TY - JOUR
AB - 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 laws. To predict Hamiltonian dynamics based on discrete trajectory observations, incorporation of prior knowledge about Hamiltonian structure greatly improves predictions. This is typically done by learning the system's Hamiltonian and then integrating the Hamiltonian vector field with a symplectic integrator. For this, however, Hamiltonian data needs to be approximated based on the trajectory observations. Moreover, the numerical integrator introduces an additional discretisation error. In this paper, we show that an inverse modified Hamiltonian structure adapted to the geometric integrator can be learned directly from observations. A separate approximation step for the Hamiltonian data avoided. The inverse modified data compensates for the discretisation error such that the discretisation error is eliminated. The technique is developed for Gaussian Processes.
AU - Offen, Christian
AU - Ober-Blöbaum, Sina
ID - 23382
JF - Chaos: An Interdisciplinary Journal of Nonlinear Science
TI - Symplectic integration of learned Hamiltonian systems
VL - 32(1)
ER -
TY - JOUR
AB - In backward error analysis, an approximate solution to an equation is compared to the exact solution to a nearby ‘modified’ equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the differential equation has a geometric property then the modified equation may share it. In this way, known properties of differential equations can be applied to the approximation. But for partial differential equations, the known modified equations are of higher order, limiting applicability of the theory. Therefore, we study symmetric solutions of discretized
partial differential equations that arise from a discrete variational principle. These symmetric solutions obey infinite-dimensional functional equations. We show that these equations admit second-order modified equations which are Hamiltonian and also possess first-order Lagrangians in modified coordinates. The modified equation and its associated structures are computed explicitly for the case of rotating travelling waves in the nonlinear wave equation.
AU - McLachlan, Robert I
AU - Offen, Christian
ID - 19941
IS - 3
JF - Journal of Geometric Mechanics
TI - Backward error analysis for variational discretisations of partial differential equations
VL - 14
ER -
TY - JOUR
AB - The principle of least action is one of the most fundamental physical principle. It says that among all possible motions connecting two points in a phase space, the system will exhibit those motions which extremise an action functional. Many qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equations, are related to the existence of an action functional. Incorporating variational structure into learning algorithms for dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features with the exact physical system. In this paper we show how to incorporate variational principles into trajectory predictions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position data of observed trajectories. Velocities or conjugate momenta do not need to be observed or approximated and no prior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward error analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the learned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this,
we introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of variational backward error analysis. (3) Finally, we introduce a method to perform system identification from position observations only, based on variational backward error analysis.
AU - Ober-Blöbaum, Sina
AU - Offen, Christian
ID - 29240
JF - Journal of Computational and Applied Mathematics
KW - Lagrangian learning
KW - variational backward error analysis
KW - modified Lagrangian
KW - variational integrators
KW - physics informed learning
TI - Variational Learning of Euler–Lagrange Dynamics from Data
ER -
TY - JOUR
AB - The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients.
AU - McLachlan, Robert
AU - Offen, Christian
ID - 29236
JF - Journal of Geometric Mechanics
KW - variational integrators
KW - backward error analysis
KW - Euler--Lagrange equations
KW - multistep methods
KW - conjugate symplectic methods
TI - Backward error analysis for conjugate symplectic methods
ER -
TY - CONF
AB - The first order optimality conditions of optimal control problems (OCPs) can
be regarded as boundary value problems for Hamiltonian systems. Variational or
symplectic discretisation methods are classically known for their excellent
long term behaviour. As boundary value problems are posed on intervals of
fixed, moderate length, it is not immediately clear whether methods can profit
from structure preservation in this context. When parameters are present,
solutions can undergo bifurcations, for instance, two solutions can merge and
annihilate one another as parameters are varied. We will show that generic
bifurcations of an OCP are preserved under discretisation when the OCP is
either directly discretised to a discrete OCP (direct method) or translated
into a Hamiltonian boundary value problem using first order necessary
conditions of optimality which is then solved using a symplectic integrator
(indirect method). Moreover, certain bifurcations break when a non-symplectic
scheme is used. The general phenomenon is illustrated on the example of a cut
locus of an ellipsoid.
AU - Offen, Christian
AU - Ober-Blöbaum, Sina
ID - 22894
KW - optimal control
KW - catastrophe theory
KW - bifurcations
KW - variational methods
KW - symplectic integrators
SN - 2405-8963
TI - Bifurcation preserving discretisations of optimal control problems
VL - 54(19)
ER -
TY - CONF
AU - Ridderbusch, Steffen
AU - Offen, Christian
AU - Ober-Blöbaum, Sina
AU - Goulart, Paul
ID - 21572
T2 - 2021 60th IEEE Conference on Decision and Control (CDC)
TI - Learning ODE Models with Qualitative Structure Using Gaussian Processes
ER -
TY - GEN
AB - Many problems in science and engineering require an efficient numerical approximation of integrals
or solutions to differential equations. For systems with rapidly changing dynamics, an equidistant
discretization is often inadvisable as it either results in prohibitively large errors or computational effort. To this end, adaptive schemes, such as solvers based on Runge–Kutta pairs, have been developed
which adapt the step size based on local error estimations at each step. While the classical schemes
apply very generally and are highly efficient on regular systems, they can behave sub-optimal when
an inefficient step rejection mechanism is triggered by structurally complex systems such as chaotic
systems. To overcome these issues, we propose a method to tailor numerical schemes to the problem
class at hand. This is achieved by combining simple, classical quadrature rules or ODE solvers with
data-driven time-stepping controllers. Compared with learning solution operators to ODEs directly,
it generalises better to unseen initial data as our approach employs classical numerical schemes as
base methods. At the same time it can make use of identified structures of a problem class and,
therefore, outperforms state-of-the-art adaptive schemes. Several examples demonstrate superior
efficiency. Source code is available at https://github.com/lueckem/quadrature-ML.
