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Neural Numerical Homogenization Based on Deep Ritz Corrections

https://doi.org/10.2139/ssrn.5366225
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Generalized multiscale finite element methods for space–time heterogeneous parabolic equations
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Numerical methods for stochastic partial differential equations with multiple scales
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Generalized multiscale finite element methods (GMsFEM)
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Gamblets for opening the complexity-bottleneck of implicit schemes for hyperbolic and parabolic ODEs/PDEs with rough coefficients
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Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
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When and why PINNs fail to train: A neural tangent kernel perspective
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The variational multiscale method—a paradigm for computational mechanics
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Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization
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Approximation theory of the MLP model in neural networks
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A mathematical view of automatic differentiation
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The heterogeneous multiscale method
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Numerical homogenization beyond scale separation
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Learning representations by back-propagating errors
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Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization
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A generalized finite element method for linear thermoelasticity
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Finite element quasi-interpolation and best approximation
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Multiscale scattering in nonlinear Kerr-type media
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Error estimates for discrete generalized FEMs with locally optimal spectral approximations
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Analysis of the heterogeneous multiscale method for parabolic homogenization problems
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Localization of elliptic multiscale problems
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Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs
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Artificial neural networks for solving ordinary and partial differential equations
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A Multiscale Finite Element Method for Numerical Homogenization
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Homogenization of Parabolic Equations with a Continuum of Space and Time Scales
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Numerical Homogenization by Localized Orthogonal Decomposition
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Optimal Local Approximation Spaces for Generalized Finite Element Methods with Application to Multiscale Problems
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Oversampling for the Multiscale Finite Element Method
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Multigrid with Rough Coefficients and Multiresolution Operator Decomposition from Hierarchical Information Games
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Optimization Methods for Large-Scale Machine Learning
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Numerical Homogenization of Elliptic PDEs with Similar Coefficients
inferredcites →https://doi.org/10.1137/19m1278211openalex-snapshot
Numerical Upscaling of Perturbed Diffusion Problems
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A Priori Error Analysis of a Numerical Stochastic Homogenization Method
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Optimal Local Approximation Spaces for Parabolic Problems
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Novel Design and Analysis of Generalized Finite Element Methods Based on Locally Optimal Spectral Approximations
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A Space-Time Multiscale Method for Parabolic Problems
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Randomized Quasi-Optimal Local Approximation Spaces in Time
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A Simple Collocation-Type Approach to Numerical Stochastic Homogenization
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COUPLING HETEROGENEOUS MULTISCALE FEM WITH RUNGE–KUTTA METHODS FOR PARABOLIC HOMOGENIZATION PROBLEMS: A FULLY DISCRETE SPACETIME ANALYSIS
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Algorithm 778: L-BFGS-B
inferredcites →https://doi.org/10.2139/ssrn.4086448openalex-snapshot
Self-Adaptive Physics-Informed Neural Networks
inferredcites →https://doi.org/10.4208/cicp.oa-2020-0193openalex-snapshot
On the Convergence of Physics Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs
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Computational Multiscale Methods for Linear Heterogeneous Poroelasticity
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Automatic differentiation in machine learning: a survey
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The Heterognous Multiscale Methods
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Numerical stochastic homogenization by quasilocal effective diffusion tensors
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Neural Tangent Kernel: Convergence and Generalization in Neural Networks
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Fully discrete Heterogeneous Multiscale Method for parabolic problems with multiple spatial and temporal scales
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