Related papers: Random Fixed Boundary Flows
We consider the classification problem and focus on nonlinear methods for classification on manifolds. For multivariate datasets lying on an embedded nonlinear Riemannian manifold within the higher-dimensional ambient space, we aim to…
Normalizing flows have shown great promise for modelling flexible probability distributions in a computationally tractable way. However, whilst data is often naturally described on Riemannian manifolds such as spheres, torii, and hyperbolic…
The boundary conditions prescribing the constant traction or the so-called do-nothing conditions are frequently taken on artificial boundaries in the numerical simulations of steady flow of incompressible fluids, despite the fact that they…
We study the boundary rigidity problem with partial data consisting of determining locally the Riemannian metric of a Riemannian manifold with boundary from the distance function measured at pairs of points near a fixed point on the…
We present variational approximations of boundary value problems for curvature flow (curve shortening flow) and elastic flow (curve straightening flow) in two-dimensional Riemannian manifolds that are conformally flat. For the evolving open…
We study a continuous-time system that solves optimization problems over the set of orthonormal matrices, which is also known as the Stiefel manifold. The resulting optimization flow follows a path that is not always on the manifold but…
A machine learning method to predict steady external fluid flows using elliptic input features is introduced. Using data from as few as one high-fidelity simulation, the proposed method produces models generalizable under changes to…
We consider a model of axisymmetric flows for a free boundary vortex embedded in a statically stable fluid at rest. We identify the boundary of the vortex by solving a variational problem. Then, we reduce the analysis of the dynamics of the…
This paper proposes a theoretical framework for modeling and optimizing the bounded functions based on the Fourier series approximation and Ricci flow. Specifically, the initial manifold, $\mathcal{M}_0$ is approximated using Fourier series…
Uncertainty often plays an important role in dynamic flow problems. In this paper, we consider both, a stationary and a dynamic flow model with uncertain boundary data on networks. We introduce two different ways how to compute the…
The potential applications of boundary functionals of random processes, such as the extreme values of these processes, the moment of first reaching a fixed level, the value of the process at the moment of reaching the level, the moment of…
Normalizing flows are invertible neural networks with tractable change-of-volume terms, which allow optimization of their parameters to be efficiently performed via maximum likelihood. However, data of interest are typically assumed to live…
Rectified flow (Liu et al., 2022; Liu, 2022; Wu et al., 2023) is a method for defining a transport map between two distributions, and enjoys popularity in machine learning, although theoretical results supporting the validity of these…
Diffusion and flow models have become the dominant paradigm for generative modeling on Riemannian manifolds, with successful applications in protein backbone generation and DNA sequence design. However, these methods require tens to…
This brief note addresses the free boundary problem arising from the steady two-dimensional seepage flow through a rectangular dam. The flow problem consists in finding the free boundary location, and the velocity and pressure fields. The…
Inspired by the idea of Colding-Minicozzi in [CM1], we define (mean curvature flow) entropy for submanifolds in a general ambient Riemannian manifold. In particular, this entropy is equivalent to area growth of a closed submanifold in a…
Normalizing Flows (NFs) are flexible explicit generative models that have been shown to accurately model complex real-world data distributions. However, their invertibility constraint imposes limitations on data distributions that reside on…
In this study we revisit the problem of computing steady Navier-Stokes flows in two-dimensional unbounded domains. Precise quantitative characterization of such flows in the high-Reynolds number limit remains an open problem of theoretical…
A new statistical definition for the mean turbulent boundary layer thickness is introduced, based on identification of the point where the streamwise velocity skewness changes sign, from negative to positive, in the outermost region of the…
Linear optimal gains are computed for the subcritical two-dimensional separated boundary-layer flow past a bump. Very large optimal gain values are found, making it possible for small-amplitude noise to be strongly amplified and to…