arXiv research

Data-driven approach learns effective equations for phase field interfaces.

New proof confirms De Giorgi's conjecture about phase-field approximation of Willmore functional.

We explore a computational model of an incompressible fluid with a multi-phase field in three-dimensional Euclidean space. By investigating an incompressible fluid with a two-phase field geometrically, we reformulate the expression of the surface tension for the two-phase field found by Lafaurie, Nardone, Scardovelli, …

We study the phase field method for the volume preserving mean curvature flow. Given an initial $C^1$ hypersurface we proved the existence of the weak solution for the volume preserving mean curvature flow via the reaction diffusion equation with a nonlocal term. We also show the monotonicity formula and the density up…

Generates tubular and membranous shapes using curvature functionals.

Machine learning predicts failure in brittle materials with high accuracy.

Study proposes a nonlocal approximation of the Willmore functional using fractional Allen-Cahn energies.

Universal model for soft tissue mechanics under shock waves.

The homotopy theory of topological defects in ordered media fails to completely characterize systems with broken translational symmetry. We argue that the problem can be understood in terms of the lack of rotational Goldstone modes in such systems and provide an alternate approach that correctly accounts for the intera…

Optimizes structure topology for ductile and brittle fracture resistance.

Continuum mechanics theory describes skin's complex anisotropic behavior.

Variational approximations for curve flows on Riemannian manifolds.

We prove nonexistence of nonconstant local minimizers for a class of functionals, which typically appears in the scalar two-phase field model, over a smooth N-dimensional Riemannian manifold without boundary with non-negative Ricci curvature. Conversely for a class of surfaces possessing a simple closed geodesic along …

We consider the sharp interface limit of the Allen-Cahn equation with Dirichlet or dynamic boundary conditions and give a varifold characterization of its limit which is formally a mean curvature flow with Dirichlet or dynamic boundary conditions. In order to show the existence of the limit, we apply the phase field me…

In order to study a one-dimensional analogue of the spontaneous curvature model for two-component lipid bilayer membranes we consider planar curves that are made of a material with two phases. Each phase induces a preferred curvature to the curve, and these curvatures as well as phase boundaries may lead to the develop…

Common models for two-phase lipid bilayer membranes are based on an energy that consists of an elastic term for each lipid phase and a line energy at interfaces. Although such an energy controls only the length of interfaces, the membrane surface is usually assumed to be at least $C^1$ across phase boundaries. We consi…

Study quantizes energy distribution in inhomogeneous phase transitions.

The paper studies momentum-based minimization for Ginzburg-Landau on Euclidean spaces and graphs.

We present a new physics informed neural network (PINN) algorithm for solving brittle fracture problems. While most of the PINN algorithms available in the literature minimize the residual of the governing partial differential equation, the proposed approach takes a different path by minimizing the variational energy o…

The study provides conditions for approximating Riemannian manifolds with polyhedral metrics.

Geometric Gaussian approximations capture any distribution.

Deep learning networks are approximated using dynamical systems theory.

Method approximates Riemannian barycenter on manifolds.

We consider in this paper the optimal approximations of convex univariate functions with feed-forward Relu neural networks. We are interested in the following question: what is the minimal approximation error given the number of approximating linear pieces? We establish the necessary and sufficient conditions and uniqu…

Efficiently reduces tensor ranks using mean-field approximation.

Study approximates unknown function levels with queries.

We study sparse approximate solutions to convex optimization problems. It is known that in many engineering applications researchers are interested in an approximate solution of an optimization problem as a linear combination of elements from a given system of elements. There is an increasing interest in building such …

Softmax attention approximates complex functions and subsumes many known universal approximators.

Improved matrix approximation using randomized algorithms.

Deviation inequalities for stochastic approximation methods.

Neural approximate computing gains enormous energy-efficiency at the cost of tolerable quality-loss. A neural approximator can map the input data to output while a classifier determines whether the input data are safe to approximate with quality guarantee. However, existing works cannot maximize the invocation of the a…

Approximate symmetries of geodesic equations on 2-spheres are studied. These are the symmetries of the perturbed geodesic equations which represent approximate path of a particle rather than exact path. After giving the exact symmetries of the geodesic equations, two different approaches to study the approximate symmet…

We are concerned with an approximation problem for a symmetric positive semidefinite matrix due to motivation from a class of nonlinear machine learning methods. We discuss an approximation approach that we call {matrix ridge approximation}. In particular, we define the matrix ridge approximation as an incomplete matri…

Transformers use ReLUs to approximate softmax efficiently.

Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By certified, we mean that the approximation can approximate the given curve to any given pr…

Recently, variational approximations such as the mean field approximation have received much interest. We extend the standard mean field method by using an approximating distribution that factorises into cluster potentials. This includes undirected graphs, directed acyclic graphs and junction trees. We derive generaliz…

Adaptive approximations improve variational inference for complex models.

Non-negative $L_1$-approximating polynomials for Gaussian distributions are proven for certain classes of sets.

Paper introduces new approximations for lognormal sums, matching comonotonicity and moments.

Paper analyzes normal approximation for two-timescale stochastic algorithms, revealing interaction between fast and slow timescales.

One-pass algorithm finds small subset for $\ell_p$ subspace approximation with additive error.

We are interested in approximation of a multivariate function $f(x_1,\dots,x_d)$ by linear combinations of products $u^1(x_1)\cdots u^d(x_d)$ of univariate functions $u^i(x_i)$, $i=1,\dots,d$. In the case $d=2$ it is a classical problem of bilinear approximation. In the case of approximation in the $L_2$ space the bili…

A new method for efficient Gaussian process inference using sparse approximations.

In this paper, we propose a low-rank approximation method based on discrete least-squares for the approximation of a multivariate function from random, noisy-free observations. Sparsity inducing regularization techniques are used within classical algorithms for low-rank approximation in order to exploit the possible sp…

Neural network based approximate computing is a universal architecture promising to gain tremendous energy-efficiency for many error resilient applications. To guarantee the approximation quality, existing works deploy two neural networks (NNs), e.g., an approximator and a predictor. The approximator provides the appro…

The paper approximates supply curves using a one-step basis method.

The paper defines a new concept of approximability for Lagrangian submanifolds.

Boosting Nyström improves accuracy of matrix approximations.