Graph neural networks are explained through energy gradient flow and framelet decomposition.
problem Understanding and improving graph neural networks.
method Viewing framelet-based models as gradient flows of energy, proposing a generalized energy via framelet decomposition.
result The proposed model leads to more flexible dynamics, enhancing graph neural networks.
Gradient flow of elastic energy converges to elastica.
problem Optimizing closed curves to minimize elastic energy.
method Proving the existence of a unique global solution and convergence via Łojasiewicz--Simon gradient inequality.
result Convergence to elastica established for the H2(ds)-gradient flow of modified elastic energy. Gradient flows for surface energies with tensor fields are derived and analyzed.
problem Deriving consistent gradient flows for surface energies involving tensor fields.
method Introducing different gauges of surface independence and demonstrating their effects on energy decrease.
result Consistent choice of gauge and time derivative is necessary for energy decrease.
Gradient flows for knot energies ensure long-term existence of knotted loops.
problem Ensuring long-term existence of knotted loops under various energies.
method Banach gradient flows, curves of maximal slope, logarithmic strain control.
result Established long-time existence of gradient flows for knot energies.
A new parametric method studies Willmore flows and energy quantization.
problem Understanding Willmore flows and their singularities.
method Parametric approach to Willmore gradient flows.
result For small-energy weak immersions, a unique solution exists.
Study curves evolving by gradient flow of elastic energy, proving existence, smoothing, and convergence.
problem Evolution of curves with fixed length and clamped boundary conditions.
method Negative L2-gradient flow of elastic energy, existence, parabolic smoothing, constrained Lojasiewicz-Simon gradient inequality. result Convergence to a critical point as time tends to infinity.
Gradient flow studies Spin(7)-structures on compact 8-manifolds.
problem Formulating and studying the gradient flow of Spin(7)-structures.
method Negative gradient flow of an energy functional of Spin(7)-structures.
result Short-time existence and uniqueness of solutions to the flow.
The paper proves a reverse isoperimetric inequality and applies it to analyze surface flows.
problem Analyzing the negative gradient flow of the Willmore energy plus volume.
method Proved a quantitative reverse isoperimetric inequality and applied it to the flow.
result Initial surfaces converge to a round point in finite or infinite time.
Gradient flow method solves isoperimetric inequality for maps.
problem Finding maps with optimal enclosed area.
method Sobolev gradient flow for area-normalised Dirichlet energy.
result Solutions converge to a circle as time goes to infinity.
Study gradient flow of phase transitions with fixed contact angle.
problem Understanding phase transitions with fixed contact angle.
method Gradient flow of the Allen-Cahn equation with fixed boundary contact angle.
result Established interior and boundary convergence properties for solutions and energy measures.
Morse theory connects low energy submanifolds in 3-sphere.
problem Understanding low energy submanifolds in the 3-sphere.
method Morse-theoretic techniques and negative gradient flow.
result Constructs connections between low energy critical submanifolds.
Li-Yau inequality applied to curves in 2D space.
problem Curves in 2D space with low elastic energy.
method Classical Li-Yau inequality applied to curves.
result Analogous results for curves in 2D space with low elastic energy.
The article analyzes the stability of a curve shortening flow for planar networks.
problem Stability analysis of anisotropic curve shortening flow for planar networks.
method Used Lojasiewicz-Simon gradient inequality to derive stability results.
result For initial data close to an energy minimizer, the flow exists globally and converges to a different energy minimum.
We investigate the low-energy behavior of the gradient flow of the L2 norm of the Riemannian curvature on four-manifolds. Specifically, we show long time existence and exponential convergence to a metric of constant sectional curvature when the initial metric has positive Yamabe constant and small initial energy.
In this monograph, we develop results on global existence and convergence of solutions to abstract gradient flows on Banach spaces for a potential function that obeys the Lojasiewicz-Simon gradient inequality. We prove a Lojasiewicz-Simon gradient inequality for the Yang-Mills energy functional over closed, smooth Riem…
Study of geometric analysis on asymmetric metric spaces, including heat flow and Sobolev spaces.
problem Analysis of geometric properties on asymmetric metric measure spaces.
method Introduction of upper gradients, q-Laplacian, and q-heat flow in asymmetric settings. result Extension of concepts from symmetric to asymmetric metric measure spaces.
