# DRM Revisited: A Complete Error Analysis

arXiv: https://arxiv.org/abs/2407.09032 · HTML: https://mlsanity.com/paper/2407.09032/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 116 references that are in the map.
- First posted on arXiv 12 Jul 2024.

## Builds on (most cited first, 6 of 6)

- [Error Estimates for the Deep Ritz Method with Boundary Penalty](https://mlsanity.com/paper/2103.01007/) (2021)
- [Deeper or Wider: A Perspective from Optimal Generalization Error with Sobolev Loss](https://mlsanity.com/paper/2402.00152/) (2024)
- [Exponential Tail Local Rademacher Complexity Risk Bounds Without the Bernstein Condition](https://mlsanity.com/paper/2202.11461/) (2022)
- [Analysis of the expected L2 error of an over-parametrized deep neural network estimate learned by gradient descent without regularization](https://mlsanity.com/paper/2311.14609/) (2023)
- [On the rate of convergence of an over-parametrized Transformer classifier learned by gradient descent](https://mlsanity.com/paper/2312.17007/) (2023)
- [Error Analysis of Three-Layer Neural Network Trained with PGD for Deep Ritz Method](https://mlsanity.com/paper/2405.11451/) (2024)

## Built on by (most cited first, 4 of 4)

- [Adaptive-Growth Randomized Neural Networks for Level-Set Computation of Multivalued Nonlinear First-Order PDEs with Hyperbolic Characteristics](https://mlsanity.com/paper/2603.01093/) (2026)
- [Feature Learning for the High Dimensional Stationary Sch"odinger Equation with Deep Ritz Method](https://mlsanity.com/paper/2607.06369/) (2026)
- [Error bounds in Sobolev norms for approximations with norm constrained ReLU neural networks](https://mlsanity.com/paper/2609.19937/) (2026)
- [A Hybrid Iterative Deep Ritz Method for Elliptic Interface Problems](https://mlsanity.com/paper/2609.27325/) (2026)

## Abstract (arXiv)

In this work, we address a foundational question in the theoretical analysis of the Deep Ritz Method (DRM) under the over-parameteriztion regime: Given a target precision level, how can one determine the appropriate number of training samples, the key architectural parameters of the neural networks, the step size for the projected gradient descent optimization procedure, and the requisite number of iterations, such that the output of the gradient descent process closely approximates the true solution of the underlying partial differential equation to the specified precision?