# Error Estimates for the Deep Ritz Method with Boundary Penalty

arXiv: https://arxiv.org/abs/2103.01007 · HTML: https://mlsanity.com/paper/2103.01007/

## Key facts

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

## Builds on (most cited first, 18 of 18)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [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)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics](https://mlsanity.com/paper/1707.09571/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Sobolev Training for Neural Networks](https://mlsanity.com/paper/1706.04859/) (2017)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [Error bounds for approximations with deep ReLU neural networks in Ws,p norms](https://mlsanity.com/paper/1902.07896/) (2019)
- [Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning](https://mlsanity.com/paper/2008.13333/) (2020)
- [An overview on deep learning-based approximation methods for partial differential equations](https://mlsanity.com/paper/2012.12348/) (2020)
- [NVIDIA SimNet^{TM}: an AI-accelerated multi-physics simulation framework](https://mlsanity.com/paper/2012.07938/) (2020)
- [Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theory](https://mlsanity.com/paper/2006.15733/) (2020)
- [The Barron Space and the Flow-induced Function Spaces for Neural Network Models](https://mlsanity.com/paper/1906.08039/) (2019)
- [A 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 coefficients](https://mlsanity.com/paper/1809.07321/) (2018)
- [A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equations](https://mlsanity.com/paper/2101.01708/) (2021)
- [Partition of unity networks: deep hp-approximation](https://mlsanity.com/paper/2101.11256/) (2021)

## Built on by (most cited first, 14 of 14)

- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Nearly Optimal VC-Dimension and Pseudo-Dimension Bounds for Deep Neural Network Derivatives](https://mlsanity.com/paper/2305.08466/) (2023)
- [NeuFENet: Neural Finite Element Solutions with Theoretical Bounds for Parametric PDEs](https://mlsanity.com/paper/2110.01601/) (2021)
- [Achieving High Accuracy with PINNs via Energy Natural Gradients](https://mlsanity.com/paper/2302.13163/) (2023)
- [Deep Neural Networks with General Activations: Super-Convergence in Sobolev Norms](https://mlsanity.com/paper/2508.05141/) (2025)
- [Solving Elliptic Problems with Singular Sources using Singularity Splitting Deep Ritz Method](https://mlsanity.com/paper/2209.02931/) (2022)
- [DRM Revisited: A Complete Error Analysis](https://mlsanity.com/paper/2407.09032/) (2024)
- [Convergence analysis of unsupervised Legendre-Galerkin neural networks for linear second-order elliptic PDEs](https://mlsanity.com/paper/2211.08900/) (2022)
- [Solutions to Elliptic and Parabolic Problems via Finite Difference Based Unsupervised Small Linear Convolutional Neural Networks](https://mlsanity.com/paper/2311.00259/) (2023)
- [Error Analysis of Three-Layer Neural Network Trained with PGD for Deep Ritz Method](https://mlsanity.com/paper/2405.11451/) (2024)
- [Geometry-aware framework for deep energy method: an application to structural mechanics with hyperelastic materials](https://mlsanity.com/paper/2405.03427/) (2024)
- [Improving PINNs By Algebraic Inclusion of Boundary and Initial Conditions](https://mlsanity.com/paper/2407.20741/) (2024)
- [THINNs: Thermodynamically Informed Neural Networks](https://mlsanity.com/paper/2509.19467/) (2025)
- [Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solvers](https://mlsanity.com/paper/2607.22004/) (2026)

## 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.