We investigate the gradient flow of the L2 norm of the Riemannian curvature on surfaces. We show long time existence with arbitrary initial data, and exponential convergence of the volume normalized flow to a constant scalar curvature metric when the initial energy is below a constant determined by the Euler charact…
This work develops a particle system to approximate Fisher-Rao gradient flows in mean-field optimization.
problem Optimizing probability measures in neural network contexts.
method Constructing an interacting particle system approximating Fisher-Rao gradient flows.
result Propagation of chaos for the Fisher-Rao gradient flow in entropic mean-field optimization.
On the universal bundle of unit spinors we study a natural energy functional whose critical points, if dim M \geq 3, are precisely the pairs (g, φ) consisting of a Ricci-flat Riemannian metric g together with a parallel g-spinor φ. We investigate the basic properties of this functional and study its negative gradient f…
Study on dynamic curves with elastic energy and spontaneous curvature.
problem Modeling and analyzing dynamic planar curves with elastic energy.
method Gradient flow of inclination angle, nonlocal quasilinear system, local well-posedness, global existence, convergence.
result Local well-posedness, global existence, convergence of the flow for weak regularity initial data.
Study sesqui-harmonic map flow from Riemannian surfaces
problem Investigate sesqui-harmonic map flow from Riemannian surfaces
method L2-gradient flow of an energy functional
result Generalizes Struwe's regularity result for harmonic maps
Paper proposes CoopFlow, a two-flow generator for energy-based models.
problem Training energy-based models with Langevin flow and normalizing flow.
method CoopFlow trains an energy-based model using a normalizing flow initialization and a short-run Langevin flow revision.
result CoopFlow converges to a moment matching estimator and synthesizes realistic images.
New method learns discrete graph diffusion via free-energy gradient flows.
problem Challenges in translating continuous diffusion models to discrete spaces.
method Proposes a novel computational approach using a specific metric on the simplex.
result Recover the underlying functional for various graph classes.
We analyze a gradient flow of closed planar curves minimizing the anisoperimetric ratio. For such a flow the normal velocity is a function of the anisotropic curvature and it also depends on the total interfacial energy and enclosed area of the curve. In contrast to the gradient flow for the isoperimetric ratio, we sho…
Gradient flow on diffeomorphisms for image registration, with well-posedness proven.
problem Image registration with metric tensor deformation penalization.
method Gradient flow on Sobolev diffeomorphisms for a specific energy functional.
result Well-posedness of the gradient flow established.
Scheme minimizes p-elastic energy of curves over time.
problem Minimizing p-elastic energy of curves over time. method Minimizing movement scheme with approximate normal graphs.
result Short-time existence and lower bound on solution's lifetime.
The elastic flow of curves converges smoothly to a critical point.
problem Smooth convergence of elastic flow of curves.
method Application of Lojasiewicz-Simon inequality.
result Smooth convergence to a critical point.
JKO scheme adds deceleration in rapidly changing metric curvature directions.
problem Understanding the implicit bias of the JKO scheme in Wasserstein gradient flow.
method Characterized the implicit bias of the JKO scheme at second order in η, modifying the energy functional.
result JKO scheme adds deceleration in directions where metric curvature of J is rapidly changing.
Study of tori of revolution under Willmore flow converges to Clifford Torus.
problem Long-time behavior and convergence of Willmore flow for tori of revolution.
method Gradient flow of Willmore energy for tori of revolution, analyzing energy threshold and convergence to Clifford Torus.
result Convergence of Willmore flow to Clifford Torus for initial energy below 8π.
Flow preserves isoperimetric ratio for immersed surfaces.
problem Preserving isoperimetric ratio in Willmore flow.
method Non-local L2-gradient flow for Willmore energy. result Long-time existence and convergence for spherical initial data.
In this paper we develop a Morse theory for the uniform energy. We use the one-sided directional derivative of the distance function to study the minimizing properties of variations through closed geodesics. This derivative is then used to define a one-sided directional derivative for the uniform energy which allows us…
We show that the concept of H2-gradient flow for the Willmore energy and other functionals that depend at most quadratically on the second fundamental form is well-defined in the space of immersions of Sobolev class W2,p from a compact, n-dimensional manifold into Euclidean space, provided that p≥2 and…
We study the evolution of closed inextensible planar curves under a second order flow that decreases the p-elastic energy. A short time existence result for p∈(1,∞) is obtained via a minimizing movements method. For p=2, that is in the case of the classic elastic energy, long-time existence is retrieve…
The paper analyzes flows related to Higgs energies on manifolds.
problem Analyzing flows related to Higgs energies on manifolds.
method Developing asymptotic analysis for gradient flow of self-dual U(1)-Higgs energies. result Solutions converge to codimension-two mean curvature flows.
