303 citations · 796 across the 37 of their papers we have counts for
29 papers · 1 filter
1-Lipschitz Neural Networks on Hadamard Manifolds
Davide Murari, Marta Ghirardelli, Ben Adcock +3
Controlling the Lipschitz constant of a neural network is a standard way to promote robustness and stability. Most existing constraining strategies are designed for Euclidean space…
A Unified Discrete Gradient-SAV Framework for Structure-Preserving Integration
Elena Celledoni, David Martín de Diego, Brynjulf Owren +1
We present a framework combining discrete gradient (DG) methods with the Scalar Auxiliary Variable (SAV) approach to construct structure-preserving integrators for dissipative and…
Approximation properties of neural ODEs
Arturo De Marinis, Davide Murari, Elena Celledoni +3
We study the approximation properties of neural ordinary differential equations (neural ODEs) in the space of continuous functions. Since a neural ODE requires input and output dim…
Conditional Stability of the Euler Method on Riemannian Manifolds
Marta Ghirardelli, Brynjulf Owren, Elena Celledoni
We derive nonlinear stability results for numerical integrators on Riemannian manifolds, by imposing conditions on the ODE vector field and the step size that makes the numerical s…
Neural networks for the approximation of Euler's elastica
Elena Celledoni, Ergys Çokaj, Andrea Leone +5
Euler's elastica is a classical model of flexible slender structures, relevant in many industrial applications. Static equilibrium equations can be derived via a variational princi…
B-stability of numerical integrators on Riemannian manifolds
Martin Arnold, Elena Celledoni, Ergys Çokaj +2
We propose a generalization of nonlinear stability of numerical one-step integrators to Riemannian manifolds in the spirit of Butcher's notion of B-stability. Taking inspiration fr…
Predictions Based on Pixel Data: Insights from PDEs and Finite Differences
Elena Celledoni, James Jackaman, Davide Murari +1
As supported by abundant experimental evidence, neural networks are state-of-the-art for many approximation tasks in high-dimensional spaces. Still, there is a lack of a rigorous t…
Using aromas to search for preserved measures and integrals in Kahan's method
Geir Bogfjellmo, Elena Celledoni, Robert McLachlan +2
The numerical method of Kahan applied to quadratic differential equations is known to often generate integrable maps in low dimensions and can in more general situations exhibit pr…
Learning Hamiltonians of constrained mechanical systems
Elena Celledoni, Andrea Leone, Davide Murari +1
Recently, there has been an increasing interest in modelling and computation of physical systems with neural networks. Hamiltonian systems are an elegant and compact formalism in c…
Dynamics of the N-fold Pendulum in the framework of Lie Group Integrators
Elena Celledoni, Ergys Çokaj, Andrea Leone +2
Since their introduction, Lie group integrators have become a method of choice in many application areas. Various formulations of these integrators exist, and in this work we focus…
Lie Group integrators for mechanical systems
Elena Celledoni, Ergys Çokaj, Andrea Leone +2
Since they were introduced in the 1990s, Lie group integrators have become a method of choice in many application areas. These include multibody dynamics, shape analysis, data scie…
Detecting and determining preserved measures and integrals of birational maps
Elena Celledoni, Charalambos Evripidou, David McLaren +3
In this paper we use the method of discrete Darboux polynomials to calculate preserved measures and integrals of rational maps. The approach is based on the use of cofactors and Da…
The Magnus expansion and Post-Lie algebras
Charles Curry, Kurusch Ebrahimi-Fard, Brynjulf Owren
We relate the classical and post-Lie Magnus expansions. Intertwining algebraic and geometric arguments allows to placing the classical Magnus expansion in the context of Lie group…
Energy preserving methods on Riemannian manifolds
Elena Celledoni, Sølve Eidnes, Brynjulf Owren +1
The energy preserving discrete gradient methods are generalized to finite-dimensional Riemannian manifolds by definition of a discrete approximation to the Riemannian gradient, a r…
Geometric and integrability properties of Kahan's method
Elena Celledoni, David McLaren, Brynjulf Owren +1
Given a quadratic vector field on \mathbb{R}^n possessing a quadratic first integral depending on two of the independent variables, we give a constructive proof that Kahan's discre…
Dissipative numerical schemes on Riemannian manifolds with applications to gradient flows
Elena Celledoni, Sølve Eidnes, Brynjulf Owren +1
This paper concerns an extension of discrete gradient methods to finite-dimensional Riemannian manifolds termed discrete Riemannian gradients, and their application to dissipative…
Variable stepsize commutator free Lie group integrators
Charles Curry, Brynjulf Owren
We introduce variable stepsize commutator free Lie group integrators, where the error control is achieved using embedded Runge-Kutta pairs.
Two classes of quadratic vector fields for which the Kahan discretization is integrable
Elena Celledoni, Robert I. McLachlan, David I. McLaren +2
Applying Kahan's discretization to the reduced Nahm equations, we obtain two classes of integrable mappings.
Lie group integrators
Brynjulf Owren
In this survey we discuss a wide variety of aspects related to Lie group integrators. These numerical integration schemes for differential equations on manifolds have been studied…
Adaptive Energy Preserving Methods for Partial Differential Equations
Sølve Eidnes, Brynjulf Owren, Torbjørn Ringholm
A method for constructing first integral preserving numerical schemes for time-dependent partial differential equations on non-uniform grids is presented. The method can be used wi…
Discretization of polynomial vector fields by polarization
Elena Celledoni, Robert I. McLachlan, David I. McLaren +2
A novel integration method for quadratic vector fields was introduced by Kahan in 1993. Subsequently, it was shown that Kahan's method preserves a (modified) measure and energy whe…
Geometric integration of non-autonomous Hamiltonian problems
Håkon Marthinsen, Brynjulf Owren
Symplectic integration of autonomous Hamiltonian systems is a well-known field of study in geometric numerical integration, but for non-autonomous systems the situation is less cle…
Preserving first integrals with symmetric Lie group methods
Elena Celledoni, Brynjulf Owren
The discrete gradient approach is generalized to yield integral preserving methods for differential equations in Lie groups.
Geometric properties of Kahan's method
Elena Celledoni, Robert I McLachlan, Brynjulf Owren +1
We show that Kahan's discretization of quadratic vector fields is equivalent to a Runge--Kutta method. When the vector field is Hamiltonian on either a symplectic vector space or a…