mlsanityData to 14 Oct 2026

Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control

arXiv 2105.14094 ↗First posted 28 May 2021cs.LG cs.NA math.NAcited by 24

Not on the map. Why?

Abstract (arXiv)

We present a new approach to using neural networks to approximate the solutions of variational equations, based on the adaptive construction of a sequence of finite-dimensional subspaces whose basis functions are realizations of a sequence of neural networks. The finite-dimensional subspaces are then used to define a standard Galerkin approximation of the variational equation. This approach enjoys a number of advantages, including: the sequential nature of the algorithm offers a systematic approach to enhancing the accuracy of a given approximation; the sequential enhancements provide a useful indicator for the error that can be used as a criterion for terminating the sequential updates; the basic approach is largely oblivious to the nature of the partial differential equation under consideration; and, some basic theoretical results are presented regarding the convergence (or otherwise) of the method which are used to formulate basic guidelines for applying the method.

Key facts. Cited by 24 papers in the map (data through 14 Oct 2026). Builds on 8 of its 24 references that are in the map. First posted on arXiv 28 May 2021.

Builds on

8 of 24 references.

References · 8

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 678
  3. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
  4. 2017Solving high-dimensional partial differential equations using deep learningcited by 556
  5. 2019Variational Physics-Informed Neural Networks For Solving Partial Differential Equationscited by 148
  6. 2017A unified deep artificial neural network approach to partial differential equations in complex geometriescited by 122
  7. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 116
  8. 2019Robust Training and Initialization of Deep Neural Networks: An Adaptive Basis Viewpointcited by 42

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 3; 2022: 2; 2023: 3; 2024: 7; 2025: 2; 2026: 7.

24 papers.

Sort:

  1. 2021Spline-PINN: Approaching PDEs without Data using Fast, Physics-Informed Hermite-Spline CNNscited by 38
  2. 2024Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problemscited by 35
  3. 2022Applications of physics informed neural operatorscited by 33
  4. 2023Stacked networks improve physics-informed training: applications to neural networks and deep operator networkscited by 21
  5. 2023Neural Spectral Methods: Self-supervised learning in the spectral domaincited by 14
  6. 2023A multifidelity approach to continual learning for physical systemscited by 13
  7. 2024Multifidelity Kolmogorov-Arnold Networkscited by 9
  8. 2024First Order System Least Squares Neural Networkscited by 5
  9. 2024Multifidelity domain decomposition-based physics-informed neural networks and operators for time-dependent problemscited by 4
  10. 2021Machine-learning custom-made basis functions for partial differential equationscited by 3
  11. 2021Probabilistic partition of unity networks: clustering based deep approximationcited by 2
  12. 2024Preconditioned FEM-based Neural Networks for Solving Incompressible Fluid Flows and Related Inverse Problemscited by 2
12 more
  1. 2025S2GPT-PINNs: Sparse and Small models for PDEscited by 2
  2. 2024Neural Networks Trained by Weight Permutation are Universal Approximatorscited by 1
  3. 2024Deep Learning without Global Optimization by Random Fourier Neural Networkscited by 1
  4. 2026Sparse RBF Networks for PDEs and nonlocal equations: function space theory, operator calculus, and training algorithmscited by 1
  5. 2026Weak-Form Evolutionary Kolmogorov-Arnold Networks for Solving Partial Differential Equationscited by 1
  6. 2022Unsupervised Legendre-Galerkin Neural Network for Singularly Perturbed Partial Differential Equationscited by 0
  7. 2025Data-Driven Self-Supervised Learning for the Discovery of Solution Singularity for Partial Differential Equationscited by 0
  8. 2026Sparse FEONet: A Low-Cost, Memory-Efficient Operator Network via Finite-Element Local Sparsity for Parametric PDEscited by 0
  9. 2026Improving the accuracy of physics-informed neural networks via last-layer retrainingcited by 0
  10. 2026Stochastic-Dimension Frozen Sampled Neural Network for High-Dimensional Gross-Pitaevskii Equations on Unbounded Domainscited by 0
  11. 2026Reference-free logged energy-oracle recovery for neural approximations of symmetric coercive variational problems: conforming Riesz reconstruction and archive-level selectioncited by 0
  12. 2026Multi-stage neural operator learning with application for convolutionscited by 0