Related papers: A class of mesh-free algorithms for some problems …

Inspired by the matching of supply to demand in logistical problems, the optimal transport (or Monge--Kantorovich) problem involves the matching of probability distributions defined over a geometric domain such as a surface or manifold. In…

Kernel based methods provide a way to reconstruct potentially high-dimensional functions from meshfree samples, i.e., sampling points and corresponding target values. A crucial ingredient for this to be successful is the distribution of the…

We propose a mesh-free policy iteration framework based on physics-informed neural networks (PINNs) for solving entropy-regularized stochastic control problems. The method iteratively alternates between soft policy evaluation and…

We propose a low-rank tensor approach to approximate linear transport and nonlinear Vlasov solutions and their associated flow maps. The approach takes advantage of the fact that the differential operators in the Vlasov equation is tensor…

A stable numerical solution of the steady Stokes problem requires compatibility between the choice of velocity and pressure approximation that has traditionally proven problematic for meshless methods. In this work, we present a…

In this paper, we consider the problem of minimizing a smooth function, given as finite sum of black-box functions, over a convex set. In order to advantageously exploit the structure of the problem, for instance when the terms of the…

Physics-informed Neural Networks (PINNs) have recently emerged as a principled way to include prior physical knowledge in form of partial differential equations (PDEs) into neural networks. Although PINNs are generally viewed as mesh-free,…

In this paper we study continuum-marginal optimal transport. Given a time-continuous family of probability marginals, the problem is to recover the minimum-energy velocity field whose flow reproduces every marginal. This problem is the…

We propose a quality-based optimization strategy to reduce the total number of degrees of freedom associated to a discrete problem defined over a polygonal tessellation with the Virtual Element Method. The presented Quality Agglomeration…

We propose a data-driven approach to quantify the uncertainty of models constructed by kernel methods. Our approach minimizes the needed distributional assumptions, hence, instead of working with, for example, Gaussian processes or…

Meshless methods are commonly used to determine numerical solutions to partial differential equations (PDEs) for problems involving free surfaces and/or complex geometries, approximating spatial derivatives at collocation points via local…

In this paper, we develop a novel mesh-free framework, termed physics-informed neural networks with invariant measure score matching (PINN-IMSM), for reconstructing dynamical systems from unlabeled point-cloud data that capture the system's…

We present a unified theoretical and computational framework for constructing reproducing kernels tailored to transport equations and adapted to Koopman eigenfunctions of nonlinear dynamical systems. These eigenfunctions satisfy a…

This work presents a structure-preserving, high-order, unconditionally stable numerical method for approximating the solution to the Fisher-Kolmogorov equation on polytopic meshes, with a particular focus on its application in simulating…

This study introduces an uncertainty-aware, mesh-free numerical method for solving Kolmogorov PDEs. In the proposed method, we use Gaussian process regression (GPR) to smoothly interpolate pointwise solutions that are obtained by Monte…

We present a unified framework for solving trajectory optimization problems in a derivative-free manner through the use of sequential convex programming. Traditionally, nonconvex optimization problems are solved by forming and solving a…

We propose algorithms for solving convective-diffusion partial differential equations (PDEs), which model surfactant concentration and heat transport on evolving surfaces, based on intrinsic kernel-based meshless collocation methods. The…

We establish numerical methods for solving the martingale optimal transport problem (MOT) - a version of the classical optimal transport with an additional martingale constraint on transport's dynamics. We prove that the MOT value can be…

Learning to sample from intractable distributions over discrete sets without relying on corresponding training data is a central problem in a wide range of fields, including Combinatorial Optimization. Currently, popular deep learning-based…

We develop a new method to solve the Fokker-Planck or Kolmogorov's forward equation that governs the time evolution of the joint probability density function of a continuous-time stochastic nonlinear system. Numerical solution of this…