mlsanityData to 7 Oct 2026

Error Estimates for the Deep Ritz Method with Boundary Penalty

arXiv 2103.01007 ↗First posted 1 Mar 2021math.NA cs.LG cs.NAcited by 14

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Abstract (arXiv)

We estimate the error of the Deep Ritz Method for linear elliptic equations. For Dirichlet boundary conditions, we estimate the error when the boundary values are imposed through the boundary penalty method. Our results apply to arbitrary sets of ansatz functions and estimate the error in dependence of the optimization accuracy, the approximation capabilities of the ansatz class and -- in the case of Dirichlet boundary values -- the penalization strength λ. To the best of our knowledge, our results are presently the only ones in the literature that treat the case of Dirichlet boundary conditions in full generality, i.e., without a lower order term that leads to coercivity on all of H1(Omega). Further, we discuss the implications of our results for ansatz classes which are given through ReLU networks and the relation to existing estimates for finite element functions. For high dimensional problems our results show that the favourable approximation capabilities of neural networks for smooth functions are inherited by the Deep Ritz Method.

Key facts. Cited by 14 papers in the map (data through 7 Oct 2026). Builds on 18 of its 51 references that are in the map. First posted on arXiv 1 Mar 2021.

Builds on

18 of 51 references.

References · 18

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  3. 2017Solving high-dimensional partial differential equations using deep learningcited by 549
  4. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 261
  5. 2016Understanding Deep Neural Networks with Rectified Linear Unitscited by 253
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  7. 2017Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equationscited by 188
  8. 2017Sobolev Training for Neural Networkscited by 136
  9. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 101
  10. 2019Error bounds for approximations with deep ReLU neural networks in Ws,p normscited by 95
  11. 2020Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learningcited by 77
  12. 2020An overview on deep learning-based approximation methods for partial differential equationscited by 65
6 more
  1. 2020NVIDIA SimNet^{TM}: an AI-accelerated multi-physics simulation frameworkcited by 58
  2. 2020Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theorycited by 41
  3. 2019The Barron Space and the Flow-induced Function Spaces for Neural Network Modelscited by 39
  4. 2018A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficientscited by 31
  5. 2021A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equationscited by 18
  6. 2021Partition of unity networks: deep hp-approximationcited by 15

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 1; 2022: 2; 2023: 3; 2024: 5; 2025: 2; 2026: 1.

14 papers.

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  1. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  2. 2023Nearly Optimal VC-Dimension and Pseudo-Dimension Bounds for Deep Neural Network Derivativescited by 30
  3. 2021NeuFENet: Neural Finite Element Solutions with Theoretical Bounds for Parametric PDEscited by 10
  4. 2023Achieving High Accuracy with PINNs via Energy Natural Gradientscited by 10
  5. 2025Deep Neural Networks with General Activations: Super-Convergence in Sobolev Normscited by 5
  6. 2022Solving Elliptic Problems with Singular Sources using Singularity Splitting Deep Ritz Methodcited by 4
  7. 2024DRM Revisited: A Complete Error Analysiscited by 4
  8. 2022Convergence analysis of unsupervised Legendre-Galerkin neural networks for linear second-order elliptic PDEscited by 3
  9. 2023Solutions to Elliptic and Parabolic Problems via Finite Difference Based Unsupervised Small Linear Convolutional Neural Networkscited by 2
  10. 2024Error Analysis of Three-Layer Neural Network Trained with PGD for Deep Ritz Methodcited by 1
  11. 2024Geometry-aware framework for deep energy method: an application to structural mechanics with hyperelastic materialscited by 0
  12. 2024Improving PINNs By Algebraic Inclusion of Boundary and Initial Conditionscited by 0
2 more
  1. 2025THINNs: Thermodynamically Informed Neural Networkscited by 0
  2. 2026Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solverscited by 0