# A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equations

arXiv: https://arxiv.org/abs/2101.01708 · HTML: https://mlsanity.com/paper/2101.01708/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 48 references that are in the map.
- First posted on arXiv 5 Jan 2021.

## Builds on (most cited first, 20 of 20)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't](https://mlsanity.com/paper/2009.10713/) (2020)
- [Limitations of Lazy Training of Two-layers Neural Networks](https://mlsanity.com/paper/1906.08899/) (2019)
- [A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks](https://mlsanity.com/paper/2007.07442/) (2020)
- [Approximation by Combinations of ReLU and Squared ReLU Ridge Functions with  ℓ1  and  ℓ0  Controls](https://mlsanity.com/paper/1607.07819/) (2016)
- [A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations](https://mlsanity.com/paper/1809.02362/) (2018)
- [Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equations](https://mlsanity.com/paper/1809.03062/) (2018)
- [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)
- [Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach](https://mlsanity.com/paper/2002.02600/) (2020)
- [Solving for high dimensional committor functions using artificial neural networks](https://mlsanity.com/paper/1802.10275/) (2018)

## Built on by (most cited first, 18 of 18)

- [Tackling the Curse of Dimensionality with Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.12306/) (2023)
- [DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2112.14038/) (2021)
- [When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?](https://mlsanity.com/paper/2109.09444/) (2021)
- [Deep Nonparametric Estimation of Operators between Infinite Dimensional Spaces](https://mlsanity.com/paper/2201.00217/) (2022)
- [On the Representation of Solutions to Elliptic PDEs in Barron Spaces](https://mlsanity.com/paper/2106.07539/) (2021)
- [A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr"odinger Eigenvalue Problems](https://mlsanity.com/paper/2105.01228/) (2021)
- [IAE-Net: Integral Autoencoders for Discretization-Invariant Learning](https://mlsanity.com/paper/2203.05142/) (2022)
- [Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspective](https://mlsanity.com/paper/2210.12101/) (2022)
- [Error Estimates for the Deep Ritz Method with Boundary Penalty](https://mlsanity.com/paper/2103.01007/) (2021)
- [Solving PDEs on Unknown Manifolds with Machine Learning](https://mlsanity.com/paper/2106.06682/) (2021)
- [Sobolev Acceleration and Statistical Optimality for Learning Elliptic Equations via Gradient Descent](https://mlsanity.com/paper/2205.07331/) (2022)
- [Stationary Density Estimation of Ito Diffusions Using Deep Learning](https://mlsanity.com/paper/2109.03992/) (2021)
- [Convergence Analysis of the Deep Galerkin Method for Weak Solutions](https://mlsanity.com/paper/2302.02405/) (2023)
- [Generalization Error Analysis of Neural networks with Gradient Based Regularization](https://mlsanity.com/paper/2107.02797/) (2021)
- [Convergence analysis of a quasi-Monte Carlo-based deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/2210.16196/) (2022)
- [Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Models](https://mlsanity.com/paper/2303.17109/) (2023)
- [Benign overfitting in Fixed Dimension via Physics-Informed Learning with Smooth Inductive Bias](https://mlsanity.com/paper/2406.09194/) (2024)
- [Parameterized Representations via Implicit Stochastic Modulation for High-Dimensional and High-Order Neural PDE Solvers](https://mlsanity.com/paper/2606.22150/) (2026)

## Abstract (arXiv)

This paper concerns the a priori generalization analysis of the Deep Ritz Method (DRM) [W. E and B. Yu, 2017], a popular neural-network-based method for solving high dimensional partial differential equations. We derive the generalization error bounds of two-layer neural networks in the framework of the DRM for solving two prototype elliptic PDEs: Poisson equation and static Schr"odinger equation on the d-dimensional unit hypercube. Specifically, we prove that the convergence rates of generalization errors are independent of the dimension d, under the a priori assumption that the exact solutions of the PDEs lie in a suitable low-complexity space called spectral Barron space. Moreover, we give sufficient conditions on the forcing term and the potential function which guarantee that the solutions are spectral Barron functions. We achieve this by developing a new solution theory for the PDEs on the spectral Barron space, which can be viewed as an analog of the classical Sobolev regularity theory for PDEs.