AU - Dellnitz, Michael
AU - Hüllermeier, Eyke
AU - Lücke, Marvin
AU - Ober-Blöbaum, Sina
AU - Offen, Christian
AU - Peitz, Sebastian
AU - Pfannschmidt, Karlson
ID - 21600
T2 - arXiv:2104.03562
TI - Efficient time stepping for numerical integration using reinforcement learning
ER -
TY - JOUR
AB - We show that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not. We provide a universal description of the breaking of umbilic bifurcations by nonysmplectic integrators. We discover extra structure induced from certain types of boundary value problems, including classical Dirichlet problems, that is useful to locate bifurcations. Geodesics connecting two points are an example of a Hamiltonian boundary value problem, and we introduce the jet-RATTLE method, a symplectic integrator that easily computes geodesics and their bifurcations. Finally, we study the periodic pitchfork bifurcation, a codimension-1 bifurcation arising in integrable Hamiltonian systems. It is not preserved by either symplectic on nonsymplectic integrators, but in some circumstances symplecticity greatly reduces the error.
AU - McLachlan, Robert I
AU - Offen, Christian
ID - 19938
IS - 6
JF - Foundations of Computational Mathematics
TI - Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation
VL - 20
ER -
TY - JOUR
AU - Kreusser, Lisa Maria
AU - McLachlan, Robert I
AU - Offen, Christian
ID - 19939
IS - 5
JF - Nonlinearity
SN - 0951-7715
TI - Detection of high codimensional bifurcations in variational PDEs
VL - 33
ER -
TY - THES
AB - Ordinary differential equations (ODEs) and partial differential equations (PDEs) arise
in most scientific disciplines that make use of mathematical techniques. As exact solutions are in general not computable, numerical methods are used to obtain approximate
solutions. In order to draw valid conclusions from numerical computations, it is crucial
to understand which qualitative aspects numerical solutions have in common with the
exact solution. Symplecticity is a subtle notion that is related to a rich family of geometric properties of Hamiltonian systems. While the effects of preserving symplecticity
under discretisation on long-term behaviour of motions is classically well known, in this
thesis
(a) the role of symplecticity for the bifurcation behaviour of solutions to Hamiltonian
boundary value problems is explained. In parameter dependent systems at a bifurcation
point the solution set to a boundary value problem changes qualitatively. Bifurcation
problems are systematically translated into the framework of classical catastrophe theory. It is proved that existing classification results in catastrophe theory apply to
persistent bifurcations of Hamiltonian boundary value problems. Further results for
symmetric settings are derived.
(b) It is proved that to preserve generic bifurcations under discretisation it is necessary and sufficient to preserve the symplectic structure of the problem.
(c) The catastrophe theory framework for Hamiltonian ODEs is extended to PDEs
with variational structure. Recognition equations for A-series singularities for functionals on Banach spaces are derived and used in a numerical example to locate high-codimensional bifurcations.
(d) The potential of symplectic integration for infinite-dimensional Lie-Poisson systems (Burgers’ equation, KdV, fluid equations, . . . ) using Clebsch variables is analysed.
It is shown that the advantages of symplectic integration can outweigh the disadvantages of integrating over a larger phase space introduced by a Clebsch representation.
(e) Finally, the preservation of variational structure of symmetric solutions in multisymplectic PDEs by multisymplectic integrators on the example of (phase-rotating)
travelling waves in the nonlinear wave equation is discussed.
AU - Offen, Christian
ID - 19947
TI - Analysis of Hamiltonian boundary value problems and symplectic integration
ER -
TY - JOUR
AB - Many PDEs (Burgers' equation, KdV, Camassa-Holm, Euler's fluid equations, …) can be formulated as infinite-dimensional Lie-Poisson systems. These are Hamiltonian systems on manifolds equipped with Poisson brackets. The Poisson structure is connected to conservation properties and other geometric features of solutions to the PDE and, therefore, of great interest for numerical integration. For the example of Burgers' equations and related PDEs we use Clebsch variables to lift the original system to a collective Hamiltonian system on a symplectic manifold whose structure is related to the original Lie-Poisson structure. On the collective Hamiltonian system a symplectic integrator can be applied. Our numerical examples show excellent conservation properties and indicate that the disadvantage of an increased phase-space dimension can be outweighed by the advantage of symplectic integration.
