22 research outputs found
Order theory for discrete gradient methods
We present a subclass of the discrete gradient methods, which are integrators
designed to preserve invariants of ordinary differential equations. From a
formal series expansion of the methods, we derive conditions for arbitrarily
high order. We devote considerable space to the average vector field discrete
gradient, from which we get P-series methods in the general case, and B-series
methods for canonical Hamiltonian systems. Higher order schemes are presented
and applied to the H\'enon-Heiles system and a Lotka-Volterra system.Comment: 45 pages, 5 figure
Adaptive Energy Preserving Methods for Partial Differential Equations
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 with both finite difference and partition of
unity approaches, thereby also including finite element approaches. The schemes
are then extended to accommodate -, - and -adaptivity. The method is
applied to the Korteweg-de Vries equation and the Sine-Gordon equation and
results from numerical experiments are presented.Comment: 27 pages; some changes to notation and figure
Shape analysis on Lie groups and homogeneous spaces
In this paper we are concerned with the approach to shape analysis based on
the so called Square Root Velocity Transform (SRVT). We propose a
generalisation of the SRVT from Euclidean spaces to shape spaces of curves on
Lie groups and on homogeneous manifolds. The main idea behind our approach is
to exploit the geometry of the natural Lie group actions on these spaces.Comment: 8 pages, Contribution to the conference "Geometric Science of
Information '17
Dissipative numerical schemes on Riemannian manifolds with applications to gradient flows
This paper concerns an extension of discrete gradient methods to
finite-dimensional Riemannian manifolds termed discrete Riemannian gradients,
and their application to dissipative ordinary differential equations. This
includes Riemannian gradient flow systems which occur naturally in optimization
problems. The Itoh--Abe discrete gradient is formulated and applied to gradient
systems, yielding a derivative-free optimization algorithm. The algorithm is
tested on two eigenvalue problems and two problems from manifold valued
imaging: InSAR denoising and DTI denoising.Comment: Post-revision version. To appear in SIAM Journal on Scientific
Computin
Pseudo-Hamiltonian neural networks for learning partial differential equations
Pseudo-Hamiltonian neural networks (PHNN) were recently introduced for
learning dynamical systems that can be modelled by ordinary differential
equations. In this paper, we extend the method to partial differential
equations. The resulting model is comprised of up to three neural networks,
modelling terms representing conservation, dissipation and external forces, and
discrete convolution operators that can either be learned or be given as input.
We demonstrate numerically the superior performance of PHNN compared to a
baseline model that models the full dynamics by a single neural network.
Moreover, since the PHNN model consists of three parts with different physical
interpretations, these can be studied separately to gain insight into the
system, and the learned model is applicable also if external forces are removed
or changed.Comment: 33 pages, 14 figures; v2: minor changes to text, updated numerical
experiment
Learning Dynamical Systems from Noisy Data with Inverse-Explicit Integrators
We introduce the mean inverse integrator (MII), a novel approach to increase
the accuracy when training neural networks to approximate vector fields of
dynamical systems from noisy data. This method can be used to average multiple
trajectories obtained by numerical integrators such as Runge-Kutta methods. We
show that the class of mono-implicit Runge-Kutta methods (MIRK) has particular
advantages when used in connection with MII. When training vector field
approximations, explicit expressions for the loss functions are obtained when
inserting the training data in the MIRK formulae, unlocking symmetric and
high-order integrators that would otherwise be implicit for initial value
problems. The combined approach of applying MIRK within MII yields a
significantly lower error compared to the plain use of the numerical integrator
without averaging the trajectories. This is demonstrated with experiments using
data from several (chaotic) Hamiltonian systems. Additionally, we perform a
sensitivity analysis of the loss functions under normally distributed
perturbations, supporting the favorable performance of MII.Comment: 23 pages, 10 figure
Port-Hamiltonian Neural Networks with State-Dependent Ports
Hybrid machine learning based on Hamiltonian formulations has recently been
successfully demonstrated for simple mechanical systems, both energy conserving
and not energy conserving. We show that port-Hamiltonian neural network models
can be used to learn external forces acting on a system. We argue that this
property is particularly useful when the external forces are state dependent,
in which case it is the port-Hamiltonian structure that facilitates the
separation of internal and external forces. Numerical results are provided for
a forced and damped mass-spring system and a tank system of higher complexity,
and a symmetric fourth-order integration scheme is introduced for improved
training on sparse and noisy data.Comment: 21 pages, 12 figures; v3: restructured the paper for more clarity,
major changes to the text, updated plot
Order theory for discrete gradient methods
The discrete gradient methods are integrators designed to preserve invariants of ordinary differential equations. From a formal series expansion of a subclass of these methods, we derive conditions for arbitrarily high order. We derive specific results for the average vector field discrete gradient, from which we get P-series methods in the general case, and B-series methods for canonical Hamiltonian systems. Higher order schemes are presented, and their applications are demonstrated on the Hénon–Heiles system and a Lotka–Volterra system, and on both the training and integration of a pendulum system learned from data by a neural network