4,814 research outputs found
Oriented Associativity Equations and Symmetry Consistent Conjugate Curvilinear Coordinate Nets
This paper is devoted to description of the relationship among oriented
associativity equations, symmetry consistent conjugate curvilinear coordinate
nets, and the widest associated class of semi- Hamiltonian hydrodynamic-type
systems.Comment: 19 page
Differentiable Game Mechanics
Deep learning is built on the foundational guarantee that gradient descent on
an objective function converges to local minima. Unfortunately, this guarantee
fails in settings, such as generative adversarial nets, that exhibit multiple
interacting losses. The behavior of gradient-based methods in games is not well
understood -- and is becoming increasingly important as adversarial and
multi-objective architectures proliferate. In this paper, we develop new tools
to understand and control the dynamics in n-player differentiable games.
The key result is to decompose the game Jacobian into two components. The
first, symmetric component, is related to potential games, which reduce to
gradient descent on an implicit function. The second, antisymmetric component,
relates to Hamiltonian games, a new class of games that obey a conservation law
akin to conservation laws in classical mechanical systems. The decomposition
motivates Symplectic Gradient Adjustment (SGA), a new algorithm for finding
stable fixed points in differentiable games. Basic experiments show SGA is
competitive with recently proposed algorithms for finding stable fixed points
in GANs -- while at the same time being applicable to, and having guarantees
in, much more general cases.Comment: JMLR 2019, journal version of arXiv:1802.0564
Generalized Group Actions in a Global Setting
We study generalized group actions on differentiable manifolds in the
Colombeau framework, extending previous work on flows of generalized vector
fields and symmetry group analysis of generalized solutions. As an application,
we analyze group invariant generalized functions in this setting
Algebro-geometric approach in the theory of integrable hydrodynamic type systems
The algebro-geometric approach for integrability of semi-Hamiltonian
hydrodynamic type systems is presented. This method is significantly simplified
for so-called symmetric hydrodynamic type systems. Plenty interesting and
physically motivated examples are investigated
Darboux Transformations for SUSY Integrable Systems
Several types of Darboux transformations for supersymmetric integrable
systems such as the Manin-Radul KdV, Mathieu KdV and SUSY sine-Gordon equations
are considered. We also present solutions such as supersolitons and superkinks.Comment: 13 pages. LaTeX209 with LamuPhys and EPSF packages, 3 figures.
Contribution to the proceedings of the "Integrable Models and Supersymmetry"
meeting held at Chicago on July'9
- …