arXiv research

New variational principle found for non-variational differential equations.

Internal Lagrangians derived from variational principles.

New variational principles found for conformal geodesics.

We give a brief introduction to some of the recent works on finding geometric structures on triangulated surfaces using variational principles.

New mass inequalities and proofs for causal variational principles.

Variational reduction simplifies Lagrangian systems with scaling symmetries.

Introduces a restricted Chen-Nagano variational principle for the Einstein-Hilbert functional.

The paper studies the dimension of limit sets using variational principles and stationary measures.

The paper proves rigidity of bordered polyhedral surfaces using variational principles.

We provide a variational description of any Liouville (i.e. volume preserving) autonomous vector fields on a smooth manifold. This is obtained via a ``maximal degree'' variational principle; critical sections for this are integral manifolds for the Liouville vector field. We work in coordinates and provide explicit for…

We discuss a recently proposed variational principle for deriving the variational equations associated to any Lagrangian system. The principle gives simultaneously the Lagrange and the variational equations of the system. We define a new Lagrangian in an extended configuration space ---which we call D'Alambert's--- com…

Variationality of conformal geodesics fails in higher dimensions.

In this paper, we provide an information-theoretic interpretation of the Vector Quantized-Variational Autoencoder (VQ-VAE). We show that the loss function of the original VQ-VAE can be derived from the variational deterministic information bottleneck (VDIB) principle. On the other hand, the VQ-VAE trained by the Expect…

PRI-VAE learns disentangled representations by optimizing principle-of-relevant-information.

We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature.

A variational principle is proposed for obtaining the Jacobi equations in systems admitting a Lagrangian description. The variational principle gives simultaneously the Lagrange equations of motion and the Jacobi variational equations for the system. The approach can be of help in finding constants of motion in the Jac…

Paper variates Navier-Stokes-Fourier system for thermodynamic consistency.

The paper explores variational principles for equations of maximal symmetry, providing new insights and results.

We prove that under certain assumptions a partial differential equation can be derived from a variational principle. It is well-known from Noether's theorem that symmetries of a variational functional lead to conservation laws of the corresponding Euler-Lagrange equation. We reverse this statement and prove that a diff…

The variational principle and the corresponding differential equation for geodesic circles in two dimensional (pseudo)-Riemannian space are being discovered. The relationship with the physical notion of uniformly accelerated relativistic particle is emphasized. The known form of spin-curvature interaction emerges due t…

The calculus of variations for lagrangians which are not functions on the tangent bundle, but sections certain affine bundles is developed. We follow a general approach to variational principles which admits boundary terms of variations.

Variational autoencoders provide a principled framework for learning deep latent-variable models and corresponding inference models. In this work, we provide an introduction to variational autoencoders and some important extensions.

In this paper, we develop several related finite dimensional variational principles for discrete optimal transport (DOT), Minkowski type problems for convex polytopes and discrete Monge-Ampere equation (DMAE). A link between the discrete optimal transport, discrete Monge-Ampere equation and the power diagram in computa…

New approach to Lagrangian systems using intrinsic geometry.

Dirac structures on tangent bundles provide a unified framework for Lagrange--Dirac dynamical systems.

Based on conservation laws for surface layer integrals for critical points of causal variational principles, it is shown how jet spaces can be endowed with an almost-complex structure. We analyze under which conditions the almost-complex structure can be integrated to a canonical complex structure. Combined with the sc…

Study nonholonomic systems with collisions using variational principles.

A variational inequality for pricing the perpetual American option and the corresponding difference equation are considered. First, the maximum principle and uniqueness of the solution to variational inequality for pricing the perpetual American option are proved. Then the maximum principle, the existence and uniquenes…

We present a variational formulation of electrodynamics using de Rham even and odd differential forms. Our formulation relies on a variational principle more complete than the Hamilton principle and thus leads to field equations with external sources and permits the derivation of the constitutive relations. We interpre…

Let $M$ be smooth $n$-dimensional manifold, fibered over a $k$-dimensional submanifold $B$ as $π:M \to B$, and $\vartheta \in Λ^k (M)$; one can consider the functional on sections $φ$ of the bundle $π$ defined by $\int_D φ^* (\vartheta)$, with $D$ a domain in $B$. We show that for $k = n-2$ the variational principle ba…

The geometry of jets of submanifolds is studied, with special interest in the relationship with the calculus of variations. A new intrinsic geometric formulation of the variational problem on jets of submanifolds is given. Working examples are provided.

By resorting to Noether's Second Theorem, we relate the generalized Bianchi identities for Lagrangian field theories on gauge-natural bundles with the kernel of the associated gauge-natural Jacobi morphism. A suitable definition of the curvature of gauge-natural variational principles can be consequently formulated in …

New method prevents forgetting in learning new tasks.

New method uses adiabatic principles to improve ground-state preparation in quantum computing.

The paper explores variational problems on Riemannian manifolds with special foliations, proving existence results.

Unified framework for planning under uncertainty using variational inference.

Derives new optimization methods using variational integrators.

Paper explores SVGD for Bayesian inference, linking deterministic and stochastic dynamics.

We study rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvature like quantities for polyhedral surfaces are introduced. Many of them are shown to determine the polyhedral metric up to isometry. The action functionals in the variational approaches are derived …

Extends geometrical description of tensor manifolds in tree-based formats.

The main goal of this paper is to extend the so-called Dirac-Frenkel Variational Principle in the framework of tensor Banach spaces. To this end we observe that a tensor product of normed spaces can be described as a union of disjoint connected components. Then we show that each of these connected components, composed …

A Delaunay cell decomposition of a surface with constant curvature gives rise to a circle pattern, consisting of the circles which are circumscribed to the facets. We treat the problem whether there exists a Delaunay cell decomposition for a given (topological) cell decomposition and given intersection angles of the ci…

Framework identifies causal direction from single data setting.

Survey of Fermat principle in general relativity and beyond.

The exactness equation for Lepage 2-forms, associated with variational systems of ordinary differential equations on smooth manifolds, is analyzed with the aim to construct a concrete global variational principle. It is shown that locally variational systems defined by homogeneous functions of degree $c \neq 0, 1$ are …

A complete error analysis of variational integrators is obtained, by blowing up the discrete variational principles, all of which have a singularity at zero time-step. Divisions by the time step lead to an order that is one less than observed in simulations, a deficit that is repaired with the help of a new past-future…

VES-Gamma adapts EI using information-theoretic principles.

Geometric derivation of Einstein equations from causal fermion systems.