Entropy measures geodesic flow complexity.
problem Measuring complexity of geodesic flows on manifolds.
method Introduced barcode entropy to measure exponential growth rate of not-too-short bars in Morse-theoretic barcodes.
result Barcode entropy bounds topological entropy and vice versa.
New spinorial functional connects Perelman's W- and F-functionals.
problem Unifying Perelman's functionals for spin manifolds.
method Introduced a new energy functional on spin manifolds, computed its first variation, and established a gradient flow.
result Critical points of the functional are twisted Ricci solitons and eigen-spinsors.
We study the negative gradient flow of the spinorial energy functional (introduced by Ammann, Weiß, and Witt) on 3-dimensional Berger spheres. For a certain class of spinors we show that the Berger spheres collapse to a 2-dimensional sphere. Moreover, for special cases, we prove that the volume-normalized standard 3-sp…
Forward-Euler fails for simulating Wasserstein gradient flows with KL divergence.
problem Simulating Wasserstein gradient flows with forward-Euler discretization fails for KL divergence.
method Forward-Euler discretization for Wasserstein gradient flows with KL divergence.
result Forward-Euler discretization can be incorrect for Wasserstein gradient flows with KL divergence.
This paper explores gradient flows for sampling distributions without normalization constants.
problem Sampling from distributions with unknown normalization constants.
method Gradient flows in the space of probability measures, focusing on Kullback-Leibler divergence, Fisher-Rao metric, and affine invariance.
result Gradient flows derived from Kullback-Leibler divergence do not depend on the normalization constant.
Study eternal solutions to Allen-Cahn equation on 3-sphere, connecting Clifford tori to equatorial spheres.
problem Understanding eternal solutions to the Allen-Cahn equation on the 3-sphere.
method Realization of Brakke's motion by mean curvature as a singular limit of Allen-Cahn gradient flows, using classifications and rigidity results.
result Construction of eternal integral Brakke flows connecting Clifford tori to equatorial spheres.
The paper defines and analyzes higher-order Yang-Mills-Higgs functionals and their gradient flows.
problem Analyzing the behavior of higher-order Yang-Mills-Higgs functionals and their gradient flows.
method Gauge fixing technique, L2-bound of the Higgs field, local L2-derivative estimates, energy estimates, blow-up analysis. result Solutions to the gradient flow do not hit finite time singularities under certain conditions.
The paper connects Ricci flow and harmonic spinors, proving new inequalities.
problem Understanding the behavior of harmonic spinors under Ricci flow.
method Introduced a weighted monopole equations and used Perelman's entropy.
result Ricci flow is the gradient flow of energy related to harmonic spinors.
For triangulated surfaces locally embedded in the standard hyperbolic space, we introduce combinatorial Calabi flow as the negative gradient flow of combinatorial Calabi energy. We prove that the flow produces solutions which converge to ZCCP-metric (zero curvature circle packing metric) if the initial energy is small …
We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the L2-space produces the same evolution as the gradient flow of the relative entropy in the L2-Wasserstein space. This means that the heat flow is well defined by either one of the two gradient fl…
Gradient flow in a potential energy (or Euclidean action) landscape provides a natural set of paths connecting different saddle points. We apply this method to General Relativity, where gradient flow is Ricci flow, and focus on the example of 4-dimensional Euclidean gravity with boundary S^1 x S^2, representing the can…
The J-flow is a parabolic flow on Kahler manifolds. It was defined by Donaldson in the setting of moment maps and by Chen as the gradient flow of the J-functional appearing in his formula for the Mabuchi energy. It is shown here that under a certain condition on the initial data, the J-flow converges to a critical metr…
Straight lines are a basin of attraction for the elastic flow at least to level 1.9615π.
problem Understanding the basin of attraction for the free boundary free elastic flow.
method Steepest descent gradient flow for elastic energy, numerical evidence.
result Straight lines have a basin of attraction at least to level 1.9615π.
This paper studies gradient flows for sampling using various metrics and their affine invariance.
problem Sampling from probability distributions with unknown normalizations.
method Gradient flows in the space of probability measures, focusing on Kullback-Leibler divergence and affine invariance of metrics.
result Gradient flows of Kullback-Leibler divergence do not depend on the normalization constant, and affine invariance is achieved for certain metrics.