Related papers: Variational multiscale analysis: the fine-scale Gr…
This paper addresses the variational multiscale stabilization of standard finite element methods for linear partial differential equations that exhibit multiscale features. The stabilization is of Petrov-Galerkin type with a standard finite…
We analyze the recent Multi-index Stochastic Collocation (MISC) method for computing statistics of the solution of a partial differential equation (PDEs) with random data, where the random coefficient is parametrized by means of a countable…
Models incorporating uncertain inputs, such as random forces or material parameters, have been of increasing interest in PDE-constrained optimization. In this paper, we focus on the efficient numerical minimization of a convex and smooth…
This paper presents a new stochastic finite element method for computing structural stochastic responses. The method provides a new expansion of stochastic response and decouples the stochastic response into a combination of a series of…
Factorization machines (FMs) are a powerful tool for regression and classification in the context of sparse observations, that has been successfully applied to collaborative filtering, especially when side information over users or items is…
In this work, the uncertainty associated with the finite element discretization error is modeled following the Bayesian paradigm. First, a continuous formulation is derived, where a Gaussian process prior over the solution space is updated…
In this contribution, we discuss the finite-element discrete variable representation (FE-DVR) of the nonequilibrium Green's function and its implications on the description of strongly inhomogeneous quantum systems. In detail, we show that…
Numerical resolution of high-dimensional nonlinear PDEs remains a huge challenge due to the curse of dimensionality. Starting from the weak formulation of the Lawson-Euler scheme, this paper proposes a stochastic particle method (SPM) by…
We introduce a guided stochastic sampling method that augments sampling from diffusion models with physics-based guidance derived from partial differential equation (PDE) residuals and observational constraints, ensuring generated samples…
We develop VSD, a method for conditioning a generative model of discrete, combinatorial designs on a rare desired class by efficiently evaluating a black-box (e.g. experiment, simulation) in a batch sequential manner. We call this task…
A dynamic 3D Green's function for the homogeneous, isotropic and viscoelastic (of the Zener type) half-space is derived in a closed form. The results obtained here can be used as either stand-alone solutions for simple problems or in…
Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…
A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, and…
In this article, we extend the Variational Multi-scale method with spectral approximation of the sub-scales to two-dimensional advection-diffusion problems. The spectral VMS method is cast for low-order elements as a standard VMS method…
This is a study of certain finite element methods designed for convection-dominated, time-dependent partial differential equations. Specifically, we analyze high order space-time tensor product finite element discretizations, used in a…
This paper studies an unsupervised deep learning-based numerical approach for solving partial differential equations (PDEs). The approach makes use of the deep neural network to approximate solutions of PDEs through the compositional…
We consider the primal and dual forms of the optimality conditions for PDE-contrained optimization problems arising in Data-Driven Computational Mechanics when specialized to the reaction-diffusion context. Starting with the continuous…
The multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to…
In this paper we propose a new model-based unsupervised learning method, called VarNet, for the solution of partial differential equations (PDEs) using deep neural networks (NNs). Particularly, we propose a novel loss function that relies…
The potential energy surface (PES) is crucial for interpreting a variety of chemical reaction processes. However, predicting accurate PESs with high-level electronic structure methods is a challenging task due to the high computational…