We give a brief introduction to some of the recent works on finding geometric structures on triangulated surfaces using variational principles.
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48 results for “Geometric Variational Principles”
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.
Extends geometrical description of tensor manifolds in tree-based formats.
problem Geometrical description of tensor manifolds in tree-based formats.
method Provided a new geometrical description of manifolds of tensors in tree-based format.
result Geometrical description compatible with Tucker format.
Geometric derivation of Einstein equations from causal fermion systems.
problem Deriving Einstein's equations from a new theoretical framework.
method Analysis of causal fermion systems and causal action principle.
result Einstein equations derived from causal action principle.
Dirac structures on tangent bundles provide a unified framework for Lagrange--Dirac dynamical systems.
problem Unified geometric framework for Lagrange--Dirac dynamical systems
method Introducing a Lagrange--Dirac structure on the tangent bundle
result Unified framework for nonholonomic, degenerate Lagrangian, and symmetric systems
Investigates invariant hulls of functionals on manifolds.
problem Understanding meaningful functionals on manifolds through reparameterizations.
method Uses inner-variations to transform arbitrary functionals into invariant realizations.
result Explicit computations for volume functional in -dimensional manifolds, especially in .
Proposes variational Wasserstein barycenters for geometric clustering.
problem Geometric clustering problems, especially K-means and co-clustering.
method Solves for Monge maps using variational principle, explores connections to K-means and co-clustering.
result Demonstrates feasibility and use of variational Wasserstein barycenters in clustering.
Optimal tracking of nonholonomic systems using geometric methods.
problem Tracking a trajectory for nonholonomic mechanical systems.
method Geometric optimal control, Pontryagin Maximum Principle, variational approach.
result Optimal control solutions for nonholonomic systems validated by examples and simulations.
Geometric framework for dissipative systems on Lie algebroids.
problem Formulating geometric framework for dissipative systems.
method Herglotz-type variational principle on Lie algebroids.
result Recover classical equations as special cases.
We use neural networks as control variates with geometric integration techniques.
problem Analytic integration of neural network approximations for variance reduction.
method Integration domain subdivision using computational geometry for MLPs with continuous piecewise linear activation functions.
result Neural networks can be used as control variates with geometric integration methods.
Reductions of higher tangent bundles of Lie groupoids provide natural examples of geometric structures which we would like to call higher algebroids. Such objects can be also constructed abstractly starting from an arbitrary almost Lie algebroid. A higher algebroid is, in principle, a graded bundle equipped with a diff…
Derives new optimization methods using variational integrators.
problem Optimization methods in machine learning.
method Variational integrators and principles of Hamilton and Lagrange-d'Alembert.
result Derives two families of optimization methods, including Nesterov's accelerated gradient method.
Locally maximizing orbits studied in twist maps and billiards.
problem Characterize orbits in locally maximizing class for twist maps.
method Geometric and variational analysis of orbits in the cotangent bundle of a torus or ball bundle over a sphere.
result Two generating functions for the Birkhoff billiard map have the same class of locally maximizing orbits.
Develops an analytic theory for quantum imaginary time evolution.
problem Lack of a first-principle understanding of quantum imaginary time evolution.
method Interprets QITE as a form of VQA trained with QNGD and connects it to the geometric geodesic distance in the quantum Fisher information metric.
result QITE converges faster than vanilla gradient descent-based VQAs, though the advantage is suppressed by Hilbert space dimensionality.
Finite energy pluripotential theory accommodates the variational theory of equations of complex Monge-Ampère type arising in Kähler geometry. Recently it has been discovered that many of the potential spaces involved have a rich metric geometry, effectively turning the variational problems in question into problems of …
Log-Aesthetic Curves: Similarity Geometry, Integrable Discretization and Variational Principlesnlin.SI
In this paper, we consider a class of plane curves called log-aesthetic curves and their generalization which are used in computer aided geometric design. We consider these curves in the framework of the similarity geometry and characterize them as invariant curves under the integrable flow on plane curves which is gov…
Unified approach classifies stable and minimal elastic curves.
problem Classifying stable and minimal elastic curves under various conditions.
method Unified geometric approach using a `cut-and-paste` trick.
result Complete classification of stable closed -elasticae and stable pinned -elasticae.
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…
A coordinate-free proof of the Maximum Principle is provided in the specific case of an optimal control problem with fixed time. Our treatment heavily relies on a special notion of variation of curves that consist of a concatenation of integral curves of time-dependent vector fields with unit time component, and on the…
Hasse principle applied to area-minimizing submanifolds across different homology types.
problem Understanding the behavior of area-minimizing submanifolds in various homology contexts.
method Extending the Hasse principle from number theory to geometric variational problems.
result Recovering information about area-minimizing submanifolds in integral homology from those in real and mod homology.
On the role of geometry in statistical mechanics and thermodynamics II: Thermodynamic perspectivecond-mat.stat-mech
Rate GENERIC extends thermodynamics principles to non-equilibrium systems.
problem Understanding non-equilibrium thermodynamics and its relation to equilibrium thermodynamics.
method Developed a geometrical framework for rate GENERIC, extending Onsager's variational principle.
result Rate GENERIC structure provides a new perspective on thermodynamics in non-equilibrium systems.
This paper studies the combinatorial Yamabe flow on hyperbolic surfaces with boundary. It is proved by applying a variational principle that the length of boundary components is uniquely determined by the combinatorial conformal factor. The combinatorial Yamabe flow is a gradient flow of a concave function. The long ti…
We introduce a new geometric evolution equation for hypersurfaces in asymptotically flat spacetime initial data sets, that unites the theory of marginally outer trapped surfaces (MOTS) with the study of inverse mean curvature flow in asymptotically flat Riemannian manifolds. A theory of weak solutions is developed usin…
New variational principle found for non-variational differential equations.
problem Non-variational differential equations without variational multipliers.
method Connecting functional forms with antiexact differential forms to identify obstructions.
result Formulation of variational problem for non-variational equations.
