Related papers: A Hamiltonian approach to implicit systems, genera…
This paper is devoted to the study of symplectic manifolds and their connection with Hamiltonian dynamical systems. We review some properties and operations on these manifolds and see how they intervene when studying the complete…
Adaptive methods do not have a direct generalization to manifolds as the adaptive term is not invariant. Momentum methods on manifolds suffer from efficiency problems stemming from the curvature of the manifold. We introduce a framework to…
Diverse optimization algorithms correctly identify, in finite time, intrinsic constraints that must be active at optimality. Analogous behavior extends beyond optimization to systems involving partly smooth operators, and in particular to…
Interior-point methods offer a highly versatile framework for convex optimization that is effective in theory and practice. A key notion in their theory is that of a self-concordant barrier. We give a suitable generalization of…
Many important physical systems can be described as the evolution of a Hamiltonian system, which has the important property of being conservative, that is, energy is conserved throughout the evolution. Physics Informed Neural Networks and…
Equations of Hammerstein type cover large variety of areas and are of much interest to a wide audience due to the fact that they have applications in numerous areas. Suitable conditions are imposed to obtain a strong convergence result for…
Kahan's method and a two-step generalization of the discrete gradient method are both linearly implicit methods that can preserve a modified energy for Hamiltonian systems with a cubic Hamiltonian. These methods are here investigated and…
Generalized models provide a framework for the study of evolution equations without specifying all functional forms. The generalized formulation of problems has been shown to facilitate the analytical investigation of local dynamics and has…
We study the set of continuous functions that admit no spurious local optima (i.e. local minima that are not global minima) which we term \textit{global functions}. They satisfy various powerful properties for analyzing nonconvex and…
Paper is devoted to maintaining the simple objective: We want to provide Hamiltonian canonical form for autonomous dynamical system reducible to even-dimensional one. Along the road we construct new class of conserved quantities, called…
In the paper, we improve our earlier results concerning the existence, uniqueness and differentiability of a global implicit function. Some application to a Cauchy problem for an integro-differential Volterra system of nonconvolution type,…
We investigate the presence of localized solutions in models described by a single real scalar field with generalized dynamics. The study offers a method to solve very intricate nonlinear ordinary differential equations, and we illustrate…
The geometric formulation of the Hamilton-Jacobi theory enables us to generalize it to systems of higher-order ordinary differential equations. In this work we introduce the unified Lagrangian-Hamiltonian formalism for the geometric…
Hamiltonian systems are differential equations which describe systems in classical mechanics, plasma physics, and sampling problems. They exhibit many structural properties, such as a lack of attractors and the presence of conservation…
Motivated by recent developments in Hamiltonian variational principles, Hamiltonian variational integrators, and their applications such as to optimization and control, we present a new Type II variational approach for Hamiltonian systems,…
We present the symplectic algorithm in the Lagrangian formalism for the Hamiltonian systems by virtue of the noncommutative differential calculus with respect to the discrete time and the Euler--Lagrange cohomological concepts. We also show…
The past few years have witnessed an increased interest in learning Hamiltonian dynamics in deep learning frameworks. As an inductive bias based on physical laws, Hamiltonian dynamics endow neural networks with accurate long-term…
This work explores the tensor and combinatorial constructs underlying the linearised higher-order variational equations of a generic autonomous system along a particular solution. The main result of this paper is a compact yet explicit and…
Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\Omega)$, and optimizing functionals arising from some…
Optimization problems with convex quadratic cost and polyhedral constraints are ubiquitous in signal processing, automatic control and decision-making. We consider here an enlarged problem class that allows to encode logical conditions and…