# A Deep Double Ritz Method (D2RM) for solving Partial Differential Equations using Neural Networks

arXiv: https://arxiv.org/abs/2211.03627 · HTML: https://mlsanity.com/paper/2211.03627/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 58 references that are in the map.
- First posted on arXiv 7 Nov 2022.

## Builds on (most cited first, 23 of 23)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems](https://mlsanity.com/paper/1603.04467/) (2016)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Horovod: fast and easy distributed deep learning in TensorFlow](https://mlsanity.com/paper/1802.05799/) (2018)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Geometric Understanding of Deep Learning](https://mlsanity.com/paper/1805.10451/) (2018)
- [VarNet: Variational Neural Networks for the Solution of Partial Differential Equations](https://mlsanity.com/paper/1912.07443/) (2019)
- [MeshingNet: A New Mesh Generation Method based on Deep Learning](https://mlsanity.com/paper/2004.07016/) (2020)
- [Numerical Solution of Inverse Problems by Weak Adversarial Networks](https://mlsanity.com/paper/2002.11340/) (2020)
- [Non-Euclidean Universal Approximation](https://mlsanity.com/paper/2006.02341/) (2020)
- [Quasi-optimal hp-finite element refinements towards singularities via deep neural network prediction](https://mlsanity.com/paper/2209.05844/) (2022)
- [Neural Control of Discrete Weak Formulations: Galerkin, Least-Squares and Minimal-Residual Methods with Quasi-Optimal Weights](https://mlsanity.com/paper/2206.07475/) (2022)
- [r-Adaptive Deep Learning Method for Solving Partial Differential Equations](https://mlsanity.com/paper/2210.10900/) (2022)
- [RAR-PINN algorithm for the data-driven vector-soliton solutions and parameter discovery of coupled nonlinear equations](https://mlsanity.com/paper/2205.10230/) (2022)
- [An automatic-adaptivity stabilized finite element method via residual minimization for heterogeneous, anisotropic advection-diffusion-reaction problems](https://mlsanity.com/paper/2011.11264/) (2020)

## Built on by (most cited first, 3 of 3)

- [STAResNet: a Network in Spacetime Algebra to solve Maxwell's PDEs](https://mlsanity.com/paper/2408.13619/) (2024)
- [A Learning-Based Ansatz Satisfying Boundary Conditions in Variational Problems](https://mlsanity.com/paper/2505.12430/) (2025)
- [A Novel Tensor Product-Based Neural Network for Solving Partial Differential Equations](https://mlsanity.com/paper/2605.29688/) (2026)

## Abstract (arXiv)

Residual minimization is a widely used technique for solving Partial Differential Equations in variational form. It minimizes the dual norm of the residual, which naturally yields a saddle-point (min-max) problem over the so-called trial and test spaces. In the context of neural networks, we can address this min-max approach by employing one network to seek the trial minimum, while another network seeks the test maximizers. However, the resulting method is numerically unstable as we approach the trial solution. To overcome this, we reformulate the residual minimization as an equivalent minimization of a Ritz functional fed by optimal test functions computed from another Ritz functional minimization. We call the resulting scheme the Deep Double Ritz Method (D2RM), which combines two neural networks for approximating trial functions and optimal test functions along a nested double Ritz minimization strategy. Numerical results on different diffusion and convection problems support the robustness of our method, up to the approximation properties of the networks and the training capacity of the optimizers.