Special Issue "Lie Group Machine Learning and Lie Group Structure Preserving Integrators"
A special issue of Entropy (ISSN 1099-4300).
Deadline for manuscript submissions: closed (2 March 2020).
Interests: Lie group thermodynamics; exponential families; representation theory; Lie group statistics; Lie group machine learning
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Interests: numerical analysis; structure preserving algorithms; differential equations; geometric numerical integration
Interests: geometric methods for the modeling and numerical discretization of partial differential equations arising in fluid dynamics; nonlinear elasticity, and nonequilibrium thermodynamics
Interests: geometric mechanics; Lie group theory; acoustics; numerical sound synthesis; finite and boundary element methods
Machine/deep learning is exploring use-cases extensions for more abstract spaces such as graphs, differential manifolds, and structured data. The most recent fruitful exchanges between geometric science of information and Lie group theory have opened new perspectives to extend machine learning on Lie groups. After the Lie group’s foundation by Sophus Lie, Felix Klein, and Henri Poincaré, based on the Wilhelm Killing study of Lie algebra, Elie Cartan achieved the classification of simple real Lie algebras and introduced affine representation of Lie groups/algebras applied systematically by Jean-Louis Koszul. In parallel, the noncommutative harmonic analysis for non-Abelian groups has been addressed with the orbit method (coadjoint representation of group) with many contributors (Jacques Dixmier, Alexander Kirillov, etc.). In physics, Valentine Bargmann, Jean-Marie Souriau, and Bertram Kostant provided the basic concepts of Symplectic Geometry to Geometric Mechanics, such as the KKS symplectic form on coadjoint orbits and the notion of Momentum map associated to the action of a Lie group. Using these tools Souriau also developed the theory of Lie Group Thermodynamics based on coadjoint representations. These set of tools could be revisited in the framework of Lie group machine learning to develop new schemes for processing structured data.
Structure preserving integrators are numerical algorithms that are specifically designed to preserve the geometric properties of the flow of the differential equation, such as invariants, (multi-)symplecticity, volume preservation, as well as the configuration manifold. As a consequence, such algorithms have proven to be highly superior in correctly reproducing the global qualitative behavior of the system. Structure-preserving methods have recently undergone significant development and constitute today a privileged road in building numerical algorithms with high reliability and robustness in various areas of computational mathematics. In particular, the capability for long-term computation makes these methods particularly well adapted to deal with the new opportunities and challenges offered by scientific computations. Among the different ways to construct such numerical integrators, the application of variational principles (such as Hamilton’s variational principle and its generalizations) has appeared to be very powerful, since it is very constructive and because of its wide range of applicability.
An important specific situation encountered in a wide range of applications going from multibody dynamics to nonlinear beam dynamics and fluid mechanics is the case of ordinary and partial differential equations on Lie groups. In this case, one can additionally take advantage of the rich geometric structure of the Lie group and its Lie algebra for the construction of the integrators. Structure preserving integrators that preserve the Lie group structure have been studied from many points of view and with several extensions to a wide range of situations, including forced, controlled, constrained, nonsmooth, stochastic, or multiscale systems, in both the finite and infinite dimensional Lie group setting. They also naturally find applications in the extension of machine learning and deep learning algorithms to Lie group data.
This Special Issue will collect long versions of papers from contributions presented during the GSI'19 "Geometric Science of Information" conference (www.gsi2019.org) but will not be limited to these authors and is open to international communities involved in research on Lie groups machine learning and Lie group structure-preserving integrators.
Prof. Frédéric Barbaresco
Prof. Elena Cellodoni
Prof. François Gay-Balmaz
Prof. Joël Bensoam
Manuscript Submission Information
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- Lie groups machine learning
- orbits method
- symplectic geometry
- geometric integrator
- symplectic integrator
- Hamilton’s variational principle