mlsanityData to 7 Oct 2026

Convergence Analysis of the Deep Galerkin Method for Weak Solutions

arXiv 2302.02405 ↗First posted 5 Feb 2023math.NA cs.LG cs.NAcited by 4

Not on the map. Why?

Abstract (arXiv)

This paper analyzes the convergence rate of a deep Galerkin method for the weak solution (DGMW) of second-order elliptic partial differential equations on ℝd with Dirichlet, Neumann, and Robin boundary conditions, respectively. In DGMW, a deep neural network is applied to parametrize the PDE solution, and a second neural network is adopted to parametrize the test function in the traditional Galerkin formulation. By properly choosing the depth and width of these two networks in terms of the number of training samples n, it is shown that the convergence rate of DGMW is O(n-1/d), which is the first convergence result for weak solutions. The main idea of the proof is to divide the error of the DGMW into an approximation error and a statistical error. We derive an upper bound on the approximation error in the H1 norm and bound the statistical error via Rademacher complexity.

Key facts. Cited by 4 papers in the map (data through 7 Oct 2026). Builds on 9 of its 30 references that are in the map. First posted on arXiv 5 Feb 2023.

Builds on

9 of 30 references.

References · 9

  1. 2014Sequence to Sequence Learning with Neural Networkscited by 2,974
  2. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  3. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  4. 2017Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equationscited by 188
  5. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 115
  6. 2021Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimalitycited by 40
  7. 2020On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEscited by 31
  8. 2020Friedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learningcited by 18
  9. 2021A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equationscited by 18

Built on by

Cited by year

Citing papers per year: 2024: 2; 2025: 2.

4 papers.

  1. 2024Error Analysis of Three-Layer Neural Network Trained with PGD for Deep Ritz Methodcited by 1
  2. 2024Geometry-aware framework for deep energy method: an application to structural mechanics with hyperelastic materialscited by 0
  3. 2025Error analysis for the deep Kolmogorov methodcited by 0
  4. 2025Multi-Scale Finite Expression Method for PDEs with Oscillatory Solutions on Complex Domainscited by 0