This paper tackles continuous domain generalization, improving model performance across unseen domains.
problem Existing domain generalization approaches fail to capture the complex, multidimensional nature of real-world variation.
method Introduces Continuous Domain Generalization (CDG), a principled framework grounded in geometric and algebraic theories. Proposes a Neural Lie Transport Operator (NeuralLio) for structure-preserving parameter transitions and a gating mechanism for robust generalization.
result Demonstrates significant improvement in generalization accuracy and robustness across various datasets.
Geometry of Mechanicsmath-ph
Studies geometric mechanics for autonomous and nonautonomous systems.
problem Understanding the geometric basis of mechanics.
method Geometric descriptions, Lagrangian, Hamiltonian, unified formalisms, symmetries, variational principles.
result Characterization of dynamical systems' properties and characteristics.
Internal Lagrangians derived from variational principles.
problem Reproducing the principle of stationary action in variational geometry.
method Introducing stationary points of internal Lagrangians, establishing connections with symmetries and conservation laws, and investigating relations between non-degenerate and internal Lagrangians.
result Noether's theorem reformulated in terms of internal Lagrangians.
In this paper we are investigating variational homogeneous second order differential equations by considering the questions of how many different variational principles exist for a given spray. We focus our attention on h(2)-variationality; that is, the regular Lagrange function is homogeneous of degree two in the dire…
New variational principles found for conformal geodesics.
problem Challenges in Lagrangian formulation for conformal geodesics.
method Enlarging the class of variations leads to a variational formulation with a third-order conformally invariant Lagrangian.
result Some integral curves of the fourth-order ODE system are spirals.
This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle. In parti…
New mass inequalities and proofs for causal variational principles.
problem Proving new mass inequalities for causal variational principles.
method Proved a new inequality for minimizers of causal variational principles and applied it to prove the positive mass theorem.
result Introduced a positive quasilocal mass and proved new mass inequalities.
Variational reduction simplifies Lagrangian systems with scaling symmetries.
problem Simplifying Lagrangian systems with scaling symmetries.
method Defining a variational reduction procedure for homogenous Lagrangian systems.
result Reconstructing trajectories from critical points of reduced variational principle.
Differential geometric mechanisms in Ostrohrads'kyj relativistic spherical top dynamicsphysics.class-ph
Some intrinsic tools from the formal theory of variational equations are being demonstrated at work in application to one concrete example of the third-order evolution equation of free relativistic top in three-dimensional space-time. The main goal is to introduce a combined approach consisting in the simultaneous util…
Introduces a restricted Chen-Nagano variational principle for the Einstein-Hilbert functional.
problem Deriving critical metrics for the Einstein-Hilbert functional on compact Riemannian manifolds.
method Restricts the variational problem to an infinite-dimensional subspace.
result Derives a novel structural characterization of critical metrics.
On the dimension of limit sets on via stationary measures: variational principles and applicationsmath.DS
The paper studies the dimension of limit sets using variational principles and stationary measures.
problem Calculating the Hausdorff dimension of limit sets of Anosov representations and the Rauzy gasket.
method Established variational principles for affinity exponents and Rauzy gaskets, combined with dimension formulas of stationary measures.
result Yields the equality between the Hausdorff dimensions and affinity exponents in both settings.
The paper proves rigidity of bordered polyhedral surfaces using variational principles.
problem Determining the rigidity of bordered polyhedral surfaces.
method Using the variational principle, the paper shows that bordered polyhedral surfaces are determined by boundary values and discrete curvatures on interior edges.
result The paper re-proves the classical result that two Euclidean or hyperbolic cyclic polygons are congruent if their side lengths are equal.
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…
This report provides an in-depth overview over the implications and novelty Generalized Variational Inference (GVI) (Knoblauch et al., 2019) brings to Deep Gaussian Processes (DGPs) (Damianou & Lawrence, 2013). Specifically, robustness to model misspecification as well as principled alternatives for uncertainty quantif…
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.
problem The variationality of conformal geodesics in higher dimensions.
method Analysis of conformal geodesics in three and higher dimensions.
result Variationality fails in both parametrized and un-parametrized conformal geodesics 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.
problem Learning disentangled representations under VAE framework remains unknown.
method Proposes PRI-VAE, a novel learning objective to optimize disentanglement.
result Demonstrates effectiveness of PRI-VAE on four benchmark datasets.
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…
Population Annealing as a Discrete-Time Schrödinger Bridgecond-mat.stat-mech
PA reinterpreted as SB problem, unifying thermodynamics and optimal transport.
problem Optimizing discrete-time paths between probability distributions.
method Schrödinger Bridge theory, solving variational problems.
result PA's reweighting step derived from Schrödinger system.
Some model reduction techniques for multiple time-scale dynamical systems make use of the identification of low dimensional slow invariant attracting manifolds (SIAM) in order to reduce the dimensionality of the phase space by restriction to the slow flow. The focus of this work is on a proposition and discussion of a …
Geodesics become an essential element of the geometry of a semi-Riemannian manifold. In fact, their differences and similarities with the (positive definite) Riemannian case, constitute the first step to understand semi-Riemannian Geometry. The progress in the last two decades has become impressive, being especially re…
Paper variates Navier-Stokes-Fourier system for thermodynamic consistency.
problem Modeling compressible fluid dynamics with thermodynamic constraints.
method Variational discretization with discrete exterior calculus.
result Derives a nonholonomic variational integrator for NSF system.