# RANG: A Residual-based Adaptive Node Generation Method for Physics-Informed Neural Networks

arXiv: https://arxiv.org/abs/2205.01051 · HTML: https://mlsanity.com/paper/2205.01051/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 50 references that are in the map.
- First posted on arXiv 2 May 2022.

## Builds on (most cited first, 15 of 15)

- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Respecting causality is all you need for training physics-informed neural networks](https://mlsanity.com/paper/2203.07404/) (2022)
- [Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks](https://mlsanity.com/paper/2007.04542/) (2020)
- [PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs](https://mlsanity.com/paper/1909.10145/) (2019)
- [Parallel Physics-Informed Neural Networks via Domain Decomposition](https://mlsanity.com/paper/2104.10013/) (2021)
- [Efficient training of physics-informed neural networks via importance sampling](https://mlsanity.com/paper/2104.12325/) (2021)
- [NVIDIA SimNet^{TM}: an AI-accelerated multi-physics simulation framework](https://mlsanity.com/paper/2012.07938/) (2020)
- [A Dual-Dimer Method for Training Physics-Constrained Neural Networks with Minimax Architecture](https://mlsanity.com/paper/2005.00615/) (2020)
- [SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEs](https://mlsanity.com/paper/2102.13037/) (2021)
- [State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learning](https://mlsanity.com/paper/2202.06416/) (2022)
- [IDRLnet: A Physics-Informed Neural Network Library](https://mlsanity.com/paper/2107.04320/) (2021)
- [On generation of node distributions for meshless PDE discretizations](https://mlsanity.com/paper/1812.03160/) (2018)

## Built on by (most cited first, 14 of 14)

- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [PINNACLE: PINN Adaptive ColLocation and Experimental points selection](https://mlsanity.com/paper/2404.07662/) (2024)
- [Auto-PINN: Understanding and Optimizing Physics-Informed Neural Architecture](https://mlsanity.com/paper/2205.13748/) (2022)
- [Failure-informed adaptive sampling for PINNs, Part II: combining with re-sampling and subset simulation](https://mlsanity.com/paper/2302.01529/) (2023)
- [A Novel Adaptive Causal Sampling Method for Physics-Informed Neural Networks](https://mlsanity.com/paper/2210.12914/) (2022)
- [Residual-Quantile Adjustment for Adaptive Training of Physics-informed Neural Network](https://mlsanity.com/paper/2209.05315/) (2022)
- [Robust Regression with Highly Corrupted Data via Physics Informed Neural Networks](https://mlsanity.com/paper/2210.10646/) (2022)
- [Physics-informed neural networks modeling for systems with moving immersed boundaries: application to an unsteady flow past a plunging foil](https://mlsanity.com/paper/2306.13395/) (2023)
- [Fixed-budget online adaptive learning for physics-informed neural networks. Towards parameterized problem inference](https://mlsanity.com/paper/2212.11776/) (2022)
- [VI-PINNs: Variance-involved Physics-informed Neural Networks for Fast and Accurate Prediction of Partial Differential Equations](https://mlsanity.com/paper/2211.16753/) (2022)
- [Efficient Training of Physics-Informed Neural Networks with Direct Grid Refinement Algorithm](https://mlsanity.com/paper/2306.08293/) (2023)
- [Geometry-aware framework for deep energy method: an application to structural mechanics with hyperelastic materials](https://mlsanity.com/paper/2405.03427/) (2024)
- [Provably Accurate Adaptive Sampling for Collocation Points in Physics-informed Neural Networks](https://mlsanity.com/paper/2504.00910/) (2025)
- [Quasi-Random Physics-informed Neural Networks](https://mlsanity.com/paper/2507.08121/) (2025)

## Abstract (arXiv)

Learning solutions of partial differential equations (PDEs) with Physics-Informed Neural Networks (PINNs) is an attractive alternative approach to traditional solvers due to its flexibility and ease of incorporating observed data. Despite the success of PINNs in accurately solving a wide variety of PDEs, the method still requires improvements in terms of computational efficiency. One possible improvement idea is to optimize the generation of training point sets. Residual-based adaptive sampling and quasi-uniform sampling approaches have been each applied to improve the training effects of PINNs, respectively. To benefit from both methods, we propose the Residual-based Adaptive Node Generation (RANG) approach for efficient training of PINNs, which is based on a variable density nodal distribution method for RBF-FD. The method is also enhanced by a memory mechanism to further improve training stability. We conduct experiments on three linear PDEs and three nonlinear PDEs with various node generation methods, through which the accuracy and efficiency of the proposed method compared to the predominant uniform sampling approach is verified numerically.