# Deep Ritz revisited

arXiv: https://arxiv.org/abs/1912.03937 · HTML: https://mlsanity.com/paper/1912.03937/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Dec 2019.

## Built on by (most cited first, 10 of 10)

- [Enhanced Physics-Informed Neural Networks with Augmented Lagrangian Relaxation Method (AL-PINNs)](https://mlsanity.com/paper/2205.01059/) (2022)
- [Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems](https://mlsanity.com/paper/2212.03130/) (2022)
- [Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problems](https://mlsanity.com/paper/2112.03749/) (2021)
- [Lagrangian dual framework for conservative neural network solutions of kinetic equations](https://mlsanity.com/paper/2106.12147/) (2021)
- [Convergence analysis of unsupervised Legendre-Galerkin neural networks for linear second-order elliptic PDEs](https://mlsanity.com/paper/2211.08900/) (2022)
- [Enforcing hidden physics in physics-informed neural networks](https://mlsanity.com/paper/2511.14348/) (2025)
- [Solving the Poisson Equation with Dirichlet data by shallow ReLUα-networks: A regularity and approximation perspective](https://mlsanity.com/paper/2412.07728/) (2024)
- [Learning Energy-Based Generative Models via Potential Flow: A Variational Principle Approach to Probability Density Homotopy Matching](https://mlsanity.com/paper/2504.16262/) (2025)
- [Unsupervised Random Quantum Networks for PDEs](https://mlsanity.com/paper/2312.14975/) (2023)
- [Variational Potential Flow: A Novel Probabilistic Framework for Energy-Based Generative Modelling](https://mlsanity.com/paper/2407.15238/) (2024)

## Abstract (arXiv)

Recently, progress has been made in the application of neural networks to the numerical analysis of partial differential equations (PDEs). In the latter the variational formulation of the Poisson problem is used in order to obtain an objective function - a regularised Dirichlet energy - that was used for the optimisation of some neural networks. In this notes we use the notion of Gamma-convergence to show that ReLU networks of growing architecture that are trained with respect to suitably regularised Dirichlet energies converge to the true solution of the Poisson problem. We discuss how this approach generalises to arbitrary variational problems under certain universality assumptions of neural networks and see that this covers some nonlinear stationary PDEs like the p-Laplace.