# Generalized Multiscale Finite Element Methods (GMsFEM)

arXiv: https://arxiv.org/abs/1301.2866 · HTML: https://mlsanity.com/paper/1301.2866/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jan 2013.

## Built on by (most cited first, 21 of 21)

- [Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems](https://mlsanity.com/paper/2203.00003/) (2022)
- [Mitigating spectral bias for the multiscale operator learning](https://mlsanity.com/paper/2210.10890/) (2022)
- [A data-driven peridynamic continuum model for upscaling molecular dynamics](https://mlsanity.com/paper/2108.04883/) (2021)
- [Defining Foundation Models for Computational Science: A Call for Clarity and Rigor](https://mlsanity.com/paper/2505.22904/) (2025)
- [Subspace Decomposition based DNN algorithm for elliptic type multi-scale PDEs](https://mlsanity.com/paper/2112.06660/) (2021)
- [A Gradient-based Deep Neural Network Model for Simulating Multiphase Flow in Porous Media](https://mlsanity.com/paper/2105.02652/) (2021)
- [A Physics-Constrained Deep Learning Model for Simulating Multiphase Flow in 3D Heterogeneous Porous Media](https://mlsanity.com/paper/2105.09467/) (2021)
- [A Neural Network Approach for Homogenization of Multiscale Problems](https://mlsanity.com/paper/2206.02032/) (2022)
- [Bayesian Nonlocal Operator Regression (BNOR): A Data-Driven Learning Framework of Nonlocal Models with Uncertainty Quantification](https://mlsanity.com/paper/2211.01330/) (2022)
- [Towards a unified nonlocal, peridynamics framework for the coarse-graining of molecular dynamics data with fractures](https://mlsanity.com/paper/2301.04540/) (2023)
- [A Forward Backward Greedy approach for Sparse Multiscale Learning](https://mlsanity.com/paper/2102.07068/) (2021)
- [AMS-Net: Adaptive Multiscale Sparse Neural Network with Interpretable Basis Expansion for Multiphase Flow Problems](https://mlsanity.com/paper/2207.11735/) (2022)
- [Prediction of Multiscale Features Using Deep Learning-based Preconditioner-Solver Architecture for Darcy Equation in High-Contrast Media](https://mlsanity.com/paper/2411.02431/) (2024)
- [Multiscale Corrections by Continuous Super-Resolution](https://mlsanity.com/paper/2411.07576/) (2024)
- [Stochastic approach for elliptic problems in perforated domains](https://mlsanity.com/paper/2403.11385/) (2024)
- [Learning-based Multi-continuum Model for Multiscale Flow Problems](https://mlsanity.com/paper/2403.14084/) (2024)
- [Embedded Nonlocal Operator Regression (ENOR): Quantifying model error in learning nonlocal operators](https://mlsanity.com/paper/2410.20331/) (2024)
- [Dual-Domain Deep Learning Method to Accelerate Local Basis Functions Computation for Reservoir Simulation in High-Contrast Porous Media](https://mlsanity.com/paper/2511.20685/) (2025)
- [Applying Two-Grid Preconditioner for Subsurface Flow Simulation using Attention-enhanced Hybrid Network to Accelerate Multiscale Discretization in High-contrast Media](https://mlsanity.com/paper/2606.02582/) (2026)
- [The Frame Kernel Method for Multiscale Operator Learning](https://mlsanity.com/paper/2608.25084/) (2026)
- [Gaussian-process surrogate indicators for residual-based adaptive GMsFEM](https://mlsanity.com/paper/2609.26843/) (2026)

## Abstract (arXiv)

In this paper, we propose a general approach called Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations for problems without scale separation over a complex input space. As in multiscale finite element methods (MsFEMs), the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space. In the proposed approach, we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage. The enrichment in the online stage is performed based on a spectral decomposition of the offline space. In the online stage, for any input parameter, a multiscale space is constructed to solve the global problem on a coarse grid. The online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues. The computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re-used for solving the forward problem with any forcing and boundary condition. Compared with the other approaches where global snapshots are used, the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse-grid level. We present various examples in the paper and some numerical results to demonstrate the effectiveness of our method.