Related papers: Shape flows for spectral optimization problems
We study topology optimization governed by the incompressible Navier-Stokes flows using a phase field model. Novel stabilized semi-implicit schemes for the gradient flows of Allen-Cahn and Cahn-Hilliard types are proposed for solving the…
Evolutionary PDEs for geometric order parameters that admit propagating singular solutions are introduced and discussed. These singular solutions arise as a result of the competition between nonlinear and nonlocal processes in various…
The latent space of normalizing flows must be of the same dimensionality as their output space. This constraint presents a problem if we want to learn low-dimensional, semantically meaningful representations. Recent work has provided…
We focus on a family of nonlinear continuity equations for the evolution of a non-negative density $\rho$ with a continuous and compactly supported nonlinear mobility $\mathrm{m}(\rho)$ not necessarily concave. The velocity field is the…
Optimization under structural constraints is typically analyzed through projection or penalty methods, obscuring the geometric mechanism by which constraints shape admissible dynamics. We propose an operator-theoretic formulation in which…
A freely falling stream of weakly cohesive granular particles is modeled and analysed with help of event driven simulations and continuum hydrodynamics. The former show a breakup of the stream into droplets, whose size is measured as a…
A relaxed notion of displacement convexity is defined and used to establish short time existence and uniqueness of Wasserstein gradient flows for higher order energy functionals. As an application, local and global well-posedness of…
We prove that the standard gradient flow in parameter space that underlies many training algorithms in deep learning can be continuously deformed into an adapted gradient flow which yields (constrained) Euclidean gradient flow in output…
Despite the widespread use of gradient-based algorithms for optimizing high-dimensional non-convex functions, understanding their ability of finding good minima instead of being trapped in spurious ones remains to a large extent an open…
We study the equation of one-dimensional quasistatic nonlinear viscoelasticity with Dirichlet boundary conditions, in the particular case that the underlying dissipation geometry (provided by the viscosity) is comparable to the Bhattacharya…
We derive pointwise curvature estimates for graphical mean curvature flows in higher codimensions. To the best of our knowledge, this is the first such estimates without assuming smallness of first derivatives of the defining map. An…
This master thesis looks at the gradient flow of the length functional on embedded loops. The space of embedded loops is endowed with a scale structure so that the length functional becomes scale smooth. For certain underlying manifolds,…
Applications of normalizing flows to the sampling of field configurations in lattice gauge theory have so far been explored almost exclusively in two space-time dimensions. We report new algorithmic developments of gauge-equivariant flow…
In this paper, we consider the following general evolution equation $$ u_t=\Delta_fu+au\log^\alpha u+bu $$ on smooth metric measure spaces $(M^n, g, e^{-f}dv)$. We give a local gradient estimate of Souplet-Zhang type for positive smooth…
In this article we consider the length functional defined on the space of immersed planar curves. The $L^2(ds)$ Riemannian metric gives rise to the curve shortening flow as the gradient flow of the length functional. Motivated by the…
We investigate the gradient flow of the $L^2$ 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…
This paper exploits the theory of geometric gradient flows to introduce an alternative regularization of the thin-film equation. The solution properties of this regularization are investigated via a sequence of numerical simulations whose…
This paper is concerned with the optimal shape design of the newtonian viscous incompressible fluids driven by the stationary nonhomogeneous Navier-Stokes equations. We use three approaches to derive the structures of shape gradients for…
In this work, we present a new approach to analyze the gradient flow for a positive semi-definite matrix denoising problem in an extensive-rank and high-dimensional regime. We use recent linear pencil techniques of random matrix theory to…
Accurate polyp segmentation remains challenging due to irregular lesion morphologies, ambiguous boundaries, and heterogeneous imaging conditions. While U-Net variants excel at local feature fusion, they often lack explicit mechanisms to…