AU - McLachlan, Robert I
AU - Offen, Christian
AU - Tapley, Benjamin K
ID - 19945
IS - 1
JF - Journal of Computational Dynamics
SN - 2158-2505
TI - Symplectic integration of PDEs using Clebsch variables
VL - 6
ER -
TY - JOUR
AB - A bifurcation is a qualitative change in a family of solutions to an equation produced by varying parameters. In contrast to the local bifurcations of dynamical systems that are often related to a change in the number or stability of equilibria, bifurcations of boundary value problems are global in nature and may not be related to any obvious change in dynamical behaviour. Catastrophe theory is a well-developed framework which studies the bifurcations of critical points of functions. In this paper we study the bifurcations of solutions of boundary-value problems for symplectic maps, using the language of (finite-dimensional) singularity theory. We associate certain such problems with a geometric picture involving the intersection of Lagrangian submanifolds, and hence with the critical points of a suitable generating function. Within this framework, we then study the effect of three special cases: (i) some common boundary conditions, such as Dirichlet boundary conditions for second-order systems, restrict the possible types of bifurcations (for example, in generic planar systems only the A-series beginning with folds and cusps can occur); (ii) integrable systems, such as planar Hamiltonian systems, can exhibit a novel periodic pitchfork bifurcation; and (iii) systems with Hamiltonian symmetries or reversing symmetries can exhibit restricted bifurcations associated with the symmetry. This approach offers an alternative to the analysis of critical points in function spaces, typically used in the study of bifurcation of variational problems, and opens the way to the detection of more exotic bifurcations than the simple folds and cusps that are often found in examples.
AU - McLachlan, Robert I
AU - Offen, Christian
ID - 19935
JF - Nonlinearity
SN - 0951-7715
TI - Bifurcation of solutions to Hamiltonian boundary value problems
ER -
TY - JOUR
AB - Symplectic integrators can be excellent for Hamiltonian initial value problems. Reasons for this include their preservation of invariant sets like tori, good energy behaviour, nonexistence of attractors, and good behaviour of statistical properties. These all refer to {\em long-time} behaviour. They are directly connected to the dynamical behaviour of symplectic maps φ:M→M' on the phase space under iteration. Boundary value problems, in contrast, are posed for fixed (and often quite short) times. Symplecticity manifests as a symplectic map φ:M→M' which is not iterated. Is there any point, therefore, for a symplectic integrator to be used on a Hamiltonian boundary value problem? In this paper we announce results that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not.
AU - McLachlan, Robert I
AU - Offen, Christian
ID - 19937
JF - Numerical Algorithms
SN - 1017-1398
TI - Symplectic integration of boundary value problems
ER -
TY - GEN
AB - Two smooth map germs are right-equivalent if and only if they generate two
Lagrangian submanifolds in a cotangent bundle which have the same contact with
the zero-section. In this paper we provide a reverse direction to this
classical result of Golubitsky and Guillemin. Two Lagrangian submanifolds of a
symplectic manifold have the same contact with a third Lagrangian submanifold
if and only if the intersection problems correspond to stably right equivalent
map germs. We, therefore, obtain a correspondence between local Lagrangian
intersection problems and catastrophe theory while the classical version only
captures tangential intersections. The correspondence is defined independently
of any Lagrangian fibration of the ambient symplectic manifold, in contrast to
other classical results. Moreover, we provide an extension of the
correspondence to families of local Lagrangian intersection problems. This
gives rise to a framework which allows a natural transportation of the notions
of catastrophe theory such as stability, unfolding and (uni-)versality to the
geometric setting such that we obtain a classification of families of local
Lagrangian intersection problems. An application is the classification of
Lagrangian boundary value problems for symplectic maps.
AU - Offen, Christian
ID - 19940
T2 - arXiv:1811.10165
TI - Local intersections of Lagrangian manifolds correspond to catastrophe theory
ER -
TY - JOUR
AB - In this paper we continue our study of bifurcations of solutions of boundary-value problems for symplectic maps arising as Hamiltonian diffeomorphisms. These have been shown to be connected to catastrophe theory via generating functions and ordinary and reversal phase space symmetries have been considered. Here we present a convenient, coordinate free framework to analyse separated Lagrangian boundary value problems which include classical Dirichlet, Neumann and Robin boundary value problems. The framework is then used to prove the existence of obstructions arising from conformal symplectic symmetries on the bifurcation behaviour of solutions to Hamiltonian boundary value problems. Under non-degeneracy conditions, a group action by conformal symplectic symmetries has the effect that the flow map cannot degenerate in a direction which is tangential to the action. This imposes restrictions on which singularities can occur in boundary value problems. Our results generalise classical results about conjugate loci on Riemannian manifolds to a large class of Hamiltonian boundary value problems with, for example, scaling symmetries.
AU - McLachlan, Robert I
AU - Offen, Christian
ID - 19943
JF - New Zealand Journal of Mathematics
KW - Hamiltonian boundary value problems
KW - singularities
KW - conformal symplectic geometry
KW - catastrophe theory
KW - conjugate loci
TI - Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci
VL - 48
ER -