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We extend the method of layer potentials to manifolds with boundary and cylindrical ends. To obtain this extension along the classical lines, we have to deal with several technical difficulties due to the non-compactness of the boundary,…
This paper presents an improved approach for preserving sharp interfaces in multiphase Lattice Boltzmann Method (LBM) simulations using Physics-Informed Neural Networks (PINNs). Interface diffusion is a common challenge in multiphase LBM,…
This paper is dedicated to proving the complete integrability of the Benjamin--Ono (BO) equation on the line when restricted to every $N$-soliton manifold, denoted by $\mathcal{U}_N$. We construct generalized action--angle coordinates which…
The purpose of this paper is to study the property of the resolvent of the Laplace-Beltrami operator on a noncompact complete Riemannian manifold with various ends each of which has a different limit of the growth rate of the Riemannian…
This paper introduces a novel boundary integral approach of shape uncertainty quantification for the Helmholtz scattering problem in the framework of the so-called parametric method. The key idea is to construct an integration grid whose…
Physics informed neural networks (PINNs) have recently been proposed as surrogate models for solving process optimization problems. However, in an active learning setting collecting enough data for reliably training PINNs poses a challenge.…
This paper studies large-scale optimization problems on Riemannian manifolds whose objective function is a finite sum of negative log-probability losses. Such problems arise in various machine learning and signal processing applications. By…
We propose a path optimization method (POM) to evade the sign problem in the Monte-Carlo calculations for complex actions. Among many approaches to the sign problem, the Lefschetz-thimble path-integral method and the complex Langevin method…
In this paper, a restricted memory quasi-Newton bundle method for minimizing a locally Lipschitz continuous function over a Riemannian manifold is proposed. The curvature information of the objective function is approximated by applying a…
The Immersed Boundary Method (IBM) is one of the popular one-fluid mixed Eulerian-Lagrangian methods to simulate motion of droplets. While the treatment of a moving complex boundary is an extremely time consuming and formidable task in a…
Current point cloud processing algorithms do not have the capability to automatically extract semantic information from the observed scenes, except in very specialized cases. Furthermore, existing mesh analysis paradigms cannot be directly…
Numerical solutions of partial differential equations (PDEs) on manifolds continues to generate a lot of interest among scientists in the natural and applied sciences. On the other hand, recent developments of 3D scanning and computer…
The symmetrized Asymptotic Mean Value Laplacian $\tilde{\Delta}$, obtained as limit of approximating operators $\tilde{\Delta}_r$, is an extension of the classical Euclidean Laplace operator to the realm of metric measure spaces. We show…
Recent work in on-line Statistical Process Control (SPC) of manufactured 3-dimensional (3-D) objects has been proposed based on the estimation of the spectrum of the Laplace-Beltrami (LB) operator, a differential operator that encodes the…
In this article, we establish a $L^1$ estimate for solutions to Poisson equation with mixed boundary condition, on complete noncompact manifolds with nonnegative Ricci curvature and compact manifolds with positive Ricci curvature…
We use the averaged variational principle introduced in a recent article on graph spectra [7] to obtain upper bounds for sums of eigenvalues of several partial differential operators of interest in geometric analysis, which are analogues of…
Physics-informed neural networks (PINNs) have been proposed to learn the solution of partial differential equations (PDE). In PINNs, the residual form of the PDE of interest and its boundary conditions are lumped into a composite objective…
The paper introduces a novel non-parametric Riemannian regression method using Isometric Riemannian Manifolds (IRMs). The proposed technique, Intrinsic Local Polynomial Regression on IRMs (ILPR-IRMs), enables global data mapping between…
In this paper, we present PIETOOLS, a MATLAB toolbox for the construction and handling of Partial Integral (PI) operators. The toolbox introduces a new class of MATLAB object, opvar, for which standard MATLAB matrix operation syntax (e.g.…
We present the Physics-Informed Optimal Homotopy Analysis Method (PI-OHAM) for solving nonlinear differential equations. PI-OHAM, based on classical HAM, employs a physics-informed residual loss to optimize convergence-control parameters…