Related papers: Keep the Momentum: Conservation Laws beyond Euclid…
The Ricci flow is a partial differential equation for evolving the metric in a Riemannian manifold to make it more regular. On the other hand, neural networks seem to have similar geometric behavior for specific tasks. In this paper, we…
Understanding the dynamics of neural network parameters during training is one of the key challenges in building a theoretical foundation for deep learning. A central obstacle is that the motion of a network in high-dimensional parameter…
Physical phenomena in the real world are often described by energy-based modeling theories, such as Hamiltonian mechanics or the Landau theory, which yield various physical laws. Recent developments in neural networks have enabled the…
The existing paradox between theory and computational experiment for weak solutions of systems of conservation laws in higher space dimensions is arguably resolved. Apparently successful computations are identified with underlying…
In recent work we constructed completely general conservation laws for energy and linear and angular momentum of extended systems in general relativity based on the notion of a rigid quasilocal frame (RQF). We argued at a fundamental level…
We demonstrate, both at the Newtonian and (general) relativistic levels, the existence of a generalization of Kelvin's circulation theorem (for pure fluids) which is applicable to perfect magnetohydrodynamics. The argument is based on the…
We consider two scalar conservation laws with non-local flux functions, describing traffic flow on roads with rough conditions. In the first model, the velocity of the car depends on an averaged downstream density, while in the second model…
Conservation laws that are based on the divergence of a second-rank four-tensor are fundamentally different from conservation laws based on the divergence of a four-vector. This article investigates the consequences of this difference for…
Graph Neural Networks (GNNs) have become essential for learning from graph-structured data. However, existing GNNs do not consider the conservation law inherent in graphs associated with a flow of physical resources, such as electrical…
In the works by the author it has been shown that the conservation laws for material media (the conservation laws for energy, linear momentum, angular momentum, and mass, that establish a balance between the variation of a physical quantity…
The training of neural networks by gradient descent methods is a cornerstone of the deep learning revolution. Yet, despite some recent progress, a complete theory explaining its success is still missing. This article presents, for…
Graph neural networks (GNNs) have emerged as a versatile and efficient option for modeling the dynamic behavior of deformable materials. While GNNs generalize readily to arbitrary shapes, mesh topologies, and material parameters, existing…
Conservation laws of a class of time-dependent damped nonlinear multidimensional wave equations are derived by Noether's theorem. For arbitrary nonzero damping coefficient and nonlinear interaction term, its infinitesimal variational…
Motivated by a geometric method employed for the derivation of the Nambu bracket for ideal two-dimensional incompressible hydrodynamics, we reconstruct the reduced magnetohydrodynamic (RMHD) model by a priori imposition of its conservation…
Historically it happen so that in branches of physics connected with field theory and of physics of material systems (continuous media) the concept of "conservation laws" has a different meaning. In field theory "conservation laws" are…
Diffusion with multipole-moment conservation gives rise to transport laws that generalize Fick's law and has attracted growing attention following experimental advances in strongly tilted optical lattices. It was recently shown that…
We consider the Euler equations of incompressible inviscid fluid dynamics. We discuss a variational formulation of the governing equations in Lagrangian coordinates. We compute variational symmetries of the action functional and generate…
Two-dimensional gas dynamics equations in mass Lagrangian coordinates are studied in this paper. The equations describing these flows are reduced to two Euler-Lagrange equations. Using group classification and Noether's theorem,…
We investigate a class of nonlocal conservation laws with the nonlinear advection coupling both local and nonlocal mechanism, which arises in several applications such as the collective motion of cells and traffic flows. It is proved that…
Particles in turbulence live complicated lives. It is nonetheless sometimes possible to find order in this complexity. It was proposed in [Falkovich et al., Phys. Rev. Lett. 110, 214502 (2013)] that pairs of Lagrangian tracers at small…