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

arXiv 2205.01051 ↗First posted 2 May 2022cs.LG cs.NA math.NAcited by 14

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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.

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

15 of 50 references.

References · 15

  1. 2019DeepXDE: A deep learning library for solving differential equationscited by 619
  2. 2021Physics-informed neural networks with hard constraints for inverse designcited by 280
  3. 2020Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanismcited by 233
  4. 2021Gradient-enhanced physics-informed neural networks for forward and inverse PDE problemscited by 222
  5. 2022Respecting causality is all you need for training physics-informed neural networkscited by 156
  6. 2020Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networkscited by 123
  7. 2019PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEscited by 118
  8. 2021Parallel Physics-Informed Neural Networks via Domain Decompositioncited by 112
  9. 2021Efficient training of physics-informed neural networks via importance samplingcited by 110
  10. 2020NVIDIA SimNet^{TM}: an AI-accelerated multi-physics simulation frameworkcited by 58
  11. 2020A Dual-Dimer Method for Training Physics-Constrained Neural Networks with Minimax Architecturecited by 50
  12. 2021SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEscited by 32
3 more
  1. 2022State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learningcited by 12
  2. 2021IDRLnet: A Physics-Informed Neural Network Librarycited by 10
  3. 2018On generation of node distributions for meshless PDE discretizationscited by 2

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Citing papers per year: 2022: 7; 2023: 3; 2024: 2; 2025: 2.

14 papers.

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  1. 2022A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networkscited by 236
  2. 2024PINNACLE: PINN Adaptive ColLocation and Experimental points selectioncited by 32
  3. 2022Auto-PINN: Understanding and Optimizing Physics-Informed Neural Architecturecited by 14
  4. 2023Failure-informed adaptive sampling for PINNs, Part II: combining with re-sampling and subset simulationcited by 13
  5. 2022A Novel Adaptive Causal Sampling Method for Physics-Informed Neural Networkscited by 5
  6. 2022Residual-Quantile Adjustment for Adaptive Training of Physics-informed Neural Networkcited by 4
  7. 2022Robust Regression with Highly Corrupted Data via Physics Informed Neural Networkscited by 4
  8. 2023Physics-informed neural networks modeling for systems with moving immersed boundaries: application to an unsteady flow past a plunging foilcited by 4
  9. 2022Fixed-budget online adaptive learning for physics-informed neural networks. Towards parameterized problem inferencecited by 3
  10. 2022VI-PINNs: Variance-involved Physics-informed Neural Networks for Fast and Accurate Prediction of Partial Differential Equationscited by 1
  11. 2023Efficient Training of Physics-Informed Neural Networks with Direct Grid Refinement Algorithmcited by 0
  12. 2024Geometry-aware framework for deep energy method: an application to structural mechanics with hyperelastic materialscited by 0
2 more
  1. 2025Provably Accurate Adaptive Sampling for Collocation Points in Physics-informed Neural Networkscited by 0
  2. 2025Quasi-Random Physics-informed Neural Networkscited by 0