331 research outputs found
Learning Integrable Dynamics with Action-Angle Networks
Machine learning has become increasingly popular for efficiently modelling
the dynamics of complex physical systems, demonstrating a capability to learn
effective models for dynamics which ignore redundant degrees of freedom.
Learned simulators typically predict the evolution of the system in a
step-by-step manner with numerical integration techniques. However, such models
often suffer from instability over long roll-outs due to the accumulation of
both estimation and integration error at each prediction step. Here, we propose
an alternative construction for learned physical simulators that are inspired
by the concept of action-angle coordinates from classical mechanics for
describing integrable systems. We propose Action-Angle Networks, which learn a
nonlinear transformation from input coordinates to the action-angle space,
where evolution of the system is linear. Unlike traditional learned simulators,
Action-Angle Networks do not employ any higher-order numerical integration
methods, making them extremely efficient at modelling the dynamics of
integrable physical systems.Comment: Accepted at Machine Learning and the Physical Sciences workshop at
NeurIPS 202
Spatial Hamiltonian identities for nonlocally coupled systems
We consider a broad class of systems of nonlinear integro-differential
equations posed on the real line that arise as Euler-Lagrange equations to
energies involving nonlinear nonlocal interactions. Although these equations
are not readily cast as dynamical systems, we develop a calculus that yields a
natural Hamiltonian formalism. In particular, we formulate Noether's theorem in
this context, identify a degenerate symplectic structure, and derive
Hamiltonian differential equations on finite-dimensional center manifolds when
those exist. Our formalism yields new natural conserved quantities. For
Euler-Lagrange equations arising as traveling-wave equations in gradient flows,
we identify Lyapunov functions. We provide several applications to
pattern-forming systems including neural field and phase separation problems.Comment: 39 pages, 1 figur
An introduction to Lie group integrators -- basics, new developments and applications
We give a short and elementary introduction to Lie group methods. A selection
of applications of Lie group integrators are discussed. Finally, a family of
symplectic integrators on cotangent bundles of Lie groups is presented and the
notion of discrete gradient methods is generalised to Lie groups
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