# r-Adaptive Deep Learning Method for Solving Partial Differential Equations

arXiv: https://arxiv.org/abs/2210.10900 · HTML: https://mlsanity.com/paper/2210.10900/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 53 references that are in the map.
- First posted on arXiv 19 Oct 2022.

## Builds on (most cited first, 14 of 14)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [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)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review](https://mlsanity.com/paper/1611.00740/) (2016)
- [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)
- [Deep least-squares methods: an unsupervised learning-based numerical method for solving elliptic PDEs](https://mlsanity.com/paper/1911.02109/) (2019)
- [Automatic differentiation in ML: Where we are and where we should be going](https://mlsanity.com/paper/1810.11530/) (2018)
- [Modeling extra-deep electromagnetic logs using a deep neural network](https://mlsanity.com/paper/2005.08919/) (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)

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

- [A Deep Double Ritz Method (D2RM) for solving Partial Differential Equations using Neural Networks](https://mlsanity.com/paper/2211.03627/) (2022)
- [UGM2N: An Unsupervised and Generalizable Mesh Movement Network via M-Uniform Loss](https://mlsanity.com/paper/2508.08615/) (2025)
- [A Green-Integral-Constrained Neural Solver with Stochastic Physics-Informed Regularization](https://mlsanity.com/paper/2604.21411/) (2026)

## Abstract (arXiv)

We introduce an r-adaptive algorithm to solve Partial Differential Equations using a Deep Neural Network. The proposed method restricts to tensor product meshes and optimizes the boundary node locations in one dimension, from which we build two- or three-dimensional meshes. The method allows the definition of fixed interfaces to design conforming meshes, and enables changes in the topology, i.e., some nodes can jump across fixed interfaces. The method simultaneously optimizes the node locations and the PDE solution values over the resulting mesh. To numerically illustrate the performance of our proposed r-adaptive method, we apply it in combination with a collocation method, a Least Squares Method, and a Deep Ritz Method. We focus on the latter to solve one- and two-dimensional problems whose solutions are smooth, singular, and/or exhibit strong gradients.