303 citations · 796 across the 37 of their papers we have counts for
44 papers
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…
Learning Forced Multibody Dynamics on Lie Groups
Martine Dyring Hansen, Marta Ghirardelli, Elena Celledoni +2
We propose an architecture for learning the dynamics of mechanical systems based on discrete forced Euler-Lagrange equations on Lie groups using only position data. By formulating…
Mixed Precision Training of Neural ODEs
Elena Celledoni, Brynjulf Owren, Lars Ruthotto +1
Exploiting low-precision computations has become a standard strategy in deep learning to address the growing computational costs imposed by ever larger models and datasets. However…
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…
Designing Stable Neural Networks using Convex Analysis and ODEs
Ferdia Sherry, Elena Celledoni, Matthias J. Ehrhardt +3
Motivated by classical work on the numerical integration of ordinary differential equations we present a ResNet-styled neural network architecture that encodes non-expansive (1-Lip…
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…
Dynamical systems' based neural networks
Elena Celledoni, Davide Murari, Brynjulf Owren +2
Neural networks have gained much interest because of their effectiveness in many applications. However, their mathematical properties are generally not well understood. If there is…
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…
Equivariant neural networks for inverse problems
Elena Celledoni, Matthias J. Ehrhardt, Christian Etmann +3
In recent years the use of convolutional layers to encode an inductive bias (translational equivariance) in neural networks has proven to be a very fruitful idea. The successes of…
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…
An integral model based on slender body theory, with applications to curved rigid fibers
Helge I. Andersson, Elena Celledoni, Laurel Ohm +2
We propose a novel integral model describing the motion of curved slender fibers in viscous flow, and develop a numerical method for simulating dynamics of rigid fibers. The model…
Structure preserving deep learning
Elena Celledoni, Matthias J. Ehrhardt, Christian Etmann +4
Over the past few years, deep learning has risen to the foreground as a topic of massive interest, mainly as a result of successes obtained in solving large-scale image processing…
Computational geometric methods for preferential clustering of particle suspensions
Benjamin K. Tapley, Helge I. Andersson, Elena Celledoni +1
A geometric numerical method for simulating suspensions of spherical and non-spherical particles with Stokes drag is proposed. The method combines divergence-free matrix-valued rad…
A slender body model for thin rigid fibers: validation and comparisons
Laurel Ohm, Benjamin K. Tapley, Helge I. Andersson +2
In this paper we consider a computational model for the motion of thin, rigid fibers in viscous flows based on slender body theory. Slender body theory approximates the fluid veloc…
Deep learning as optimal control problems: models and numerical methods
Martin Benning, Elena Celledoni, Matthias J. Ehrhardt +2
We consider recent work of Haber and Ruthotto 2017 and Chang et al. 2018, where deep learning neural networks have been interpreted as discretisations of an optimal control problem…
Using discrete Darboux polynomials to detect and determine preserved measures and integrals of rational maps
E Celledoni, C Evripidou, D I McLaren +4
In this Letter we propose a systematic approach for detecting and calculating preserved measures and integrals of a rational map. The approach is based on the use of cofactors and…
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…