…
This paper studies formulations of second-order elliptic partial differential equations in nondivergence form on convex domains as equivalent variational problems. The first formulation is that of Smears \& S\"uli [SIAM J.\ Numer.\ Anal.\…
In a coordinate free form are found the (deviation) equations satisfied by the (infinitesimal) deviation vector, relative velocity, relative momentum, relative acceleration and relative energy of two point particles in a differentiable…
A variational principle is further developed for out of equilibrium dynamical systems by using the concept of maximum entropy. With this new formulation it is obtained a set of two first-order differential equations, revealing the same…
We revisit the relativistic restricted two-body problem with spin employing a perturbation scheme based on Lie series. Starting from a post-Newtonian expansion of the field equations, we develop a first-order secular theory that reproduces…
We develop a method for optimization in shape spaces, i.e., sets of surfaces modulo re-parametrization. Unlike previously proposed gradient flows, we achieve superlinear convergence rates through a subtle approximation of the shape Hessian,…
We propose new continuous-time formulations for first-order stochastic optimization algorithms such as mini-batch gradient descent and variance-reduced methods. We exploit these continuous-time models, together with simple Lyapunov analysis…
A class of second order approximations, called the weighted and shifted Gr\"{u}nwald difference operators, are proposed for Riemann-Liouville fractional derivatives, with their effective applications to numerically solving space fractional…
In recent years, much effort in designing numerical methods for the simulation and optimization of mechanical systems has been put into schemes which are structure preserving. One particular class are variational integrators which are…
Molecular Dynamics method is based on solution of Newtonian differential equations of motion. A new very accurate and efficient time-reversible explicit integrator was derived on the basis of second order Tailor expansion of force. There is…
In this study, linear second-order conformable differential equations using a proportional derivative are shown to be formally self-adjoint equations with respect to a certain inner product and the associated self-adjoint boundary…
Solving inverse problems, such as parameter estimation and optimal control, is a vital part of science. Many experiments repeatedly collect data and rely on machine learning algorithms to quickly infer solutions to the associated inverse…
In this paper we will discuss some new developments in the design of numerical methods for optimal control problems of Lagrangian systems on Lie groups. We will construct these geometric integrators using discrete variational calculus on…
We focus here on a class of fourth-order parabolic equations that can be written as a system of second-order equations by introducing an auxiliary variable. We design a novel second-order fully discrete mixed finite element method to…
Differentiable simulators promise faster computation time for reinforcement learning by replacing zeroth-order gradient estimates of a stochastic objective with an estimate based on first-order gradients. However, it is yet unclear what…
Variational inequalities represent a broad class of problems, including minimization and min-max problems, commonly found in machine learning. Existing second-order and high-order methods for variational inequalities require precise…