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

arXiv: https://arxiv.org/abs/2105.14094 · HTML: https://mlsanity.com/paper/2105.14094/

## 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 (most cited first, 8 of 8)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [A unified deep artificial neural network approach to partial differential equations in complex geometries](https://mlsanity.com/paper/1711.06464/) (2017)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Robust Training and Initialization of Deep Neural Networks: An Adaptive Basis Viewpoint](https://mlsanity.com/paper/1912.04862/) (2019)

## Built on by (most cited first, 24 of 24)

- [Spline-PINN: Approaching PDEs without Data using Fast, Physics-Informed Hermite-Spline CNNs](https://mlsanity.com/paper/2109.07143/) (2021)
- [Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problems](https://mlsanity.com/paper/2406.19662/) (2024)
- [Applications of physics informed neural operators](https://mlsanity.com/paper/2203.12634/) (2022)
- [Stacked networks improve physics-informed training: applications to neural networks and deep operator networks](https://mlsanity.com/paper/2311.06483/) (2023)
- [Neural Spectral Methods: Self-supervised learning in the spectral domain](https://mlsanity.com/paper/2312.05225/) (2023)
- [A multifidelity approach to continual learning for physical systems](https://mlsanity.com/paper/2304.03894/) (2023)
- [Multifidelity Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2410.14764/) (2024)
- [First Order System Least Squares Neural Networks](https://mlsanity.com/paper/2409.20264/) (2024)
- [Multifidelity domain decomposition-based physics-informed neural networks and operators for time-dependent problems](https://mlsanity.com/paper/2401.07888/) (2024)
- [Machine-learning custom-made basis functions for partial differential equations](https://mlsanity.com/paper/2111.05307/) (2021)
- [Probabilistic partition of unity networks: clustering based deep approximation](https://mlsanity.com/paper/2107.03066/) (2021)
- [Preconditioned FEM-based Neural Networks for Solving Incompressible Fluid Flows and Related Inverse Problems](https://mlsanity.com/paper/2409.04067/) (2024)
- [S2GPT-PINNs: Sparse and Small models for PDEs](https://mlsanity.com/paper/2506.15687/) (2025)
- [Neural Networks Trained by Weight Permutation are Universal Approximators](https://mlsanity.com/paper/2407.01033/) (2024)
- [Deep Learning without Global Optimization by Random Fourier Neural Networks](https://mlsanity.com/paper/2407.11894/) (2024)
- [Sparse RBF Networks for PDEs and nonlocal equations: function space theory, operator calculus, and training algorithms](https://mlsanity.com/paper/2601.17562/) (2026)
- [Weak-Form Evolutionary Kolmogorov-Arnold Networks for Solving Partial Differential Equations](https://mlsanity.com/paper/2602.18515/) (2026)
- [Unsupervised Legendre-Galerkin Neural Network for Singularly Perturbed Partial Differential Equations](https://mlsanity.com/paper/2207.10241/) (2022)
- [Data-Driven Self-Supervised Learning for the Discovery of Solution Singularity for Partial Differential Equations](https://mlsanity.com/paper/2506.23344/) (2025)
- [Sparse FEONet: A Low-Cost, Memory-Efficient Operator Network via Finite-Element Local Sparsity for Parametric PDEs](https://mlsanity.com/paper/2601.00672/) (2026)
- [Improving the accuracy of physics-informed neural networks via last-layer retraining](https://mlsanity.com/paper/2603.04672/) (2026)
- [Stochastic-Dimension Frozen Sampled Neural Network for High-Dimensional Gross-Pitaevskii Equations on Unbounded Domains](https://mlsanity.com/paper/2604.09361/) (2026)
- [Reference-free logged energy-oracle recovery for neural approximations of symmetric coercive variational problems: conforming Riesz reconstruction and archive-level selection](https://mlsanity.com/paper/2608.16473/) (2026)
- [Multi-stage neural operator learning with application for convolutions](https://mlsanity.com/paper/2608.18851/) (2026)

## 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.